# How residual fertility impacts the efficiency of crop pest control by the sterile insect technique

Marine A Courtois, Ludovic Mailleret, Suzanne Touzeau, Louise van

Oudenhove, Frédéric Grognard

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## To cite this version:

Marine A Courtois, Ludovic Mailleret, Suzanne Touzeau, Louise van Oudenhove, Frédéric Grognard. How residual fertility impacts the efficiency of crop pest control by the sterile insect technique. 2024. hal-04403761

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# How residual fertility impacts the efficiency of crop pest control by the sterile insect technique

Marine A Courtois, Ludovic Mailleret, Suzanne Touzeau, Louise van

Oudenhove, Frédéric Grognard

► To cite this version:

Marine A Courtois, Ludovic Mailleret, Suzanne Touzeau, Louise van Oudenhove, Frédéric Grognard. How residual fertility impacts the efficiency of crop pest control by the sterile insect technique. Bulletin of Mathematical Biology, 2025, 87 (2), pp.25. 10.1007/s11538-024-01401-1. hal-04403761v2

HAL Id: hal-04403761

https://hal.inrae.fr/hal-04403761v2

Submitted on 9 Jan 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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Distributed under a Creative Commons Attribution 4.0 International License
# HOW RESIDUAL FERTILITY IMPACTS THE EFFICIENCY OF CROP PEST CONTROL BY THE STERILE INSECT TECHNIQUE

Marine A. Courtois<fnref n="1" />,*, Ludovic Mailleret<fnref n="1" />,<fnref n="2" />, Suzanne Touzeau<fnref n="1" />,<fnref n="2" />, Louise van Oudenhove<fnref n="1" />, Frédéric Grognard<fnref n="2" />

## Highlights

* The sterile insect technique allows pest control if not eradication.

* Pest control is possible even with residual fertility.

* Fitness costs of fertile released males matter.

## Abstract

The Sterile Insect Technique (SIT) is a biological control technique based on mass-rearing, radiation-based sterilization that can induce fitness costs, and releases of the pest species targeted for population control. Sterile matings, between females and sterilized males, can reduce the overall population growth rate and cause a fall in population density. However, a proportion of irradiated males may escape sterilization, which is termed residual fertility. Our aim in this paper is to study the impact of residual fertility on pest control, by a modeling approach.

We modeled the pest population dynamics with three generic differential equations representing sterilized males, wild males and wild females. We explored the impact of residual fertility, associated or not with fitness costs, on pest control possibilities as compared to a situation in which male sterilization is flawless. We carried out a detailed mathematical analysis of the model dynamics through the computation of its equilibria and their stability. Bifurcation analyses were performed with parameters calibrated on the Mediterranean fruit fly *Ceratitis capitata*.

We showed that when residual fertility is below a threshold value, wild populations can be driven to extinction by flooding the landscape with sterilized males. This threshold is higher when residual fertility is associated with fitness costs. Too high a level of residual fertility makes SIT less effective and hinders population eradication. Nevertheless, substantial decreases in outbreak levels can still be achieved for much larger residual fertility rates.

**Keywords:** Modeling, Biological control, *Ceratitis capitata*, Bifurcation analysis, Ordinary differential equations, Stability analysis

## 1 Introduction

The Sterile Insect Technique (SIT) consists in releasing sterile males in order to decrease the number of offspring in the next generation (Dyck *et al.*, 2021). The insects of the targeted species are mass-produced, sexed (when possible) and then sterilized before being massively released in the environment. These releases dilute the population of wild males, thus the females are more likely to meet and mate with a sterile male. These matings do not produce offspring and therefore reduce the population size in the next generation.

The Sterile Insect Technique has been developed to eradicate several pests threatening human health and agriculture. Many SIT projects aim at eradicating different species of mosquitoes, vectors of numerous human diseases (Benedict, 2021). In the agricultural context, SIT has been used to eradicate the melon fly, *Bactrocera cucurbitae*, in Hawaii and the New World screw-worm fly, *Cochliomyia hominivorax*, in several places, in particular Curaçao and Mexico (Oliva *et al.*, 2022). Since its conception in the 1930s-1940s

<footnote n="1">¹Université Côte d'Azur, INRAE, CNRS, ISA, France</footnote>

<footnote n="2">²Université Côte d'Azur, Inria, INRAE, CNRS, MACBES, France</footnote>

<footnote>*Corresponding author marine.courtois@inrae.fr</footnote>
# How residual fertility impacts the efficiency of crop pest control by the Sterile Insect Technique

Marine A. Courtois<fnref n="1" />*, Ludovic Mailleret<fnref n="1" />,<fnref n="2" />, Suzanne Touzeau<fnref n="1" />,<fnref n="2" />, Louise van Oudenhove<fnref n="1" />, Frédéric Grognard<fnref n="2" />

*Corresponding author(s). E-mail(s): marine.courtois@inrae.fr; Contributing authors: ludovic.mailleret@inrae.fr; suzanne.touzeau@inrae.fr; louise.vanoudenhove@inrae.fr; frederic.grognard@inria.fr;

## Abstract

The sterile insect technique (SIT) is a biological control technique based on mass-rearing, radiation-based sterilization that can induce fitness costs, and releases of the pest species targeted for population control. Sterile matings, between females and sterilized males, can reduce the overall population growth rate and cause a fall in population density. However, a proportion of irradiated males may escape sterilization, resulting in what is called residual fertility. Our aim in this study was to examine the impact of residual fertility on pest control employing a modeling approach.

We modeled pest population dynamics with three generic differential equations representing sterilized males, wild males and wild females. We explored the impact of residual fertility, in the presence or absence of fitness costs, on potential pest control outcomes using a scenario with 100% male sterilization as our standard of reference. We carried out a detailed mathematical analysis of the model's dynamics by calculating model equilibria and the latter's stability. Bifurcation analyses were performed with parameters for the Mediterranean fruit fly *Ceratitis capitata*.

We showed that, when residual fertility is below a threshold value, wild populations can be eradicated by flooding the landscape with irradiated males. This threshold is higher when residual fertility is associated with fitness costs. Too high a level of residual fertility makes SIT less effective and hinders population eradication. That said, substantial decreases in population density can be achieved even when residual fertility is much larger than the above threshold.

<footnote n="1">¹*Université Côte d'Azur, INRAE, CNRS, ISA, France.</footnote>

<footnote n="2">²Université Côte d'Azur, Inria, INRAE, MACBES, France.</footnote>
by Knipling (1955), Bushland (Melvin and Bushland, 1936) and Serebrovsky (1940), SIT has developed rapidly and been integrated into operational area-wide integrated pest management (AW-IPM) programs (Klassen et al., 2021). The range of action of the technique extends from the plot to the regional scale, to eliminate certain insects from vast territories or prevent colonization by new exotic species (Klassen and Vreysen, 2021). SIT is often associated with insect eradication programs, especially in a human epidemiological context where the objective is to eliminate the disease. However, in agriculture, instead of targeting pest eradication, the objective is to reduce pest density in order to control its economic impact.

With regard to the Mediterranean fruit fly *Ceratitis capitata*, a polyphagous dipterous insect that infects many fruit crops (Robinson and Hooper, 1989), SIT is an efficient alternative to chemical pesticides (Dunn and Follett, 2017). When mated with sterile males, females lay non-viable eggs in the fruits, thus reducing damages and increasing crop yields. This technique has been successfully deployed in several countries. Mexico and Guatemala developed the Moscamed program in 1975, joined in 1977 by the United States, to prevent the northward spread of *C. capitata* that was progressing in Central America (Enkerlin et al., 2017). In 1986, the medfly was eradicated in northern Guatemala and Mexico. This success was explained by the release of billions of sterile insects on hundreds of thousands of hectares, and this for decades to avoid re-infestation. Spain has been using SIT since the 1990s (Sancho et al., 2021). In France, CeraTIS-Corse is the first SIT project in an agricultural context implemented in Corsica, a French island in the Mediterranean Sea, where fruit growing is an important economic resource (Odarc, 2022).

In such programs, the deployment of SIT includes several major steps: mass production of insects, sexing (with *C. capitata*, sexing can be effectively achieved thanks to a special strain (Caceres, 2002)), sterilization and finally release of sterile insects into the environment. Sterilization is a crucial step, usually carried out through the irradiation of insect pupae with gamma rays. In many insect groups, such irradiations not only sterilize the insects, but also lead to a decrease in competitiveness (Dyck et al., 2005). Indeed, it can damage many physiological processes leading to a reduction in survival, flight capacity and to the development of malformations (Guerfali et al., 2011). At best, it is then necessary to release an even greater number of sterile insects to ensure the effectiveness of the technique (Robinson et al., 2002). At worst, the sterile insects fail to attract and mate with wild insects, which makes the releases useless, especially for species such as *C. capitata*, in which females choose males to mate (Arita and Kaneshiro, 1985). However, the timing of irradiation can be adjusted to preserve sterilized insect competitiveness (Hooper, 1971). Indeed, insects are usually less damaged when irradiation is performed shortly after adult emergence (Estal et al., 1986), but the logistic of irradiating and deploying adult flies is difficult to implement. Therefore, insects are usually irradiated at pupal stage. The irradiation dose must be as low as possible to maintain the competitiveness of sterile individuals, while maintaining the sterilization efficiency (Parker and Mehta, 2007). If the insects receive too low a dose, a significant part of them may escape sterilization. This leads to the release of fertile individuals into the environment that can be extremely damaging, as they increase the reproductive potential of the wild population and reduce the rate at which the population is suppressed (Dyck et al., 2005). This proportion of non-sterile insects released is called residual fertility. The recommended sterility percentage for a SIT program is over 99.5% (FAO/IAEA/USDA, 2003). A dose inducing 100% sterility is rarely used due to the excessive damage it causes to insects (Robinson, 2005; Bakri and Mehta, 2005).

A main challenge is hence to determine how low the residual fertility rate should be to ensure pest control is achievable in the field. The question is thus to predict the residual fertility threshold, i.e. the maximal proportion of fertile irradiated males that can be accepted without threatening SIT effectiveness. Modeling represents an essential and efficient tool to tackle this issue. Indeed, modeling can guide field deployment, which is costly and time-consuming. Models of SIT have been developed for many years (Knipling, 1955; Dyck et al., 2021), but only a few models focusing on *C. capitata* have been implemented (Carey, 1982; Messoussi et al., 2007; Manoukis and Hoffman, 2014), and very few modeling studies have looked at residual fertility. Klassen and Creech (1971) constructed a numerical model in which a proportion of the released males remained fertile. They modified the model developed by Knipling (1955) and concluded that if the proportion of fertile males in the release is greater than the inverse of the population growth rate, the population can no longer be controlled by sterile releases. In another approach, Aronna and Dumont (2020) showed that SIT can only be successful if the residual fertility is below $1/R$ where $R$ is the basic reproduction number in relation to the reproductive potential of the pest population. The success of SIT is expressed as the theoretical possibility of eradicating the population by evaluating if the pest-free equilibrium is stable. This model was applied to two pest species: *Aedes albopictus* and *Bactrocera dorsalis*. A recent study focusing on *Ceratitis capitata* and examining the
**Acknowledgments**

The authors acknowledge the financial support of the CeraTIS - Corse project, within the framework of ECOPHYTO 2019, "Leviers territoriaux pour réduire l'utilisation et les risques liés aux produits phytopharmaceutiques (2020 - 2023)". The authors are also grateful to the project ANR Ecophyto-Maturation SuzuKIISS.ME (ANR-21-ECOM-0002), co-funded by the OFB (Office français de la biodiversité) via the fee for diffuse pollution under the French Ecophyto plan. M. Courtois's PhD is funded by INRAE.

# 1 Introduction

The sterile insect technique (SIT) involves releasing sterilized males of a target species into the environment to provoke the decline of a wild population of that same species (Dyck et al., 2021). Insects of the target species are mass produced, sexed (when possible), and then sterilized before being released in large numbers. As a result, sterilized males mix with wild males, diluting the mating pool, and females are more likely to meet and mate with sterilized males. These matings events do not produce offspring, leading to reductions in population size in the next generation.

Conceived in the 1930s-1940s by Knipling (1955), Bushland (Melvin and Bushland, 1936) and Serebrovsky (1940), SIT was initially developed with the ambitious aim of eradicating insect pests that threaten agricultural systems and human health. In the agricultural systems, SIT has historically been used to eradicate populations of pests such as the melon fly, *Bactrocera cucurbitae*, in Hawaii and the New World screw-worm fly, *Cochliomyia hominivorax*, in places like Curaçao and Mexico (Oliva et al., 2022). With regards to human health, SIT can help manage disease vectors such as mosquitoes (Dyck et al., 2021). Over time, it has become clear that complete eradication is a challenging goal, especially in large, open environments. This reality has led the scientific community to revisit SIT's primary objective toward pest population suppression (Benedict, 2021).

Today, SIT is commonly included in operational area-wide integrated pest management (AW-IPM) programs (Klassen et al., 2021). The technique's range of action extends from the plot scale to the regional scale, with the aim of controlling certain insects spread across vast landscapes or of preventing colonization by new exotic pest species (Klassen and Vreysen, 2021). In this context, pest control focuses on maintaining insect populations below critical thresholds, which may be an economic threshold for agricultural pests (Pedigo et al., 2021), where damage is minimized to limit financial losses, or an epidemiological threshold for disease vectors (Aryaprema et al., 2023), where disease risk is reduced without requiring complete elimination. The distinction between eradication and control is important, as the choice of objective influences program design, reintroduction prevention strategies, and financial sustainability. While
<figure><img src="image_3.png" /><figcaption>Figure 1: Flow diagram of the population dynamics model (1). The compartments (and color code) correspond to sterilized males S (pink), wild females F (blue) and wild males M (green). Solid arrows correspond to flows, dotted arrows to activating (round tip) or inhibiting effects (bar tip). Irradiated males are released at rate $\sigma$. Among these males, a proportion $\delta$ (orange) or $\epsilon$ (purple) may remain fertile, without or with associated fitness costs, respectively. Reproduction is represented by the emergence rate $r$, the proportion of males among offspring $p$, and the proportion of successful matings $X$. The latter increases with the density of wild males and fertile irradiated males ($\epsilon S$), but decreases with the density of sterile males. Competition among females $C(F)$ reduces the emergence rate. Finally, all compartments are affected by specific mortality rates $\mu$.</figcaption></figure>

impact of residual fertility and female re-mating, confirmed the threshold found in Aronna and Dumont (2020) for achieving population elimination (Dumont and Oliva, 2023).

In this work, we built a generic population dynamics model of SIT with sterilized males S, wild females F and wild males M compartments, from which we derived a sub-model without residual fertility and two concurrent sub-models, both incorporating residual fertility but with or without fitness costs associated to irradiation. In order to determine whether SIT is effective or not, bifurcation analyses were carried out to study the population control capacities associated with each sub-model. Numerical simulations were performed with parameter values specific to C. capitata to best account for the biology of the species.

In section 2, the general model and the residual fertility sub-models are presented. The general model is studied to determine its equilibria, their stabilities and the associated bifurcations. Section 3 is dedicated to the study of the residual fertility sub-models, calibrated with C. capitata parameter values, highlighting the impact of residual fertility and fitness costs. Finally, in section 4, we summarize and discuss the main results obtained.

# 2 Model

## 2.1 Model description

### 2.1.1 General model

The model represents the population dynamics of the following three compartments: sterilized males S, wild females F and wild males M (Fig. 1). It is defined as follows:

$$ egin{cases} \dot{S} = -\mu_S S + (1 - \delta)\sigma, \\ \dot{F} = -\mu_F F + r(1 - p)X(S, M, \epsilon)C(F)F, \\ \dot{M} = -\mu_M M + rpX(S, M, \epsilon)C(F)F + \delta\sigma. \end{cases} \quad (1) $$

with the dot representing time derivative.

The dynamics of sterilized males S are affected by their mortality $\mu_S$ and the release rate $\sigma$. Among $\sigma$ releases, only a proportion $(1 - \delta)$ of genuinely sterilized males is added to the compartment. The dynamics of wild females F are affected by their mortality $\mu_F$ and the emergence of new females. The latter is determined by the product of the emergence rate $r$, the proportion of females among the offspring $(1 - p)$, the proportion of successful matings $X(S, M, \epsilon)$, as well as the competition between females for oviposition $C(F)$, all of it multiplied by the number of females engaged in reproduction F. Finally, the dynamics of wild males M are affected by their mortality $\mu_M$, the emergence of new males, which is
eradication may be possible in isolated or confined areas, control is often the more realistic and sustainable option given constraints associated with region size, logistical costs, reintroduction risks, and pest population resilience. Examples of local-level successes, such as the elimination of *Culex pipiens* from a village in Myanmar (Laven, 1967; Benedict, 2021) or of *Culex quinquefasciatus* from an island in Florida (Patterson et al., 1970; Benedict, 2021), illustrate that elimination is realistic in restricted, controlled settings. However, elsewhere, SIT and other integrated pest management strategies can nonetheless serve as powerful tools for maintaining pest populations at economically or epidemiologically acceptable levels without the need for eradication (Isman, 2019; Walters et al., 2008).

SIT has been an effective alternative to chemical pesticides (Dunn and Follett, 2017) in the case of the Mediterranean fruit fly (medfly) *Ceratitis capitata*, a polyphagous dipterous insect that infects many fruit crops (Robinson and Hooper, 1989). When females mate with sterilized males, they lay non-viable eggs in the fruits, resulting in reduced fruit damage and increased crop yields. SIT has been successfully deployed in several countries. In 1975, Mexico and Guatemala developed the Moscamed program, which the United States joined in 1977. The program's aim was to prevent the northward spread of *C. capitata* from Central America (Enkerlin et al., 2017). In 1986, the medfly was eradicated from northern Guatemala and Mexico. This success was the product of billions of sterilized insects being released across hundreds of thousands of hectares for decades. Spain has been using SIT since the 1990s (Sancho et al., 2021). CeraTIS-Corse is the first SIT project to be implemented in an agricultural context in France; it is focused on Corsica, a French island in the Mediterranean Sea, where fruit growing is an important economic activity (Odarc, 2022).

In such programs, SIT deployment involves several major steps: insect mass production, sexing (possible for *C. capitata* thanks to a special strain; (Caceres, 2002)), sterilization, and, finally, release into the environment. Sterilization is a crucial step that is usually carried out by irradiating insect pupae with gamma rays. In many insect groups, this form of irradiation not only sterilizes the insects, but also leads to decreased competitiveness (Dyck et al., 2005). Indeed, it can impair many physiological processes, leading to diminished survival and flight capacity as well as the development of malformations (Guerfali et al., 2011). At best, it is then necessary to release an even greater number of sterilized insects to ensure SIT effectiveness (Robinson et al., 2002). At worst, the sterilized insects fail to attract and mate with wild insects, rendering the releases useless, especially for species such as *C. capitata*, in which there is mate selection by females (Arita and Kaneshiro, 1985). However, the timing of irradiation can be adjusted to help the competitiveness of sterilized insects (Hooper, 1971). Indeed, impairments are usually less marked when irradiation is performed shortly after adult emergence (Estal et al., 1986). That said, the logistics of irradiating and releasing adult flies are complicated to implement. Therefore, insects are usually irradiated as pupae. The dose of radiation must be as low as possible to maintain competitiveness while ensuring sterilization efficiency (Parker and Mehta, 2007). If the insects receive too low of a dose, a significant proportion may escape sterilization. As a result, still fertile individuals are released into the environment, which can cause great harm as the reproductive potential of the wild population will
identical to that of females up to the proportion of male offspring (complementary to that of female offspring) and a proportion of released males. Indeed, we assumed that the proportion $\delta$ of males among $\sigma$ releases, that have been irradiated but remain fertile, directly contributes to the wild male population. We considered that wild individuals have a lower mortality rate than sterilized ones: $\mu_M \le \mu_S$. Hence, fitness costs are expressed both in higher mortality and lower attractiveness.

We assumed that the proportion of successful matings $X(S, M, \epsilon)$ depends on both the density of fertile males and the density of attractive males. Fertile males correspond to wild males and a proportion $\epsilon$ of irradiated males. Attractive males correspond to wild and irradiated males, with a potential decrease in attractiveness linked to the $\eta$ parameter for irradiated males. This corresponds to the biological characteristics of the species C. capitata, whose males aggregate and form leks in order to attract females for mating (Prokopy and Hendrichs, 1979). Thus, attractive males must be numerous enough to attract females. By being numerous and attractive, effectively sterilized males can cause a strong dilution effect, drastically reducing the number of successful matings. These biological hypotheses on the proportion of successful matings can be summarized as follows:

• $0 \le X(S, M, \epsilon) \le 1$

• $X(0, 0, \epsilon) = 0$

• $X(S, M, \epsilon)$ is increasing in $M$, concave in $M$ and such that $\lim_{M \to +\infty} X(S, M, \epsilon) = 1$

The first point signifies that $X$ is a proportion, the second one that in the absence of males, no reproduction can take place, and the third one that the more wild males there are, the more reproduction succeeds. As an example, the proportion of successful matings can be expressed as the following function:

$$X(S, M, \epsilon) = \frac{M + \epsilon\eta S}{k + M + \eta S} \quad (2)$$

where $k > 0$ is a constant representing the cost of being too few males and $\eta$ represents the attractiveness of sterilized males. Non-zero $k$ accounts for reduced mating at small male density linked to the associated difficulty of forming leks for C. capitata. In addition, we hypothesized that irradiated males suffer from a lack of attractiveness $\eta \in [0, 1]$.

The competition function $C(F)$ is a non-negative decreasing function. This feature corresponds to the competition among C. capitata females for oviposition (access to egg-laying sites) and thus affects the emergence rate (Papadopoulos et al., 2009). Possible forms for $C(F)$ include: $\frac{1}{1+\beta F}$ which is a Beverton and Holt-like function, $e^{-\beta F}$ which is a Ricker-like function and $1 - \frac{F}{K}$ which is a logistic-like competition function, where $\beta$ represents the competition strength and $K$ the biotic capacity (Hastings and Gross, 2012). All these functions have the following characteristics:

• $C(0) = 1$ and $C(F)$ is a decreasing function

• $1/C(F)$ is convex

Among such functions, we will focus on the following explicit form:

$$C(F) = \frac{1}{1 + \beta F} \quad (3)$$

where $\beta$ represents the competition strength among females. In the model, in all cases $1/C(F)$ is strictly convex or $X(S, M, \epsilon)$ is strictly concave in $M$, or both.

To determine the general model equilibria and their stability in sections 2.2.1 to 2.2.3, we considered $X(S, M, \epsilon)$ and $C(F)$ satisfying the above hypotheses. For the residual fertility sub-models in section 2.1.2 and the bifurcation analysis in section 2.2.4, we chose $X$ and $C$ as in (2) and (3) respectively.

### 2.1.2 Residual fertility sub-models

In order to study the impact of residual fertility, i.e. the proportion of non-sterile irradiated males released, associated or not with fitness costs, we considered three cases extracted from general model (1).

Firstly, the case where $\delta = \epsilon = 0$ corresponds to a basic situation with no residual fertility. The explicit form of this sub-model called “Perfect sterilization” is as follows:
increase and the rate at which the population is suppressed will decrease (Dyck et al., 2005). The proportion of irradiated insects that remain fertile is called residual fertility. In SIT programs, it is recommended that the percentage of successfully sterilized males exceed 99.5% (FAO/IAEA/USDA, 2003). Radiation doses inducing 100% sterility are rarely used because they cause excessive impairments (Robinson, 2005; Bakri and Mehta, 2005).

Therefore a major challenge is to determine the impact of residual fertility on SIT effectiveness. In this study, our objectives were to establish whether pest eradication is possible given residual fertility, and, more generally, to determine the residual fertility threshold at which SIT can still successfully maintain pest populations below a target threshold.

Modeling is an essential and efficient tool for tackling this challenge. Its results can guide SIT deployment in the field, which is both costly and time consuming. SIT models have been developed for many years (Knipling, 1955; Dyck et al., 2021), but only a few have specifically addressed *C. capitata* (Carey, 1982; Messoussi et al., 2007; Manoukis and Hoffman, 2014), and even fewer studies have examined residual fertility. Klassen and Creech (1971) created a numerical model in which a proportion of the released males remained fertile, which was an adaptation of the model developed by Knipling (1955). The researchers concluded that if, among the released males, the proportion of fertile males exceeds the inverse of the population growth rate, the proportion of sterilized males will be insufficient to control the population. Using a different approach, Aronna and Dumont (2020) demonstrated that SIT can succeed only if residual fertility remains below $1/R$, where $R$ is the basic reproduction number linked to the pest population's reproductive potential; SIT success was defined as the theoretical possibility of eradicating the population by assessing the stability of the pest-free equilibrium. This model was applied to two pest species, *Aedes albopictus* and *Bactrocera dorsalis*. A recent study on *C. capitata* (Dumont and Oliva, 2023) further examined the effects of residual fertility and female polyandry, confirming the population eradication threshold identified by Aronna and Dumont (2020). Most of these models focus on population eradication, as they generally assess the stability of the pest-free equilibrium. However, it would be highly beneficial to expand upon these analyses by exploring population control scenarios, which is one of the objectives of our study.

We built a generic model describing the population dynamics associated with SIT implementation that contained compartments for sterilized males $S$, wild females $F$, and wild males $M$. We then derived a submodel without residual fertility and two concurrent submodels with residual fertility in the presence or absence of irradiation-associated fitness costs. To evaluate SIT effectiveness, bifurcation analyses were carried out to determine the population control capacities of each submodel. Numerical simulations were performed with parameter values specific to *C. capitata* to better account for species biology.

In section 2, we present the general model and the residual fertility submodels. The general model is explored to determine its equilibria, their stability, and the associated bifurcations. Section 3 is dedicated to the residual fertility submodels, which were calibrated with the *C. capitata* parameter values; we highlight the impacts of residual
$$
\left\{
\begin{aligned}
\dot{S} &= -\mu_S S + \sigma, \\
\dot{F} &= -\mu_F F + r(1-p) \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F, \\
\dot{M} &= -\mu_M M + r p \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F.
\end{aligned}
\right.
\qquad (4)
$$

140

141

142

143

Secondly, we considered the existence of residual fertility without associated fitness costs. The fertile irradiated males are considered as fit as wild males in terms of attractiveness and survival, and are therefore introduced in the M compartment of the model. This case corresponds to $\epsilon = 0$ and $0 < \delta < 1$. The explicit form of this sub-model called “Cost-free residual fertility” is:

$$
\left\{
\begin{aligned}
\dot{S} &= -\mu_S S + (1 - \delta)\sigma, \\
\dot{F} &= -\mu_F F + r(1-p) \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F, \\
\dot{M} &= -\mu_M M + r p \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F + \delta\sigma.
\end{aligned}
\right.
\qquad (5)
$$

144

145

146

147

148

And thirdly, we considered the existence of residual fertility with associated fitness costs. All irradiated males, even if they are fertile, suffer from a lack of competitiveness (i.e. attractiveness) which is represented by parameter $\eta$ in the mating function, and have a higher mortality rate compared to wild males. Therefore, fertile irradiated males remain in the S compartment. This case corresponds to $\delta = 0$ and $0 < \epsilon < 1$. The explicit form of this sub-model called “Costly residual fertility” is:

$$
\left\{
\begin{aligned}
\dot{S} &= -\mu_S S + \sigma, \\
\dot{F} &= -\mu_F F + r(1-p) \frac{M + \epsilon\eta S}{k + M + \eta S} \frac{1}{1 + \beta F} F, \\
\dot{M} &= -\mu_M M + r p \frac{M + \epsilon\eta S}{k + M + \eta S} \frac{1}{1 + \beta F} F.
\end{aligned}
\right.
\qquad (6)
$$

149

150

These three sub-models: “Perfect sterilization” (4), “Cost-free residual fertility” (5) and “Costly residual fertility” (6), correspond to an increasing gradient of biological realism.

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## 2.2 Model analysis

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In this section, we carry out an equilibrium search and stability study on the general model (1), as the analyses on the three sub-models detailed in 2.1.2 are similar.

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Under the above mentioned hypotheses the model (1) is biologically well-posed: S, F and M remain non-negative for positive time if their initial conditions are non-negative. The $\dot{S}$ equation is actually decoupled from the other two, and we assumed that S reaches its equilibrium $S^* = \frac{(1-\delta)\sigma}{\mu_S}$ very fast. Thus, we only considered the (F, M) subsystem. For readability reasons, we rewrite $\dot{X}(S^*, M, \epsilon)$ as $X(M)$.

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### 2.2.1 Reproduction numbers

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From the second equation of the general model (1) with $\sigma = 0$, we have:

$$
\begin{aligned}
\frac{\dot{F}}{F} &= -\mu_F + r(1-p)X(M)C(F), \\
&\leq -\mu_F + r(1-p)\sup_M(X(M))C(0), \\
&\leq -\mu_F + r(1-p).
\end{aligned}
$$

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This inequation is used to introduce the maximal reproduction number for females:

$$ \mathcal{R}_F = \frac{r(1-p)}{\mu_F}. \qquad (7) $$

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$\mathcal{R}_F$ is the average number of females produced by a female during its lifetime when males M are not limiting and female competition is neglected. A necessary condition for the female population to not go extinct is that $\mathcal{R}_F$ be greater than 1.
<figure><img src="image_2.png" /><figcaption>Fig. 1 Flow diagram of the population dynamics model (1). The color-coded compartments correspond to the sterilized males S (pink), wild females F (blue), and wild males M (green). Solid arrows indicate the flows, dashed arrows indicate the activating effects (circles as tip) and inhibiting effects (bar as tip). Irradiated males are released at rate $\sigma$; a certain proportion remain fertile in the absence $\delta$ (orange) or presence $\epsilon$ (purple) of associated fitness costs. Reproduction is represented by the emergence rate $r$, the proportional representation of males among offspring $p$, and the proportion of successful matings events $X$. The latter increases with the density of wild males and fertile irradiated males ($\epsilon S$) but decreases with the density of sterilized males. Competition among females $C(F)$ reduces the emergence rate. Finally, all compartments are affected by a specific mortality rate $\mu$.</figcaption></figure>

fertility and fitness costs. Finally, in section 4, we summarize and discuss the main results obtained.

# 2 Model

## 2.1 Model description

### 2.1.1 General model

The general model represents the population dynamics of the following three compartments: sterilized males $S$, wild females $F$ and wild males $M$ (Fig. 1). It is defined as follows:

$$ 
\begin{cases}
\dot{S} = -\mu_S S + (1 - \delta)\sigma, \\
\dot{F} = -\mu_F F + r(1 - p)X(S, M, \epsilon)C(F)F, \\
\dot{M} = -\mu_M M + rpX(S, M, \epsilon)C(F)F + \delta\sigma.
\end{cases}
\quad (1) $$

with the dot representing the time derivative.

The dynamics of sterilized males $S$ are affected by their mortality rate $\mu_S$ and the release rate $\sigma$. For $\sigma$ releases, only a proportion $(1 - \delta)$ of the successfully sterilized males is added to the compartment. The dynamics of wild females $F$ are affected by their mortality rate $\mu_F$ and the emergence of new females. The latter is determined by the product of the emergence rate $r$, the proportion of females among the offspring $(1 - p)$, the proportion of successful matings events $X(S, M, \epsilon)$, as well as by the competition among females for oviposition sites $C(F)$, all multiplied by the number of females engaged in reproduction $F$. Finally, the dynamics of wild males $M$ are affected by their mortality rate $\mu_M$, the emergence of new males (identical to that of females up to the limit of the proportion of male offspring [complementary to that of female offspring]), and the proportion of released males. Indeed, the model assumes that the
From the third equation of (1), we have:

$$ \begin{aligned} \frac{\dot{M}}{M} &= -\mu_M + rp \frac{X(M)}{M} C(F)F, \\ &\leq -\mu_M + rpX'(0) \sup_{F \geq 0} (C(F)F), \end{aligned} $$

because $X(M)$ is concave in $M$. This inequation allows to introduce the maximal reproduction number for males:

$$ \mathcal{R}_M = \frac{rpX'(0) \sup_{F \geq 0} (C(F)F)}{\mu_M}. $$

With the explicit expressions of $X(M)$ (2) and $C(F)$ (3), we get:

$$ \mathcal{R}_M = \frac{rp}{\beta k \mu_M}. \qquad (8) $$

$\mathcal{R}_M$ is the average number of males produced by a male during its lifetime for the optimal number of females in the population maximizing $C(F)F$. A necessary condition for the male population to not go extinct is that $\mathcal{R}_M$ be greater than 1.

### 2.2.2 Equilibria

For the $(F, M)$ subsystem at $S = S^*$, equilibria of system (1) are the solutions of the following equations:

$$ \begin{cases} F(-\mu_F + r(1-p)X(M)C(F)) = 0, \\ -\mu_M M + rpX(M)C(F)F + \delta\sigma = 0. \end{cases} \qquad (9) $$

Population densities at equilibrium hence verify $F^* = 0$ or $X(M^*)C(F^*) = \frac{\mu_F}{r(1-p)} = \frac{1}{\mathcal{R}_F}$ from the first equation of (9).

**Pest-free equilibrium** We easily deduce the pest-free equilibrium $(F^*, M^*) = (0, \frac{\delta\sigma}{\mu_M})$. When $\delta = 0$ it is a true pest-free equilibrium of the form $(0, 0)$, whereas when $\delta > 0$ the male compartment is maintained by non-sterile males within the releases.

**Infestation equilibria** The other equilibria must satisfy:

$$ X(M)C(F) = \frac{\mu_F}{r(1-p)} = \frac{1}{\mathcal{R}_F}. \qquad (10) $$

Injecting (10) into the second equation of (9), we obtain:

$$ M(F) = \frac{p\mu_F}{(1-p)\mu_M}F + \frac{\delta\sigma}{\mu_M}. \qquad (11) $$

Replacing $M$ in (10) by its value in (11), we get an equation that only depends on the female density:

$$ X(M(F))C(F) = \frac{1}{\mathcal{R}_F}. \qquad (12) $$

As $X(M(F))$ is concave and $1/C(F)$ is convex, this equation has 2, 1 or no solutions (Appendix A). We note $G(F) = X(M(F))C(F)$ and detail these different cases below.

(i) If $G(0) < \frac{1}{\mathcal{R}_F}$ and $\max(G(F)) > \frac{1}{\mathcal{R}_F}$ (Fig. 2A) there are two solutions $F_2^* > F_1^* > 0$, such that $\frac{dG}{dF}(F_1^*) > 0$ and $\frac{dG}{dF}(F_2^*) < 0$. We then have two infestation equilibria $(F_1^*, M_1^*)$ and $(F_2^*, M_2^*)$.

(ii) If $G(0) > \frac{1}{\mathcal{R}_F}$ (Fig. 2B) there is only one solution, at which $\frac{dG}{dF} < 0$. This solution is named $F_2^*$ and corresponds to the infestation equilibrium $(F_2^*, M_2^*)$. In the perfect sterilization case corresponding to model (4): $X(M(0)) = X(0) = 0$, so $G(0) = 0$ and this second situation cannot happen.

(iii) If $G(F) < \frac{1}{\mathcal{R}_F}$, $\forall F > 0$, there is no solution, so there is no infestation equilibrium.
proportion $\delta$ of males among $\sigma$ releases that remain fertile despite irradiation enter directly into wild male compartment. We considered that wild males have a lower mortality rate than sterilized males: $\mu_M \le \mu_S$. Hence, fitness costs are expressed via both higher mortality and lower attractiveness.

The model assumes that the proportion of successful matings events $X(S, M, \epsilon)$ depends on both the density of fertile males and the density of attractive males. Fertile males correspond to the wild males plus a proportion $\epsilon$ of the irradiated males. Attractive males correspond to the wild males and the irradiated males, with a potential decrease in attractiveness linked to the $\eta$ parameter for the irradiated males. This structure reflects the biological characteristics of the species *C. capitata*, whose males aggregate and form leks to attract females for mating (Prokopy and Hendrichs, 1979). Thus, attractive males must be numerous enough to attract females. When they are numerous and attractive, successfully sterilized males can have a strong dilution effect, drastically reducing the number of successful matings events. These biological hypotheses regarding the proportion of successful matings events can be summarized as follows:

* $0 \le X(S, M, \epsilon) \le 1$

* $X(0, 0, \epsilon) = 0$

* $X(S, M, \epsilon)$ is increasing and concave in $M$, such that $\lim_{M \to +\infty} X(S, M, \epsilon) = 1$

The first point indicates that $X$ is a proportion; the second point indicates that, in the absence of males, no reproduction can take place; and the third point indicates that great numbers of wild males lead to higher reproduction success. To illustrate, the proportion of successful matings events can be expressed using the following function:

$$X(S, M, \epsilon) = \frac{M + \epsilon\eta S}{k + M + \eta S} \qquad (2)$$

where $k > 0$ is a constant representing the cost of too few males being present and $\eta$ represents the attractiveness of sterilized males. Non-zero values of $k$ are associated with reduced mating when male density is small, given the resulting difficulty in forming leks. In addition, we hypothesized that irradiated males suffer from a lack of attractiveness $\eta \in [0, 1]$.

The competition function $C(F)$ is a non-negative decreasing function. It reflects competition among *C. capitata* females for oviposition sites and thus affects the emergence rate (Papadopoulos et al., 2009). Possible forms for $C(F)$ include the following: $\frac{1}{1+\beta F}$, which is a Beverton-Holt-like function; $e^{-\beta F}$, which is a Ricker-like function; and $1 - \frac{F}{K}$, which is a logistic-like competition function. $\beta$ represents competition strength, and $K$ represents the biotic capacity (Hastings and Gross, 2012). All these functions have the following characteristics:

* $C(0) = 1$ and $C(F)$ is a decreasing function

* $1/C(F)$ is convex
<figure><img src="image_4.png" /><figcaption>Figure 2: Female density values at the infestation equilibria of model (1). They correspond to the intersections of function $G(F)$ (in black), and the horizontal dotted line $1/R_F$ (in grey). The two simulated cases correspond to different parameter values. Case A: when $G(0) < \frac{1}{R_F}$ and $\max(G(F)) > \frac{1}{R_F}$, there are two infestation equilibria $F_1^*$ and $F_2^*$. Case B: when $G(0) > \frac{1}{R_F}$, there is only one infestation equilibrium $F_2^*$.</figcaption></figure>

### 2.2.3 Local stability of equilibria

We consider the Jacobian matrix of the $(F, M)$ subsystem at $S = S^*$ of general model (1):

$$J_{(F,M)} = \begin{pmatrix} -\mu_F + r(1-p)X(M)(C'(F)F + C(F)) & r(1-p)C(F)X'(M)F \\ rpX(M)(C'(F)F + C(F)) & -\mu_M + rpC(F)X'(M)F \end{pmatrix}$$

Pest-free equilibrium At the pest-free equilibrium $(0, \frac{\delta\sigma}{\mu_M}) = (0, M(0))$:

$$J_{(0,M(0))} = \begin{pmatrix} -\mu_F + r(1-p)X(M(0)) & 0 \\ rpX(M(0)) & -\mu_M \end{pmatrix}$$

So $(0, M(0))$ is locally asymptotically stable if $X(M(0)) < \frac{1}{R_F}$, i.e. $G(0) < \frac{1}{R_F}$. This corresponds to cases (i) and (iii) above: there are no infestation equilibria or two infestation equilibria.

Infestation equilibria At the positive equilibria $(F^*, M^*)$, the Jacobian is:

$$J_{(F^*, M^*)} = \begin{pmatrix} -\mu_F + r(1-p)X(M^*)(C'(F^*)F^* + C(F^*)) & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\mu_M + rpC(F^*)X'(M^*)F^* \end{pmatrix}$$

The nullcline $r(1-p)X(M)C(F) = \mu_F$ from system (9) allows us to simplify the matrix, as follows:

$$J_{(F^*, M^*)} = \begin{pmatrix} r(1-p)X(M^*)C'(F^*)F^* & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\mu_M + rpC(F^*)X'(M^*)F^* \end{pmatrix}$$

To determine the stability of the positive equilibria, we compute the trace and the determinant of this matrix.

For the trace, the other nullcline $rpX(M)C(F)F + \delta\sigma = \mu_M M$ from system (9), allows us to obtain:

$$J_{(F^*, M^*)} = \begin{pmatrix} r(1-p)X(M^*)C'(F^*)F^* & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\frac{rpX(M^*)C(F^*)F^* + \delta\sigma}{M^*} + rpC(F^*)X'(M^*)F^* \end{pmatrix}$$

The computation of the trace yields:

$$\mathrm{Tr}(J_{(F^*, M^*)}) = r(1-p)X(M^*)C'(F^*)F^* - \frac{\delta\sigma}{M^*} + rpC(F^*)F^* \left( X'(M^*) - \frac{X(M^*)}{M^*} \right)$$

The first two terms are negative since $C(F)$ is decreasing. Furthermore, as $X(M)$ is concave, the last term is nonpositive. Indeed:

$$X(M^*) = X(0) + \int_{0}^{M^*} X'(M) dM \geq \int_{0}^{M^*} X'(M^*) dM = M^* X'(M^*)$$
We will focus on the following explicit form:

$$C(F) = \frac{1}{1 + \beta F} \qquad (3)$$

where $\beta$ represents the strength of competition among females. In the model, in all cases, $1/C(F)$ is strictly convex, $X(S, M, \epsilon)$ is strictly concave in $M$, or both co-occur.

To determine the general model equilibria and their stability (sections 2.2.1 to 2.2.3), we considered that $X(S, M, \epsilon)$ and $C(F)$ satisfied the above hypotheses. For the residual fertility submodels (section 2.1.2) and the bifurcation analysis (section 2.2.4), we used $X$ and $C$ as in (2) and (3), respectively.

## 2.1.2 Residual fertility sub-models

We studied the impacts of residual fertility (i.e., the proportion of fertile irradiated males) in the presence and absence of fitness costs using three cases extracted from the general model (1).

First, the case where $\delta = \epsilon = 0$ corresponds to a basic situation with no residual fertility. The explicit form of this submodel (hereafter called the perfect sterilization submodel) is as follows:

$$\left\{
\begin{aligned}
\dot{S} &= -\mu_S S + \sigma, \\
\dot{F} &= -\mu_F F + r(1-p) \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F, \\
\dot{M} &= -\mu_M M + r p \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F.
\end{aligned}
\right.
\qquad (4)$$

Second, we considered the case where residual fertility has no associated fitness costs. The fertile irradiated males are considered to be fit as wild males in terms of attractiveness and survival; they thus enter into the $M$ compartment of the model. This case corresponds to $\epsilon = 0$ and $0 < \delta < 1$. The explicit form of this submodel (hereafter called the cost-free residual fertility submodel) is

$$\left\{
\begin{aligned}
\dot{S} &= -\mu_S S + (1 - \delta)\sigma, \\
\dot{F} &= -\mu_F F + r(1-p) \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F, \\
\dot{M} &= -\mu_M M + r p \frac{M}{k + M + \eta S} \frac{1}{1 + \beta F} F + \delta\sigma.
\end{aligned}
\right.
\qquad (5)$$

Third, we consider the case where residual fertility has associated fitness costs. All irradiated males, even if fertile, suffer from a lack of competitiveness (i.e., attractiveness), which is represented by parameter $\eta$ in the mating function, and they have a higher mortality rate compared to wild males. Therefore, fertile irradiated males remain in the $S$ compartment. This case corresponds to $\delta = 0$ and $0 < \epsilon < 1$. The explicit form of this submodel (hereafter called the costly residual fertility submodel)
since $X(0) \ge 0$ and $X'(M) \ge X'(M^*)$, for all $M \le M^*$. Therefore, the trace is negative.

Regarding the determinant, we get (Appendix B):

$$ \mathrm{Det}(J_{(F^*, M^*)}) = -r(1-p)\mu_M F^* \cdot \frac{dG}{dF}(F^*) $$

Therefore the sign of the slope of $G(F)$ at equilibrium determines the stability.

Exploiting the properties of $F_1^*$ and $F_2^*$ defined in cases (i) and (ii), we conclude that the infestation equilibrium $(F_1^*, M_1^*)$ is unstable and $(F_2^*, M_2^*)$ is asymptotically stable.

## Summary: equilibria and local stability

(i) If $G(0) < \frac{1}{\mathcal{R}_F}$ and $\max(G(F)) > \frac{1}{\mathcal{R}_F}$, there are three equilibria: the infestation equilibrium $(F_1^*, M_1^*)$ is unstable, while the pest-free equilibrium $(0, M(0))$ and the other infestation equilibrium $(F_2^*, M_2^*)$ are asymptotically stable.

(ii) If $G(0) > \frac{1}{\mathcal{R}_F}$, there are two equilibria: the pest-free equilibrium $(0, M(0))$ is unstable and the unique infestation equilibrium $(F_2^*, M_2^*)$ is asymptotically stable. This situation cannot occur for the perfect sterilization sub-model (4).

(iii) If $G(F) < \frac{1}{\mathcal{R}_F}$ for all $F \ge 0$, there is only one equilibrium, the pest-free equilibrium $(0, M(0))$, which is asymptotically stable.

## 2.2.4 Bifurcation diagram

The aim of this section is to determine the effect of the release rate of irradiated males $\sigma$ on the long-term population dynamics. A convenient way to address this issue is to compute a bifurcation diagram. In an agricultural context, it is important to focus on the population dynamics of females, as they are the ones which cause agricultural damages. So, we chose to represent the bifurcation diagram linking the density of females $F^*$ to the release rate $\sigma$. An expression of $\sigma$ as a function of $F^*$ was deduced from equations (10) and (12), taking explicit expressions for $X(S, M, \epsilon)$ (2) and $C(F)$ (3) described in section 2.1.1. Detailed calculations can be found in Appendix C. Thus $\sigma(F^*)$ can be expressed as follows for the general model (1):

$$ \sigma(F^*) = \frac{\frac{\beta p \mu_F}{(1-p)\mu_M} (F^*)^2 + \left(\frac{p \mu_F (1-\mathcal{R}_F)}{(1-p)\mu_M} + \beta k \right) F^* + k}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \left(\frac{\delta}{\mu_M} + \frac{\eta (1-\delta)}{\mu_S}\right) - \left(\frac{\delta}{\mu_M} + \frac{\eta (1-\delta)}{\mu_S}\right) \beta F^*} \quad (13) $$

The bifurcation diagram corresponds to the locus of the points $(\sigma(F^*), F^*)$ (Fig. 3). The effect of the residual fertility can be estimated according to how parameter $\delta$ or $\epsilon$ affect the shape of this bifurcation diagram. We proceeded in two stages to study $\sigma(F^*)$: firstly, the numerator, then the denominator.

**$\sigma(F^*)$ numerator** The numerator is a second degree polynomial in $F^*$ that does not depend on $\delta$ and $\epsilon$. The roots of the numerator $(F_A^*$ and $F_B^*$) then always represent the $F^*$ equilibria for $\sigma = 0$, independently of $\delta$ and $\epsilon$, illustrated by crosses in Fig. 3. The roots were determined classically after calculating the discriminant. We show in Appendix D that the discriminant is positive and both roots are positive when (D.20) holds; a sufficient condition for this is that the harmonic mean of the reproduction number of females and males is greater than 4:

$$ H(\mathcal{R}_M, \mathcal{R}_F) > 4. $$

It is a stronger condition than $\mathcal{R}_M > 1$ and $\mathcal{R}_F > 1$, that were identified as necessary conditions to avoid extinction of male and female populations. For this bifurcation study, we assume that this condition holds. The numerator is hence negative between $F_A^*$ and $F_B^*$, positive outside.

**$\sigma(F^*)$ denominator** The denominator of (13) can cancel out which leads to the existence of an asymptote and several shapes for the $\sigma$ bifurcation diagram. The denominator equals zero for values of $F^*$, named $F_-^*$, such that:

$$ F_-^*(\delta, \epsilon) = \frac{1}{\beta} \left( \frac{\mathcal{R}_F \delta \mu_S + \mu_M \mathcal{R}_F \epsilon \eta (1 - \delta)}{\delta \mu_S + \mu_M \eta (1 - \delta)} - 1 \right) \quad (14) $$
is

$$ \left\{ \begin{aligned} \dot{S} &= -\mu_S S + \sigma, \\ \dot{F} &= -\mu_F F + r(1-p) \frac{M + \epsilon \eta S}{k + M + \eta S} \frac{1}{1 + \beta F} F, \\ \dot{M} &= -\mu_M M + r p \frac{M + \epsilon \eta S}{k + M + \eta S} \frac{1}{1 + \beta F} F. \end{aligned} \right. \qquad (6) $$

The perfect sterilization submodel (4), the cost-free residual fertility submodel (5) and the costly residual fertility submodel (6) represent a spectrum of increasingly realistic biological scenarios.

## 2.2 Model analysis

This section contains the results of an equilibrium search and stability study conducting using the general model (1), as the analyses of the three submodels yielded similar findings (see 2.1.2).

Based on the aforementioned hypotheses, the model (1) is well defined biologically: S, F, and M remain non-negative over positive time values if their initial conditions are non-negative. The $\dot{S}$ equation is decoupled from the other equations, and the model assumes that S reaches its equilibrium $S^* = \frac{(1-\delta)\sigma}{\mu_S}$ very fast. Thus, we only considered the (F, M) subsystem. To improve readability, $X(S^*, M, \epsilon)$ was rewritten as $X(M)$.

### 2.2.1 Reproduction numbers

From the second equation of the general model (1), where $\sigma = 0$, we have

$$ \begin{aligned} \frac{\dot{F}}{F} &= -\mu_F + r(1-p)X(M)C(F), \\ &\leq -\mu_F + r(1-p) \sup_M(X(M))C(0), \\ &\leq -\mu_F + r(1-p). \end{aligned} $$

This inequality is used to define the maximal reproduction number for females:

$$ \mathcal{R}_F = \frac{r(1-p)}{\mu_F}. \qquad (7) $$

where $\mathcal{R}_F$ is the average number of females produced by a female over her lifetime, when males M are not limiting and female competition is ignored. For females to persist, $\mathcal{R}_F$ must be greater than 1.

From the third equation of (1), we arrive at the following:

$$ \begin{aligned} \frac{\dot{M}}{M} &= -\mu_M + r p \frac{X(M)}{M} C(F) F, \\ &\leq -\mu_M + r p X'(0) \sup_{F \geq 0} (C(F) F), \end{aligned} $$
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It can be easily shown that $F^*_{-}$ is increasing in $\delta$ and $\epsilon$. Without residual fertility ($\epsilon = 0$ or $\delta = 0$), $F^*_{-}$ is negative and smaller than $F^*_A$. When all released males are fertile ($\epsilon = 1$ or $\delta = 1$), $F^*_{-}$ goes up to $\frac{R_F-1}{\beta}$, which is larger than $F^*_A + F^*_B$ (D.21). Therefore, when the residual fertility ($\epsilon$ or $\delta$) is large enough, $F^*_{-}$ is larger than $F^*_A$ and $F^*_B$. The denominator is negative above the asymptote.

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**Bifurcation diagram shape** A bifurcation diagram of $F^*$ as a function of the release rate $\sigma$ is determined for fixed residual fertility rates. The shape of the bifurcation diagram depends on the value of the asymptote $F^*_{-}$, and in particular its position in relation to the two roots $F^*_A$ and $F^*_B$ of the numerator of (13). The stability of the different branches is deduced from section 2.2.3: when there are two infestation equilibria, the smaller is always unstable and the larger stable; when there is only one infestation equilibrium, it is always stable (Fig. 3). The asymptotic behavior of the model is therefore extremely dependent on both the release rate and the residual fertility of released individuals (Fig. 4).

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In the perfect sterilization case, when $\delta = \epsilon = 0$ (sub-model (4)), $F^*_{-}(0,0) = -1/\beta$, the denominator of (13) is always negative, and the bifurcation diagrams generated has no non-negative asymptote. The sign of $\sigma(F^*)$ is the opposite of the sign of the numerator of (13); it is therefore positive for $F^*$ between $F^*_A$ and $F^*_B$ and negative elsewhere. A similar reasoning applies for low residual fertility rates. It corresponds to shape $A'$ in Fig. 3. With regard to stability, as the release rate $\sigma$ increases, a zone of bistability precedes a zone where only the pest-free equilibrium is stable (Fig. 4).

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As $\delta$ or $\epsilon$ increases so does $F^*_{-}(\delta, \epsilon)$, which eventually becomes positive and smaller than $F^*_A$; the denominator of (13) is then negative between $F^* = 0$ and $F^*_{-}$, as is the numerator. A second positive branch of $\sigma(F^*)$ thus appears between $F^* = 0$ and $F^*_{-}$, which converges from below towards a horizontal asymptote in $F^*_{-}$. $\sigma(F^*)$ remains positive between $F^*_A$ and $F^*_B$. It corresponds to shape $B'$ in Fig. 3. With regard to stability, as the release rate $\sigma$ increases, three zones follow one another: a bistability zone, a zone where only the pest-free equilibrium is stable, and finally a zone where only the infestation equilibrium is stable (Fig. 4).

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While $\delta$ or $\epsilon$ increases further, $F^*_{-}$ gets larger than $F^*_A$ which causes a drastic change in the bifurcation diagram. There are positive branches below $F^*_A$ and between $F^*_{-}$ and $F^*_B$ where the numerator and denominator of (13) have the same sign, with now convergence to the asymptote from above. It corresponds to shape $C'$ in Fig. 3. With regard to stability, as the release rate $\sigma$ increases, a zone of bistability precedes a zone where only the infestation equilibrium is stable (Fig. 4).

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Finally, when $\delta$ or $\epsilon$ increases even further, $F^*_{-}$ gets larger than $F^*_B$. The positive upper branch increases now from $F^*_B$ and converges to $F^*_{-}$ from below. It corresponds to shape $D'$ in Fig. 3. With regard to stability, as the release rate $\sigma$ increases, a zone of bistability precedes a zone where only the infestation equilibrium is stable (Fig. 4). This infestation equilibrium is characterized by an increase in female density as compared with a situation without releases.

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It is possible to summarise the different dynamics occurring when both a component of residual fertility ($\epsilon$ or $\sigma$) and the release rate $\sigma$ vary. This co-dimension 2 bifurcation diagram is sketched in Fig. 4.

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# 3 Application to *Ceratitis capitata*

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In this section, results focus on the analysis and comparison of the residual fertility sub-models presented in section 2.1.2. The parameter values used to perform the simulations were taken from the published literature and are listed in Table 1. Data and calculations related to the mortality rates, emergence rate and oviposition competition coefficient are detailed in Appendix E.
because $X(M)$ is concave in $M$. This inequality is used to define the maximal reproduction number for males:

$$ \mathcal{R}_M = \frac{rpX'(0) \sup_{F \ge 0} (C(F)F)}{\mu_M}. $$

From the explicit expressions for $X(M)$ (2) and $C(F)$ (3), we arrive at:

$$ \mathcal{R}_M = \frac{rp}{\beta k \mu_M}. \qquad (8) $$

where $\mathcal{R}_M$ is the average number of males produced by a male over his lifetime for the optimal number of females in the population where $C(F)F$ is maximized. For males to persist, $\mathcal{R}_M$ must be greater than 1.

### 2.2.2 Equilibria

For the $(F, M)$ subsystem at $S = S^*$, the equilibria of system (1) are the solutions of the following equations:

$$ \begin{cases} F(-\mu_F + r(1-p)X(M)C(F)) = 0, \\ -\mu_M M + rpX(M)C(F)F + \delta\sigma = 0. \end{cases} \qquad (9) $$

Hence, the population densities at equilibrium can be used to verify that $F^* = 0$ or $X(M^*)C(F^*) = \frac{\mu_F}{r(1-p)} = \frac{1}{\mathcal{R}_F}$ from the first equation of (9).

#### Pest-free equilibrium

We can easily deduce the pest-free equilibrium $(F^*, M^*) = (0, \frac{\delta\sigma}{\mu_M})$. When $\delta = 0$, it is a true pest-free equilibrium of the form $(0, 0)$. In contrast, when $\delta > 0$, the male compartment is maintained by fertile irradiated males.

#### Infestation equilibria

The other equilibria must satisfy

$$ X(M)C(F) = \frac{\mu_F}{r(1-p)} = \frac{1}{\mathcal{R}_F}. \qquad (10) $$

Incorporating (10) into the second equation of (9), we obtain

$$ M(F) = \frac{p\mu_F}{(1-p)\mu_M}F + \frac{\delta\sigma}{\mu_M}. \qquad (11) $$

Replacing $M$ in (10) by its value in (11), we arrive at an equation that solely depends on female density:

$$ X(M(F))C(F) = \frac{1}{\mathcal{R}_F}. \qquad (12) $$

As $X(M(F))$ is concave and $1/C(F)$ is convex, this equation has 2, 1 or no solutions (Appendix A). We note that $G(F) = X(M(F))C(F)$ and describe these different cases below.
<figure><img src="image_5.png" /><figcaption>Figure 3: Qualitative σ-bifurcation diagram for sub-models with cost-free residual fertility (5) or costly residual fertility (6). The diagram, representing the density of females $F^*$ as a function of the release rate $\sigma$, has four different shapes ($A'$, $B'$, $C'$ and $D'$) corresponding to increasing values of the residual fertility rate ($\delta$ or $\epsilon$ from 0 to 1). For sub-model with perfect sterilization (4), only shape $A'$ applies. Stable equilibria are represented by solid lines, and unstable by dashed lines. The red line corresponds to asymptote $F^*$ (14) and the crosses to the roots of the numerator of $\sigma(F^*)$ (13). These shapes do not reproduce parameter values, but are sketched manually for understanding purposes.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 4: Stability zones as a function of the residual fertility rate $\delta$ or $\epsilon$ and the release rate $\sigma$. Three zones are defined by the black curve: the bistability zone, both the pest-free and the infestation equilibria are stable; in the other zones, only one of the two is stable. The colored areas are associated with the different shapes of bifurcation diagrams depicted in Fig. 3. The graphic was hand-drawn.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2 Female density at the infestation equilibria of model (1), which correspond to the intersections of function G(F) (in black) and the horizontal dotted line 1/R_F (in gray). The two simulated cases utilize different parameter values. Case A: when G(0) < 1/R_F and max(G(F)) > 1/R_F, there are two infestation equilibria F_1* and F_2*. Case B: when G(0) > 1/R_F, there is only one infestation equilibrium F_2*.</figcaption></figure>

(i) If G(0) < 1/R_F and max(G(F)) > 1/R_F (Fig. 2A), there are two solutions F_2* > F_1* > 0, such that dG/dF (F_1*) > 0 and dG/dF (F_2*) < 0. We then obtain two infestation equilibria (F_1*, M_1*) and (F_2*, M_2*).

(ii) If G(0) > 1/R_F (Fig. 2B), there is only one solution, at which dG/dF < 0. This solution is named F_2* and corresponds to the infestation equilibrium (F_2*, M_2*). In the perfect sterilization submodel (4): X(M(0)) = X(0) = 0, so G(0) = 0, and this second situation is impossible.

(iii) If G(F) < 1/R_F, ∀F > 0, there is no solution, so there is no infestation equilibrium.

### 2.2.3 Local stability of equilibria

The Jacobian matrix of the (F, M) subsystem at S = S* of general model (1) is

$$J_{(F,M)} = \begin{pmatrix} -\mu_F + r(1-p)X(M)(C'(F)F + C(F)) & r(1-p)C(F)X'(M)F \\ rpX(M)(C'(F)F + C(F)) & -\mu_M + rpC(F)X'(M)F \end{pmatrix}$$

#### Pest-free equilibrium

At the pest-free equilibrium (0, δ/μ_M) = (0, M(0)):

$$J_{(0,M(0))} = \begin{pmatrix} -\mu_F + r(1-p)X(M(0)) & 0 \\ rpX(M(0)) & -\mu_M \end{pmatrix}$$

Thus, (0, M(0)) is locally asymptotically stable if X(M(0)) < 1/R_F (i.e., G(0) < 1/R_F), which corresponds to cases (i) and (iii) above: there is no infestation equilibrium, or there are two infestation equilibria.
<table><caption>Table 1: Model parameters.</caption><thead><tr><th>Parameters</th><th>Descriptions</th><th>Values & [Range]</th><th>Units</th></tr></thead><tbody><tr><td>μ<sub>F</sub></td><td>Female mortality rate<sup>1,6</sup></td><td>0.050 [0.018 - 0.083]</td><td>day<sup>-1</sup></td></tr><tr><td>μ<sub>M</sub></td><td>Male mortality rate<sup>1,6</sup></td><td>0.036 [0.014 - 0.057]</td><td>day<sup>-1</sup></td></tr><tr><td>μ<sub>S</sub></td><td>Sterilized male mortality rate<sup>2,6</sup></td><td>0.057 [0.037 - 0.077]</td><td>day<sup>-1</sup></td></tr><tr><td>p</td><td>Sex ratio<sup>3</sup></td><td>0.500 [0.450 - 0.550]</td><td>-</td></tr><tr><td>r</td><td>Emergence rate<sup>4,6</sup></td><td>23.7 [19.2 - 28.5]</td><td>day<sup>-1</sup></td></tr><tr><td>k</td><td>Mating half-saturation constant<sup>5</sup></td><td>1 [0.01 - 100]</td><td>ind.ha<sup>-1</sup></td></tr><tr><td>1 - η</td><td>Sterilization cost<sup>5</sup></td><td>0.8 [0 - 1]</td><td>-</td></tr><tr><td>β</td><td>Oviposition competition between females<sup>6</sup></td><td>0.24 [0.18 - 0.30]</td><td>(ind.ha<sup>-1</sup>)<sup>-1</sup></td></tr><tr><td>σ</td><td>Sterilized male release rate<sup>7</sup></td><td>250 [0.01 - 500]</td><td>ind.ha<sup>-1</sup>.day<sup>-1</sup></td></tr><tr><td>δ</td><td>Proportion of non-sterile males among the releases<sup>7</sup> (cost-free fertility)</td><td>0.5 [0.01 - 1]</td><td>-</td></tr><tr><td>ε</td><td>Proportion of non-sterile males among the releases<sup>7</sup> (costly fertility)</td><td>0.5 [0.01 - 1]</td><td>-</td></tr></tbody></table>

ind.ha⁻<fnref n="1" /> corresponds to individuals per hectare.

## 3.1 Population control capacities with SIT deployment

The ability to control pest populations can be deduced from the different shapes of bifurcation diagrams described in section 2.2.4, and thus depends on the residual fertility rate and the fitness cost associated with irradiated males.

**Residual fertility impact** The existence of residual fertility in the releases has a considerable impact on SIT efficiency. Without residual fertility, by adjusting the release rate $\sigma$, it is always possible to only have a stable pest-free equilibrium, and therefore to theoretically go as far as population eradication (Fig. 3, Shape A').

For very low residual fertility rates ($\delta < 5.349 \times 10^{-4}$ for sub-model (5) and $\epsilon < 4.219 \times 10^{-3}$ for sub-model (6)), Shape A' is obtained. SIT can lead to eradication of the pest population by adapting the release rate $\sigma$. Indeed, increasing $\sigma$ makes it possible to move from a bistability zone, where the pest-free equilibrium but also the highest infestation equilibrium are stable, to a zone where only the pest-free equilibrium is stable (Fig. 5, purple zones).

For intermediate residual fertility rates ($5.349 \times 10^{-4} < \delta < 5.353 \times 10^{-4}$ for sub-model (5) and $4.219 \times 10^{-3} < \epsilon < 4.222 \times 10^{-3}$ for sub-model (6)), Shape B' is obtained. SIT can theoretically still lead to pest extinction. By progressively increasing $\sigma$, three different zones follow one another: first the bistability zone, then the stable pest-free equilibrium stability zone, and finally the stable infestation equilibrium stability zone (Fig. 5, tiny pink zones). In practice, this shape represents negligible fertility rate ranges as $F_A^*$ is very close to zero.

For medium to high residual fertility rates ($5.353 \times 10^{-4} < \delta < 0.994$ for sub-model (5) and $4.222 \times 10^{-3} < \epsilon < 0.999$ for sub-model (6)), Shape C' is obtained. SIT cannot lead to the eradication of the pest population but, in the best case, to a drastic reduction in its density. Indeed, the stable infestation equilibrium zone follows the bistability zone when $\sigma$ increases (Fig. 5, blue zones). However, with high sigma values, it is possible to reduce population density by more than 80% (Fig. 6). Obviously, the higher the fertility rate, the less likely it is that the population will be drastically reduced.

For even higher residual fertility rates ($\delta > 0.994$ for sub-model (5) and $\epsilon > 0.999$ for sub-model (6)), Shape D' is obtained. Population growth is enhanced by releases, which is not realistic and has therefore not been represented in this section.

So, in summary, depending on the release rate $\sigma$ and the residual fertility rate, there are different zones with associated specific stabilities (Fig. 5), resulting in greater or lesser population reduction capacities (Fig. 6). Without residual fertility, by adjusting the value of the release rate it is always possible to

<footnote n="1">¹ Female and male mortality rates are extracted from Vargas et al. (2000) and Pieterse et al. (2020).</footnote>

<footnote n="2">² Sterilized male mortality rate is extracted from Juan-Blasco et al. (2013).</footnote>

<footnote n="3">³ Sex ratio is deduced from Juan-Blasco et al. (2013).</footnote>

<footnote n="4">⁴ Emergence rate is calculated from Pieterse et al. (2020).</footnote>

<footnote n="5">⁵ Arbitrary values.</footnote>

<footnote n="6">⁶ Detailed calculations in Appendix E.</footnote>

<footnote n="7">⁷ Studied values.</footnote>
Infestation equilibria

At the positive equilibria (F*, M*), the Jacobian matrix is

$$
J_{(F^*, M^*)} = \begin{pmatrix} -\mu_F + r(1-p)X(M^*)(C'(F^*)F^* + C(F^*)) & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\mu_M + rpC(F^*)X'(M^*)F^* \end{pmatrix}
$$

The nullcline $r(1-p)X(M)C(F) = \mu_F$ from system (9) allows us to simplify the matrix as follows:

$$
J_{(F^*, M^*)} = \begin{pmatrix} r(1-p)X(M^*)C'(F^*)F^* & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\mu_M + rpC(F^*)X'(M^*)F^* \end{pmatrix}
$$

To determine the stability of the positive equilibria, we calculated the trace and the
determinant of this matrix.

For the trace, the other nullcline $rpX(M)C(F)F + \delta\sigma = \mu_M M$ from system (9)
allows us to obtain

$$
J_{(F^*, M^*)} = \begin{pmatrix} r(1-p)X(M^*)C'(F^*)F^* & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\frac{rpX(M^*)C(F^*)F^* + \delta\sigma}{M^*} + rpC(F^*)X'(M^*)F^* \end{pmatrix}
$$

The calculation of the trace yields

$$
\mathrm{Tr} (J_{(F^*, M^*)}) = r(1-p)X(M^*)C'(F^*)F^* - \frac{\delta\sigma}{M^*} + rpC(F^*)F^* \left( X'(M^*) - \frac{X(M^*)}{M^*} \right)
$$

The first two terms are negative since C(F) is decreasing. Furthermore, as X(M) is
concave, the last term is nonpositive. Indeed

$$
X(M^*) = X(0) + \int_{0}^{M^*} X'(M) dM \geq \int_{0}^{M^*} X'(M^*) dM = M^* X'(M^*)
$$

since $X(0) \ge 0$ and $X'(M) \ge X'(M^*)$, for all $M \le M^*$. Therefore, the trace is negative.

For the determinant, we arrive at (Appendix B):

$$
\mathrm{Det}(J_{(F^*, M^*)}) = -r(1-p)\mu_M F^* \cdot \frac{dG}{dF}(F^*)
$$

Therefore, the sign of the slope of $G(F)$ at equilibrium determines stability.

Exploiting the properties of $F_1^*$ and $F_2^*$ defined in cases (i) and (ii), we can con-
clude that the infestation equilibrium ($F_1^*, M_1^*$) is unstable and that ($F_2^*, M_2^*$) is
asymptotically stable.

Summary: equilibria and local stability

(i) If $G(0) < \frac{1}{R_F}$ and $\max(G(F)) > \frac{1}{R_F}$, there are three equilibria: the infestation equilibrium ($F_1^*, M_1^*$) is unstable, while the pest-free equilibrium (0, $M(0)$) and the other infestation equilibrium ($F_2^*, M_2^*$) are asymptotically stable.
<figure><img src="image_7.png" /><figcaption>Figure 5: Stability zones as functions of residual fertility $\delta$ or $\epsilon$ and the release rate $\sigma$, for A: cost-free fertility model (5) and B: costly residual fertility model (6). The lower figures correspond to zooms of the upper figures to show the tiny pink area corresponding to Shape B'. Three zones are defined by the black curve: the bistability zone, both the pest-free and the infestation equilibria are stable; in the other zones, only one of the two is stable. The colored areas are associated with the different shapes of bifurcation diagrams depicted in Fig. 3. The scale used does not allow shape D' to be visualized.</figcaption></figure>

eradicate the population, whereas with residual fertility the population can only be eradicated for a very small range of residual fertility rate values ($\delta$ or $\epsilon < 0.5\%$).

**Fitness cost impact** Whether there is a cost associated with the sterilization process (sub-model (6)) or not (sub-model (5)), the same four bifurcation diagram shapes are obtained (Fig. 3), but for different residual fertility rates. For example, Shape A' is obtained for higher rates for costly residual fertility compared to cost-free residual fertility (Fig. 5). In terms of population control potential, with an associated cost, the population can be reduced for higher residual fertility rates (Fig. 6). However, in both cases, when residual fertility rates are too high, population density cannot be controlled, regardless of the release rate $\sigma$. Similarly, for low $\sigma$ values, population control cannot be achieved whatever the residual fertility rates.

**Residual fertility thresholds to eradicate the population** Shape A' (Fig. 3) is the (almost) only case for which the pest-free equilibrium can be the only stable equilibrium (shape B' being negligible), provided that the release rate is high enough (Fig. 5, purple zones). It is obtained when the asymptote $F_{-}^{*}$, defined in (14) is negative, which corresponds to the following residual fertility thresholds for cost-free sub-model (5):

$$ \delta < \frac{\eta\mu_M}{(\mathcal{R}_F - 1)\mu_S + \eta\mu_M} = 0.001, $$

and for costly sub-model (6):

$$ \epsilon < \frac{1}{\mathcal{R}_F} = 0.004. \qquad (15) $$

This last result generalizes the condition obtained by Aronna and Dumont (2020) for $k = 0$.

Both thresholds are very small (less than 0.5%) and arguably hardly achievable in practice. Therefore, a more realistic goal would be to drastically reduce the population and not aim for eradication.
(ii) If $G(0) > \frac{1}{\mathcal{R}_F}$, there are two equilibria: the pest-free equilibrium $(0, M(0))$ is unstable and the single infestation equilibrium $(F_2^*, M_2^*)$ is asymptotically stable. This situation is impossible in the perfect sterilization submodel (4).

(iii) If $G(F) < \frac{1}{\mathcal{R}_F}$ for all $F \ge 0$, there is only one equilibrium, the pest-free equilibrium $(0, M(0))$, which is asymptotically stable.

### 2.2.4 Bifurcation diagram

This section explores the effects of the release rate of irradiated males $\sigma$ on long-term population dynamics. A convenient approach is to generate a bifurcation diagram. In agricultural settings, it is important to focus on the density dynamics of females, as they are responsible for agricultural damage. Therefore, we chose to generate a bifurcation diagram in which female density $F^*$ is linked to the release rate $\sigma$. An expression of $\sigma$ as a function of $F^*$ was deduced from equations (10) and (12), using the explicit expressions for $X(S, M, \epsilon)$ (2) and $C(F)$ (3) described in section 2.1.1. Detailed calculations can be found in Appendix C. Thus, for the general model, $\sigma(F^*)$ can be expressed as follows (1):

$$ \sigma(F^*) = \frac{\frac{\beta p \mu_F}{(1-p)\mu_M} (F^*)^2 + \left(\frac{p \mu_F (1-\mathcal{R}_F)}{(1-p)\mu_M} + \beta k \right) F^* + k}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \left(\frac{\delta}{\mu_M} + \frac{\eta (1-\delta)}{\mu_S}\right) - \left(\frac{\delta}{\mu_M} + \frac{\eta (1-\delta)}{\mu_S}\right) \beta F^*} \quad (13) $$

The bifurcation diagram corresponds to the locus of the points $(\sigma(F^*), F^*)$ (Fig. 3). The effect of residual fertility can be estimated based on how parameters $\delta$ or $\epsilon$ affect the shape of this bifurcation diagram. We proceeded to study $\sigma(F^*)$ in two stages: we looked first at the numerator and then at the denominator.

#### $\sigma(F^*)$ numerator

The numerator is a second-degree polynomial in $F^*$ that does not depend on $\delta$ or $\epsilon$. The roots of the numerator ($F_A^*$ and $F_B^*$) thus always represent the $F^*$ equilibria for $\sigma = 0$ independently of $\delta$ and $\epsilon$ (crosses in Fig. 3). The roots were determined using a classical approach after calculating the discriminant. We show in Appendix D that the discriminant and both roots are positive when (D2) applies, such as when the harmonic mean of the reproduction numbers for the females and males is greater than 4:

$$ H(\mathcal{R}_M, \mathcal{R}_F) > 4. $$

This condition is stronger than $\mathcal{R}_M > 1$ and $\mathcal{R}_F > 1$, identified as the conditions needed to maintain the persistence of males and females. In this bifurcation work, it was assumed that this condition held. The numerator is thus negative between $F_A^*$ and $F_B^*$ but is otherwise positive.

#### $\sigma(F^*)$ denominator

The denominator of (13) can be cancelled out, leading to the existence of an asymptote and several shapes for the $\sigma$ bifurcation diagram. The denominator equals zero for the
<figure><img src="image_8.png" /><figcaption>Figure 6: Contour plot illustrating the percentage of population reduction ($1 - q$), with $q$ defined in (16), as a function of the release rate $\sigma$ and the residual fertility rate, for A: cost-free residual fertility model (sub-model (5)) and B: costly residual fertility model (sub-model (6)). The lower graphs correspond to zooms of the blue boxes in the upper graphs. The grey curve visible on the zoomed-in graphs corresponds to the limit between a 90% reduction and a 100% reduction (eradication) of the population.</figcaption></figure>
values of $F^*$, named $F^*_-$, such that

$$F^*_-(\delta, \epsilon) = \frac{1}{\beta} \left( \frac{\mathcal{R}_F \delta \mu_S + \mu_M \mathcal{R}_F \epsilon \eta (1 - \delta)}{\delta \mu_S + \mu_M \eta (1 - \delta)} - 1 \right) \quad (14)$$

It can be easily shown that $F^*_-$ increases with $\delta$ and $\epsilon$. In the absence of residual fertility ($\epsilon = 0$ or $\delta = 0$), $F^*_-$ is negative and smaller than $F^*_A$. When all released males are fertile ($\epsilon = 1$ or $\delta = 1$), $F^*_-$ increases up to $\frac{\mathcal{R}_F - 1}{\beta}$, which is larger than $F^*_A + F^*_B$ (D3). Therefore, when residual fertility ($\epsilon$ or $\delta$) is large enough, $F^*_-$ is larger than $F^*_A$ and $F^*_B$. The denominator is negative above the asymptote.

### Bifurcation diagram shape

A bifurcation diagram of $F^*$ as a function of the release rate $\sigma$ was determined for fixed residual fertility rates. The shape of the bifurcation diagram depends on the value of the asymptote $F^*_-$, and, in particular, its position in relation to the two roots, $F^*_A$ and $F^*_B$, of the numerator of (13). The stability of the different branches was deduced from the work in section 2.2.3: when there are two infestation equilibria, the smaller equilibrium is unstable and the larger equilibrium is stable; when there is only one infestation equilibrium, it is always stable (Fig. 3). The asymptotic behavior of the model is therefore highly dependent on both the release rate and residual fertility (Fig. 4).

In the perfect sterilization submodel (4), when $\delta = \epsilon = 0$, $F^*_-(0,0) = -1/\beta$ and the denominator of (13) is always negative; the resulting bifurcation diagrams do not have a nonnegative asymptote. The sign of $\sigma(F^*)$ is the opposite of the sign of the numerator of (13); it is therefore positive for $F^*$ between $F^*_A$ and $F^*_B$ and negative elsewhere. This situation is similar for low residual fertility rates (see shape $A'$ in Fig. 3). With regards to stability, as the release rate $\sigma$ increases, a zone of bistability precedes a zone where only the pest-free equilibrium is stable (Fig. 4).

As $\delta$ or $\epsilon$ increases, so does $F^*_-(\delta, \epsilon)$; it eventually becomes positive and smaller than $F^*_A$. The denominator of (13) is therefore negative between $F^* = 0$ and $F^*_-$, as is the numerator. A second positive branch of $\sigma(F^*)$ thus appears between $F^* = 0$ and $F^*_-$ and converges from below towards a horizontal asymptote in $F^*_-$; $\sigma(F^*)$ remains positive between $F^*_A$ and $F^*_B$ (see shape $B'$ in Fig. 3). With regards to stability, as the release rate $\sigma$ increases, three zones follow one another: a bistability zone, a zone where only the pest-free equilibrium is stable, and, finally, a zone where only the infestation equilibrium is stable (Fig. 4).

As $\delta$ or $\epsilon$ increases further, $F^*_-$ gets larger than $F^*_A$ which causes a drastic change in the bifurcation diagram. There are positive branches below $F^*_A$ and between $F^*_-$ and $F^*_B$ where the numerator and denominator of (13) have the same sign and where there is now convergence with the asymptote from above (see shape $C'$ in Fig. 3). With regards to stability, as the release rate $\sigma$ increases, a zone of bistability precedes a zone where only the infestation equilibrium is stable (Fig. 4).

Finally, as $\delta$ or $\epsilon$ increases even further, $F^*_-$ becomes larger than $F^*_B$. The positive upper branch now increases from $F^*_B$ and converges to $F^*_-$ from below (see shape $D'$
**Residual fertility thresholds to control the population** When eradication is not possible regardless of the release rate, the population reduction must be significant for the technique to be considered effective. It corresponds to shape C' (Fig. 3), in which the infestation equilibrium decreases towards the asymptote $F_{-}^{*}$ when the release rate increases. Thus, the maximum residual fertility rate allowing a $(1-q)$ reduction of the female population density compared to the $\sigma = 0$ case is determined by solving:

$$F_{-}^{*} = qF_{B}^{*}, \qquad (16)$$

For instance, a 90% population reduction corresponds to $q = 0.1$.

For the cost-free sub-model (5), we obtain:

$$\delta = \frac{(q\beta F_{B}^{*} + 1)\mu_{M}\eta}{\mathcal{R}_{F}\mu_{S} + (q\beta F_{B}^{*} + 1)(\mu_{M}\eta - \mu_{S})}.$$

For a 90% female population reduction, this threshold is equal to 0.014.

Finally, for the costly sub-model (6), we obtain the following tolerable residual fertility threshold:

$$\epsilon = \frac{1}{\mathcal{R}_{F}} + \frac{qF_{B}^{*}\beta}{\mathcal{R}_{F}}. \qquad (17)$$

The threshold for population control is thus $\frac{qF_{B}^{*}\beta}{\mathcal{R}_{F}}$ higher than the threshold required for population eradication, it is equal to 0.103.

For the particular case where $k = 0$, we obtain:

$$\epsilon = \frac{1}{\mathcal{R}_{F}} + q(1 - \frac{1}{\mathcal{R}_{F}}), \qquad (18)$$

so that for a 90% female population reduction, this threshold is equal to 0.104.

**Convergence time** To optimize the effectiveness of SIT, it is essential to consider not only the residual fertility rate, but also the release rate $\sigma$. In Fig. 6, for a given residual fertility rate, a range of $\sigma$ values will lead to the same population reduction range. However, $\sigma$ has an influence on the convergence time to reach a given population reduction range, so that higher $\sigma$ values allow for faster population control. For example, a release rate $\sigma$ ten times higher allows to reach extinction or population reduction much faster for all sub-models (Fig. 7): 75 days are enough for the higher release rate, but it takes between 200 and 300 days for the lower rate to get close to the equilibrium value, starting from the uncontrolled infestation equilibrium.

## 3.2 Sensitivity analysis

A global sensitivity analysis was performed to determine the most influential parameters on percentage of female population reduction $(1-q)$, with $q$ defined in (16), in order to study how to improve pest population control.

We used a variance-based method (Sobol, 1990; Wu et al., 2013). For each parameter, three values were tested, corresponding to the reference value, plus the minimum and maximum values of the range specified in Table 1). A full factorial design was used to explore the parameter space. An ANOVA was then performed to obtain the variance decomposition based on a linear model with two-way interactions between the output and the parameters. Finally, Sobol Sensitivity Indices (SI) and Total Sensitivity Indices (TSI) were calculated for each parameter as follows: the SI as the ratio between the sum of squares of the parameter main effect and the total sum of squares; the TSI as the ratio between the sum of squares of the parameter main effect plus its interactions and the total sum of squares.

The parameter with the greatest influence on female density reduction is the female mortality rate $\mu_{F}$ with a total sensitivity index greater than 50% (Fig. 8). The competition coefficient between females $\beta$ and the emergence rate $r$, which both relate to the biology of the species, also have a significant influence. Finally, parameters directly related to SIT - the residual fertility rate ($\delta$ or $\epsilon$), the release rate $\sigma$ and the cost of sterilization $\eta$ - also have a major influence on female density reduction. The other parameters, i.e. male and sterilized male mortality rates ($\mu_{M}$ and $\mu_{S}$) and the mating half-saturation constant $k$, have a negligible influence compared to the others.
<figure><img src="image_4.png" /><figcaption>**Fig. 3** Qualitative σ-bifurcation diagram for the cost-free residual fertility submodel (5) and the costly residual fertility submodel (6). The diagram represents female density $F^*$ as a function of the release rate $\sigma$ and has four different shapes ($A'$, $B'$, $C'$ and $D'$) that correspond to increasing values of the residual fertility rate (range of $\delta$ or $\epsilon$: 0 to 1). For the perfect sterilization submodel (4), only shape $A'$ applies. Stable and unstable equilibria are represented by solid and dashed lines, respectively. The red line corresponds to asymptote $F^*_-$ (14), and the crosses indicate the roots of the numerator of $\sigma(F^*)$ (13). These shapes are manually generated depictions; they were not created using parameter values.</figcaption></figure>

in Fig. 3). With regards to stability, as the release rate $\sigma$ increases, a zone of bistability precedes a zone where only the infestation equilibrium is stable (Fig. 4). This infestation equilibrium is characterized by an increase in female density as compared to a situation without any releases.

It is possible to summarize the different dynamics occurring when both a residual fertility parameter ($\epsilon$ or $\sigma$) and the release rate $\sigma$ vary. We have provided a depiction of this codimension-two bifurcation diagram (Fig. 4).

## 3 Application to *Ceratitis capitata*

In this section, we focus on the results of the comparative analyses of the residual fertility submodels presented in section 2.1.2. The parameter values used to perform the simulations were obtained from the published literature (Table 1). The data and calculations related to the mortality rates, emergence rate, and female competition coefficient are described in Appendix E.
<figure><img src="image_9.png" /><figcaption>Figure 7: Temporal simulations of female population dynamics (*F*). Four cases are considered: no SIT (black curve), SIT without residual fertility (blue curves), SIT with 1% cost-free residual fertility (orange curves) and SIT with 1% costly residual fertility (purple curves). The dotted curves were obtained with the release rate σ = 500 ind.ha⁻¹.day⁻¹ and the solid curves with σ = 5000 ind.ha⁻¹.day⁻¹.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8: Total sensitivity index per parameter, separated into main effect (black bar, SI) and two-way interactions (grey bar). Sensitivity analysis performed on female density reduction for A: cost-free residual fertility sub-model (5) and B: costly residual fertility sub-model (6).</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>**Fig. 4** Stability zones as a function of the residual fertility rate $\delta$ or $\epsilon$ and the release rate $\sigma$. Three zones are defined by the black curve. In the bistability zone, both the pest-free and infestation equilibria are stable; in the other zones, only one of the two is stable. The colored areas are associated with the different shapes of the bifurcation diagrams depicted in Fig. 3. This graph is a manually generated depiction.</figcaption></figure>

<table><caption>**Table 1** Model parameters.</caption><thead><tr><th>Parameters</th><th>Descriptions</th><th>Values[Range]</th><th>Units</th></tr></thead><tbody><tr><td>μ<sub>F</sub></td><td>Female mortality rate<sup>1,6</sup></td><td>0.050 [0.018 - 0.083]</td><td>day<sup>-1</sup></td></tr><tr><td>μ<sub>M</sub></td><td>Male mortality rate<sup>1,6</sup></td><td>0.036 [0.014 - 0.057]</td><td>day<sup>-1</sup></td></tr><tr><td>μ<sub>S</sub></td><td>Sterilized male mortality rate<sup>2,6</sup></td><td>0.057 [0.037 - 0.077]</td><td>day<sup>-1</sup></td></tr><tr><td>p</td><td>Sex ratio<sup>3</sup></td><td>0.500 [0.450 - 0.550]</td><td>-</td></tr><tr><td>r</td><td>Emergence rate<sup>4,6</sup></td><td>23.7 [19.2 - 28.5]</td><td>day<sup>-1</sup></td></tr><tr><td>k</td><td>Mating half-saturation constant<sup>5</sup></td><td>1 [0.01 - 100]</td><td>ind.ha<sup>-1</sup></td></tr><tr><td>1 - η</td><td>Cost of sterilization<sup>5</sup></td><td>0.8 [0 - 1]</td><td>-</td></tr><tr><td>β</td><td>Female competition for oviposition sites<sup>6</sup></td><td>0.24 [0.18 - 0.30]</td><td>(ind.ha<sup>-1</sup>)<sup>-1</sup></td></tr><tr><td>σ</td><td>Release rate of irradiated males<sup>7</sup></td><td>250 [0.01 - 500]</td><td>ind.ha<sup>-1</sup>.day<sup>-1</sup></td></tr><tr><td>δ</td><td>Proportion of fertile irradiated males<sup>7</sup><br>(cost-free residual fertility)</td><td>0.5 [0.01 - 1]</td><td>-</td></tr><tr><td>ε</td><td>Proportion of fertile irradiated males<sup>7</sup><br>(costly residual fertility)</td><td>0.5 [0.01 - 1]</td><td>-</td></tr></tbody></table>

ind.ha⁻<fnref n="1" /> corresponds to individuals per hectare.

<footnote n="1">¹ The female and male mortality rates come from Vargas et al. (2000) and Pieterse et al. (2020).</footnote>

<footnote n="2">² The mortality rate for sterilized male comes from Juan-Blasco et al. (2013).</footnote>

<footnote n="3">³ The sex ratio was deduced from data in Juan-Blasco et al. (2013).</footnote>

<footnote n="4">⁴ The emergence rate was calculated from data in Pieterse et al. (2020).</footnote>

<footnote n="5">⁵ Arbitrary values.</footnote>

<footnote n="6">⁶ Detailed calculations in Appendix E.</footnote>

<footnote n="7">⁷ Values from this study.</footnote>
# 4 Discussion

We investigated the impact of residual fertility on sterilized male release efficiency in the context of SIT deployment. We showed that, as hypothesized, imperfect sterilization has a significant impact on the effectiveness of SIT. In addition, when a fitness cost is associated with radiation exposure, higher residual fertility rates can be tolerated to achieve effective control of the pest. The analysis carried out shows the impact of residual fertility, and emphasizes the need to better understand the sterilization and release process. Indeed, the fertile released males threaten more the effectiveness of SIT if they present the same phenotypic and behavioral traits as the wild males (cost-free sub-model), rather than when they share their characteristics with the sterilized males (costly sub-model). This analysis shows that it is capital to apprehend the mechanisms underpinning the residual fertility and to understand how the males escaping sterilization are impacted by the different SIT processes (mass-breeding, irradiation, transport and release). Have the irradiated males lost their fitness but have some conserved viable sperm? Or have some males escaped the sterility-inducing rays and retained all their fitness? In a nutshell, our results illustrate the importance of considering not only the sterilized male release rate required to reduce the population density in a timely manner, but also the residual fertility rate and associated fitness costs.

In our study, costs were associated with irradiated males at two levels: firstly, sterilized males had a higher mortality than wild-type males and secondly, a lower mating rate (parameter $\eta$). The latter can be due to reduced attractiveness, flight and dispersal capacity (Guerfali et al., 2011), or limited competitiveness (Dyck et al., 2005). The dose needed to induce total sterility in *C. capitata* induces a significant drop in male performance (Robinson et al., 2002), so SIT programs rarely aim at 100% sterility (Parker and Mehta, 2007) and the addition of aromatherapy based on ginger root oil, known to improve the attractiveness of sterile males, is often adopted (Morelli et al., 2013). Aromatherapy has been confirmed as an essential step in the success of SIT against *C. capitata* (Dumont and Oliva, 2023). In this species, the attractiveness of sterilized males is indeed essential, as males gather in leks (Whittier et al., 1992), to court wild females, attract them and finally mate (Shelly and McInnis, 2016). According to our sensitivity analysis, the influence of this $\eta$ cost on the reduction of population density is significant, which highlights the importance of determining the traits affected by the sterilization process and of quantifying their impact on the male mating capacity. In contrast, the mortality rate of sterilized males, linked to the radiation dose they received, does not have a major influence. This point illustrates the complexity of the trade-off between radiation dose and fitness, and hence competitiveness of irradiated males. Other authors addressed this trade-off issue and suggested that lower doses than those applied to achieve high sterilization rates can optimize SIT, noting that any increase in residual fertility is more than offset by the increased competitiveness of the released insects (Parker and Mehta, 2007). In our study, the sensitivity analysis shows that residual fertility rate ($\delta$ or $\epsilon$) has a similar if slightly higher impact on SIT effectiveness compared to sterilization cost ($1 - \eta$), which suggests that increased residual fertility rates could be mitigated by increased irradiated male competitiveness. Other species-specific parameters had a significative influence on population reduction: they were either linked to density-dependence processes (e.g. competition strength) or population growth (e.g. fertility and emergence rate). This underlines the importance of understanding both qualitatively and quantitatively the demographic processes driving the pest population dynamics.

The objective of SIT programs in the agricultural context is to reduce fruit damages, and therefore population levels, below an economic damage threshold (Dyck et al., 2021). This study confirms that the effectiveness of SIT depends largely on the residual fertility rate in the releases (Dyck et al., 2005; Parker and Mehta, 2007; Dumont and Oliva, 2023). The lower the residual fertility rate, the more effective the control. In the majority of theoretical studies, the effectiveness of SIT is associated with a stable pest-free equilibrium and thus with the possibility of eradicating the population (Bliman et al., 2019; Aronna and Dumont, 2020). As Aronna and Dumont (2020), we show that the threshold allowing population eradication in the costly sub-model is $1/R_F$ (15), which is less than 0.5%. It is even lower for the cost-free residual fertility sub-model. Hence, the possibility of eradicating the population exists only for very low residual fertility rates. However, even if eradication is theoretically impossible, it does not mean that control is impossible and that the technique is useless. Indeed, in the South African SIT program against *C. capitata*, the objective was to bring fly populations below an economic threshold and then create an internationally recognised low pest prevalence area (Zavala-López and Enkerlin, 2017). This can be achieved when the pest-free equilibrium is stable, but also when it is unstable and a low pest equilibrium is stable. Thus, we show that a residual fertility threshold $\frac{1}{R_F} + \frac{qF_B\beta}{R_F}$ (17), higher than the eradication threshold, is theoretically acceptable to control the population. We showed that, up to this threshold of 10% of residual fertility, the population can be reduced by $1 - q = 90\%$ of the uncontrolled equilibrium
## 3.1 Using SIT for population control

The ability of SIT to control pest populations can be deduced from the different shapes of the bifurcation diagrams described in section 2.2.4 and thus depends on the residual fertility rate and the fitness costs experienced by irradiated males.

### Impact of residual fertility

The presence of residual fertility among the irradiated males has a considerable impact on SIT efficiency. In the absence of residual fertility, by adjusting the release rate $\sigma$, it is always possible to arrive at a stable pest-free equilibrium and, therefore, to theoretically attain population eradication (Fig. 3, Shape A').

At very low residual fertility rates ($\delta < 5.349 \times 10^{-4}$ for submodel (5) and $\epsilon < 4.219 \times 10^{-3}$ for submodel (6)), Shape A' is obtained. SIT can lead to the eradication of the pest population if the release rate $\sigma$ is adapted. Indeed, increasing $\sigma$ makes it possible to move from a bistability zone, where both the pest-free equilibrium and the highest infestation equilibrium are stable, to a zone where only the pest-free equilibrium is stable (Fig. 5, purple zones).

At intermediate residual fertility rates ($5.349 \times 10^{-4} < \delta < 5.353 \times 10^{-4}$ for submodel (5) and $4.219 \times 10^{-3} < \epsilon < 4.222 \times 10^{-3}$ for submodel (6)), Shape B' is obtained. SIT can theoretically still lead to pest eradication. As $\sigma$ is progressively increased, three different zones follow one another: first the bistability zone, then the zone where the pest-free equilibrium is stable, and finally the zone where the infestation equilibrium is stable (Fig. 5, tiny pink zones). In practice, this shape is associated with negligible residual fertility rates as $F_A^*$ is very close to zero.

At medium to high residual fertility rates ($5.353 \times 10^{-4} < \delta < 0.994$ for submodel (5) and $4.222 \times 10^{-3} < \epsilon < 0.999$ for submodel (6)), Shape C' is obtained. SIT cannot lead to the eradication of the pest population, but it can, in the best case scenario, cause drastic reductions in density. Indeed, the zone where the infestation equilibrium is stable occurs after the bistability zone as $\sigma$ increases (Fig. 5, blue zones). However, using high $\sigma$ values, it is possible to reduce population density by more than 80% (Fig. 6). Obviously, the higher the fertility rate, the less likely the population will experience drastic reductions.

At even higher residual fertility rates ($\delta > 0.994$ for submodel (5) and $\epsilon > 0.999$ for submodel (6)), Shape D' is obtained. Population growth is enhanced by the released males, which is unrealistic and is therefore not addressed in this section.

So, in summary, depending on the release rate $\sigma$ and the residual fertility rate, there are different zones with their own stability patterns (Fig. 5), such that there is a greater or lesser capacity for population control (Fig. 6). In the absence of residual fertility, by adjusting the value of the release rate, it is always possible to eradicate the population; in contrast, in the presence of residual fertility, the population can only be eradicated across a very small range of residual fertility rates ($\delta$ or $\epsilon < 0.5\%$).

### Impact of fitness costs

Whether or not fitness costs are present (submodel (6) and submodel (5), respectively), the same four bifurcation diagram shapes are obtained (Fig. 3), albeit for different residual fertility rates. For example, Shape A' is obtained at higher residual fertility
density. This 90% reduction would indeed be acceptable as it would bring the amount of males caught by 50 traps over an hectare from 982, at the uncontrolled equilibrium density, down to 98 which is below the economic threshold of about 3 males per trap and per day for *C. capitata* (Hafsi et al., 2020a). Thus, efficiently deploying this technique does not necessarily require the eradication of the species in the area of interest, where the persistence of a low population density causes negligible economic losses.

With a residual fertility threshold of 1%, which is the typical threshold for *C. capitata*, the IAEA recommends releasing about 4000 individuals per week and per hectare. Our results suggest, for a costly residual fertility of 1%, that a release of about 400 sterilized males per day per hectare would be sufficient to control the population (at least 90% reduction in Fig. 6), which corresponds to 2,800 males per week and per hectare. So, our model allows us to estimate release rates of the same order of magnitude as those recommended for *C. capitata*. According to our model, for a cost-free residual fertility of 1%, a release of more than 8,000 males per week and per hectare would be necessary to ensure a 90% population reduction. If fertile released males are as fit as wild males, our results show that the releases should be more than twice the IAEA recommendations. Therefore, release recommendations should not only take into account residual fertility but also the fitness of released individuals.

By studying residual fertility, we are considering the possibility of introducing fertile irradiated males into the environment, which raises ethical and safety issues (Oliva et al., 2021). With current IAEA recommendations, the risk that residual fertility might compromise pest population control is fairly low. However, the release of non-sterile irradiated males raises issues relative to the introduction of non endemic strains into a wild population (Lynch and Thomas, 2000; Benedict et al., 2018). Indeed, non-sterile males can reproduce and transfer their genes, potentially introducing new alleles and new phenotypes into the target population. Moreover, strains used for releases in the SIT context might be different from wild population in many regards. First, strains might be genetically selected to choose specific properties, such as the developing of genetic sexing traits (Robinson, 2002; Ramírez-Santos et al., 2016). Second, mass rearing during many generations has probably lead to adaptation to laboratory conditions (Hoffmann and Ross, 2018). And third, individuals are irradiated to induce sterilization, which can lead to numerous unknown mutations. Residual fertility is thus a threat to SIT programs in the sense that it makes self-replication possible, making sterile insects comparable to other biocontrol agents (Kapranas et al., 2022). Indeed, crosses between introduced and local biocontrol agents are possible, making hybrids with non-controlled traits that potentially increase the risk of invasion (Turgeon et al., 2011). The context of SIT is generally comparable to inundation biological control (mass introduction of macro-organisms (Eilenberg et al., 2001)), because (i) the released species is already established in the target area (since it is the target species) and (ii) sterile insects are mass released. In inundation or classical biological control, the main non-target effects concern the risk that the introduced biocontrol agents attack other species than the target pest (Lynch and Thomas, 2000). SIT is based on intra-specific competition between the released and the wild males and is thus highly specific to the target species (Abram et al., 2021). Residual fertility might thus lead to two different non-target effect: (i) when SIT is used preventively against a species not present in the target area, it might promote its earlier establishment (Kapranas et al., 2022), and (ii) it might allow gene flow between introduced individuals and wild populations. To our knowledge, no gene flows have been observed (or studied) following the release of sterile insects but further studies might be necessary for quantitative risk assessment (David et al., 2013; Kapranas et al., 2022).

The model's bistability cases show that with low release rates, it is possible to control a population quickly if the initial conditions are not too high, i.e. below the unstable equilibrium. This is why SIT is particularly effective at low population densities. In order to achieve effective population control, the idea is either to start releases particularly early, before the strong invasion phase of the pest, or to combine with other control methods to reduce densities beforehand. The most obvious method is prophylaxis, i.e. all measures designed to prevent the arrival of the pest in the crop or greatly reduce its outbreak, thus limiting the size of the pest population even before the implementation of SIT (Benedict, 2021). For fruit flies such as *C. capitata*, crushing fallen fruits and ploughing the soil over winter to expose pupae to moisture, frost and predators can be effective prophylactic measures. SIT is often part of area-wide integrated pest management programs, in which the technique is used in combination with pesticides (Vreysen et al., 2006). Indeed, pesticides can help to reduce population densities before the start of releases, but they have a detrimental effect on sterilized males released, problems of resistance in *C. capitata* have been reported (Magaña et al., 2007) and the current aim is to reduce their use. So, to eliminate a part of pests without using pesticides, SIT can be considered in combination with trapping. However, this option is subject to debate. Indeed, some authors claim that for *C. capitata*, mass trapping in a limited area is compatible with SIT without requiring an increase in release effort (Duarte et al., 2022). On the one hand, if the traps preferentially attract males (Leza et al., 2008), whether wild or
<figure><img src="image_6.png" /><figcaption>**Fig. 5** Stability zones arising from different values of residual fertility δ or ϵ and the release rate σ for A: the cost-free residual fertility submodel (5) and B: the costly residual fertility submodel (6). The lower figures are close-ups of the upper figures and show the tiny pink area corresponding to Shape B'. Three zones are defined by the black curve. The bistability zone is where both the pest-free and the infestation equilibria are stable. In the other zones, only one of the two is stable. The colored areas are associated with the different shapes of bifurcation diagrams depicted in Fig. 3. The scale used here does not allow shape D' to be depicted.</figcaption></figure>

rates when there are fitness costs than when there are not (Fig. 5). In terms of the potential for population control, the population can be reduced when residual fertility rates are higher if there are fitness costs (Fig. 6). However, in both cases, when residual fertility rates are too high, population density cannot be controlled, no matter the value of the release rate σ. Similarly, for low σ values, population control cannot be achieved regardless of the residual fertility rate.

## Residual fertility thresholds for population eradication

Shape A' (Fig. 3) is nearly the sole case in which only the pest-free equilibrium is stable (given the negligible presence of shape B'), provided that the release rate is high enough (Fig. 5, purple zones). This shape is obtained when the asymptote $F^*$ defined in (14) is negative, resulting in the following residual fertility thresholds for the submodel without fitness costs (5):

$$ \delta < \frac{\eta\mu_M}{(R_F - 1)\mu_S + \eta\mu_M} = 0.001. $$
sterile, they will be trapped, thus maintaining the proportion of wild and sterile males and preserving SIT effectiveness. On the other hand, it seems ineffective to release sterilized males doomed to be captured. So, for effective pest population control, theoretical study of this option may represent a real challenge, in order to have a prior idea of the effectiveness associated with the trap deployment effort to be envisaged.

# 5 Data accessibility and reproductibility

All analyses were performed using the Python programming language (version 3.11.0). The complete script coded in Python, specifying the versions of the libraries used, is available on the following link: https://gitlab.com/marinecourtoism2/sit_residual_fertility.

# A Existence of equilibria

The infestation equilibria are the solutions of $G(F) = X(M(F))C(F) = \frac{1}{\mathcal{R}_F}$ (12) for the female density, from which the male density is deduced using (11). Equilibria thus correspond to intersections between the curves $X(M(F))$ and $\frac{1}{\mathcal{R}_F C(F)}$. The first being concave and the second convex, there can be 2, 1 or 0 infestation equilibria.

In order to study the derivative of $G(F)$ (12) we first note that:

$$ \frac{d}{dF} \left[ \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) C(F) \right] = \frac{d}{dF} (X(M(F))C(F)). $$

Developing the derivative in the square brackets, we have:

$$ \frac{d}{dF} \left[ \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) C(F) \right] = \frac{d}{dF} \left( X(M(F)) - \left( \frac{1}{\mathcal{R}_F C(F)} \right) C(F) \right) + \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) \frac{d}{dF} C(F). $$

Merging the two equations and noting that at equilibrium $X(M(F^*)) = \frac{1}{\mathcal{R}_F C(F^*)}$, we obtain:

$$ \frac{d}{dF} (X(M(F)C(F))) = \left( \frac{d}{dF} (X(M(F))) - \frac{d}{dF} \left( \frac{1}{\mathcal{R}_F C(F)} \right) \right) C(F). $$

As $C(F) > 0$, the sign of the derivative of $X(M(F))C(F)$ depends on the sign of the difference between the derivatives of $X(M(F))$ and $\frac{1}{\mathcal{R}_F C(F)}$. The difference is non-negative for the first equilibrium called $F_1^*$ (when it exists) as $X(M(F)) < \frac{1}{\mathcal{R}_F C(F)}$ for $F < F_1^*$, the reverse is true for $F_2^*$ (Fig. A.1). From the concavity and convexity conditions, this non-negativity, respectively non-positivity, turns positive, respectively negative (Fig. A.1).

Thus, when $\frac{d}{dF} X(M(F)) > \frac{d}{dF} \left( \frac{1}{\mathcal{R}_F C(F)} \right)$, we have $\frac{dG}{dF}(F^*) > 0$ as $X(M(F)C(F) = G(F)$, it corresponds to $F_1^*$; and when $\frac{d}{dF} X(M(F)) < \frac{d}{dF} \left( \frac{1}{\mathcal{R}_F C(F)} \right)$, we have $\frac{dG}{dF}(F^*) < 0$, it corresponds to $F_2^*$.

To sum up:

(i) If the curves intersect at two positive points (Fig. A.1A): there are two solutions $F_2^* > F_1^* > 0$, such that $\frac{dG}{dF}(F_1^*) > 0$ and $\frac{dG}{dF}(F_2^*) < 0$. We then have two infestation equilibria ($F_1^*, M_1^*$) and ($F_2^*, M_2^*$).

(ii) If the curves intersect at one positive point only (Fig. A.1B): there is only one solution, at which $\frac{dG}{dF} < 0$. This solution is named $F_2^*$ and corresponds to the infestation equilibrium ($F_2^*, M_2^*$).

(iii) If there is no intersection between the curves: there is no solution, so there is no infestation equilibrium.

# B Determinant computation

In this section, the calculation of the determinant of the Jacobian matrix defined in section 2.2.3 is detailed. As a reminder, the Jacobian matrix $J_{(F^*, M^*)}$ is expressed as follows:

$$ J_{(F^*, M^*)} = \begin{pmatrix} -r(1-p)X(M^*)C(F^*) + r(1-p)X(M^*)(C'(F^*)F^* + C(F^*)) & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\mu_M + rpC(F^*)X'(M^*)F^* \end{pmatrix}. $$
<figure><img src="image_7.png" /><figcaption>Fig. 6 Contour plot illustrating the percent reduction in the population (1 − q ) as a function of the release rate σ and the residual fertility rate, where q is defined in (16). A: results for the cost-free residual fertility submodel (5) and B: results for the costly residual fertility submodel (6). The lower graphs are a close-up of the blue boxes in the upper graphs. The gray curve visible on these lower graphs is the limit between a 90% reduction and a 100% reduction (eradication) of the population.</figcaption></figure>

The result is as follows for the submodel with fitness costs (6):

$$
\epsilon < \frac{1}{\mathcal{R}_F} = 0.004. \tag{15}
$$

The latter generalizes the condition obtained by Aronna and Dumont (2020) for k = 0.

Both thresholds are very small (less than 0.5%) and seem impossible to achieve in practice. Therefore, a more realistic goal would be to drastically reduce rather than eradicate the population.

Residual fertility thresholds for population control

When eradication is not possible regardless of the release rate, there must be a significant reduction in population density for SIT to be considered effective. This sit- uation corresponds to shape C ′ (Fig. 3), in which the infestation equilibrium decreases towards the asymptote F ∗ as the release rate increases. Thus, the maximum resid- ual fertility rate allowing a (1 − q ) reduction in female density (compared to the case
<figure><img src="image_11.png" /><figcaption>Figure A.1: Female density values at the infestation equilibria of the model (1). They correspond to the intersections between concave curve ($X(M(F))$) (grey solid line) and convex curve $\frac{1}{R_F C(F)}$ (grey dotted line). Curves were sketched by hand. In case A, there are two infestation equilibria $F_1^*$ (green point) and $F_2^*$ (blue point). In case B, there is only one infestation equilibrium $F_2^*$ (blue point).</figcaption></figure>

The determinant is calculated as follows:

$$ 
\begin{aligned}
Det(J(F^*, M^*)) &= r(1-p)X(M^*)C'(F^*)F^*(-\mu_M + rpC(F^*)X'(M^*)F^*) \\
&\quad - r(1-p)C(F^*)X'(M^*)F^*rpX(M^*)(C'(F^*)F^* + C(F^*)), \\
&= r(1-p)F^*[X(M^*)C'(F^*(-\mu_M + rpC(F^*)X'(M^*)F^*) \\
&\quad - C(F^*)X'(M^*)rpX(M^*)(C'(F^*)F^* + C(F^*))], \\
&= r(1-p)F^*(-\mu_M X(M^*)C'(F^*) - C(F^*)X'(M^*)rpX(M^*)C(F^*)), \\
&= r(1-p)F^*[-\mu_M X(M^*)C'(F^*) - C(F^*)X'(M^*)\frac{p\mu_F}{1-p}],
\end{aligned}
$$

as $X(M^*)C(F^*) = \frac{\mu_F}{r(1-p)}$ (10). In addition, according to (11), $M'(F^*) = \frac{p\mu_F}{(1-p)\mu_M}$, so the determinant can be written:

$$ 
\begin{aligned}
Det(J(F^*, M^*)) &= -r(1-p)\mu_M F^* \left[ (X(M^*)C'(F^*) + \frac{p\mu_F}{(1-p)\mu_M})C(F^*)X'(M^*) \right], \\
&= -r(1-p)\mu_M F^* [X(M^*)C'(F^*) + M'(F^*)C(F^*)X'(M^*)], \\
&= -r(1-p)\mu_M F^* \frac{dG}{dF}(F^*),
\end{aligned}
$$

as $X(M^*)C'(F^*) + X'(M^*)M'(F^*)C(F^*) = \frac{dG}{dF}(F^*)$.

Thus, the sign of the determinant is equal to the sign of the slope of $G(F)$ at the equilibrium.

## C $\sigma(F^*)$ expression

In this section, the calculations leading to the explicit expression of $\sigma(F^*)$ (13) are detailed. In equation (12), we substitute $X(S, M, \epsilon)$ by explicit expression (2) and obtain:

$$ (12) \iff \frac{M(F^*)+\epsilon\eta S^*}{k+M(F^*)+\eta S^*} \cdot C(F^*) = \frac{1}{R_F}. $$
where $\sigma = 0$) is determined by solving

$$F_{-}^{*} = qF_{B}^{*}, \qquad (16)$$

For instance, a 90% population reduction in female density would corresponds to $q = 0.1$.

For the cost-free residual fertility submodel (5), we obtain

$$\delta = \frac{(q\beta F_{B}^{*} + 1)\mu_{M}\eta}{\mathcal{R}_{F}\mu_{S} + (q\beta F_{B}^{*} + 1)(\mu_{M}\eta - \mu_{S})}.$$

For a 90% reduction in female density, this threshold is equal to 0.014.

Finally, for the costly residual fertility submodel (6), we obtain the following threshold for tolerable residual fertility:

$$\epsilon = \frac{1}{\mathcal{R}_{F}} + \frac{qF_{B}^{*}\beta}{\mathcal{R}_{F}}. \qquad (17)$$

The threshold for population control is thus $\frac{qF_{B}^{*}\beta}{\mathcal{R}_{F}}$ higher than the threshold required for population eradication, it is equal to 0.103.

For the particular case where $k = 0$, we obtain:

$$\epsilon = \frac{1}{\mathcal{R}_{F}} + q(1 - \frac{1}{\mathcal{R}_{F}}), \qquad (18)$$

Thus, for a 90% reduction in female density, the threshold is equal to 0.104.

### Convergence time

To optimize SIT effectiveness, it is essential to consider not only the residual fertility rate, but also the release rate $\sigma$. In Fig. 6, for a given residual fertility rate, a range of $\sigma$ values will lead to the same range of population reduction values. However, $\sigma$ has an influence on the convergence time needed to reach a given range in population reduction values: higher $\sigma$ values allow for faster population control. For example, when the release rate $\sigma$ is ten times higher, population eradication or reduction occurs much more quickly, for all the submodels (Fig. 7). Starting from the uncontrolled infestation equilibrium, the eradication/reduction equilibrium is reached within 75 days at the higher release rate. It takes between 200 and 300 days at the lower rate.

## 3.2 Sensitivity analysis

A global sensitivity analysis was performed to determine the parameters with the greatest influence on the percent reduction in female density $(1-q)$, where $q$ is defined in (16). The goal was to study how to improve pest population control.

We used a variance-based method (Sobol, 1990; Wu et al., 2013). For each parameter, three values were tested: the reference value and the minimum and maximum values of the range specified in Table 1. A full factorial design was used to explore
524 Substituting $S^* = \frac{(1-\delta)\sigma}{\mu_S}$ and $M(F^*) = \frac{p\mu_F}{(1-p)\mu_M}F^* + \frac{\delta\sigma}{\mu_M}$ from equation (11), we then obtain:

$$ \iff \frac{\frac{p\mu_F}{(1-p)\mu_M}F^* + \frac{\delta\sigma}{\mu_M} + \frac{\epsilon\eta(1-\delta)\sigma}{\mu_S}}{k + \frac{p\mu_F}{(1-p)\mu_M}F^* + \frac{\delta\sigma}{\mu_M} + \eta\frac{(1-\delta)\sigma}{\mu_S}} \cdot C(F^*) = \frac{1}{\mathcal{R}_F}, $$

$$ \iff \frac{\mathcal{R}_F p \mu_F}{(1-p)\mu_M} F^* + \frac{\mathcal{R}_F \delta \sigma}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta) \sigma}{\mu_S} = \frac{1}{C(F^*)} \left( k + \frac{p \mu_F}{(1-p) \mu_M} F^* + \frac{\delta \sigma}{\mu_M} + \eta \frac{(1-\delta) \sigma}{\mu_S} \right), $$

$$ \iff \frac{\mathcal{R}_F \delta \sigma}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta) \sigma}{\mu_S} - \frac{1}{C(F^*)} \frac{\delta \sigma}{\mu_M} - \frac{1}{C(F^*)} \left( \eta \frac{(1-\delta) \sigma}{\mu_S} \right) = \frac{1}{C(F^*)} \left( k + \frac{p \mu_F}{(1-p) \mu_M} F^* \right) - \frac{\mathcal{R}_F p \mu_F}{(1-p) \mu_M} F^*, $$

$$ \iff \sigma \left( \frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \frac{1}{C(F^*)} \frac{\delta}{\mu_M} - \frac{1}{C(F^*)} \left( \eta \frac{(1-\delta)}{\mu_S} \right) \right) = \frac{1}{C(F^*)} \left( k + \frac{p \mu_F}{(1-p) \mu_M} F^* \right) - \frac{\mathcal{R}_F p \mu_F}{(1-p) \mu_M} F^*, $$

$$ \iff \sigma(F^*) = \frac{\left(\frac{1}{C(F^*)}\right)\left(k + \frac{p\mu_F}{(1-p)\mu_M}F^*\right) - \frac{\mathcal{R}_F p\mu_F}{(1-p)\mu_M}F^*}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \frac{1}{C(F^*)} \frac{\delta}{\mu_M} - \frac{1}{C(F^*)} \left(\eta \frac{(1-\delta)}{\mu_S}\right)}. $$

525 By explicit $C(F^*)$ using expression (3), we have:

$$ \begin{aligned} \sigma(F^*) &= \frac{(1+\beta F^*) \left( k + \frac{p\mu_F}{(1-p)\mu_M} F^* \right) - \frac{\mathcal{R}_F p \mu_F}{(1-p)\mu_M} F^*}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - (1+\beta F^*) \frac{\delta}{\mu_M} - (1+\beta F^*) \left( \eta \frac{(1-\delta)}{\mu_S} \right)}, \\ &= \frac{\frac{\beta p \mu_F}{(1-p)\mu_M} (F^*)^2 + \left( \frac{p \mu_F (1-\mathcal{R}_F)}{(1-p)\mu_M} + \beta k \right) F^* + k}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \frac{\delta(1+\beta F^*)}{\mu_M} - \frac{\eta(1-\delta)(1+\beta F^*)}{\mu_S}}. \end{aligned} $$

526 This expression corresponds to the equation (13) in section 2.2.4. By replacing $C(F^*)$ in the same way with the expressions proposed in Section 2.1.1 in $\sigma(F^*)$ (13), we obtain bifurcation diagrams of a shape similar to those described in Section 2.2.4.

529 D $\sigma(F^*)$ numerator study

530 The numerator of $\sigma(F^*)$ is expressed as follows:

$$ \frac{\beta p \mu_F}{(1-p)\mu_M} (F^*)^2 + \left( \frac{p \mu_F (1-\mathcal{R}_F)}{(1-p)\mu_M} + \beta k \right) F^* + k, $$

531 which can be rewritten as:

$$ \frac{k}{\mathcal{R}_F} \left( \mathcal{R}_M (\beta F^*)^2 + (\mathcal{R}_M + \mathcal{R}_F - \mathcal{R}_M \mathcal{R}_F) (\beta F^*) + \mathcal{R}_F \right). $$

532 It does not depend on residual fertility rates, unlike the denominator. The discriminant of the $\beta F^*$ second order polynomial is:

$$ \begin{aligned} \Delta &= (\mathcal{R}_M + \mathcal{R}_F - \mathcal{R}_M \mathcal{R}_F)^2 - 4 \mathcal{R}_F \mathcal{R}_M, \\ &= \mathcal{R}_M^2 + \mathcal{R}_F^2 + \mathcal{R}_F^2 \mathcal{R}_M^2 + 2 \mathcal{R}_F \mathcal{R}_M - 2 (\mathcal{R}_M + \mathcal{R}_F) \mathcal{R}_M \mathcal{R}_F - 4 \mathcal{R}_F \mathcal{R}_M. \end{aligned} $$

534 Thus the discriminant is positive if and only if:

$$ (\mathcal{R}_M - \mathcal{R}_F)^2 + \mathcal{R}_M^2 \mathcal{R}_F^2 > 2 \mathcal{R}_M \mathcal{R}_F (\mathcal{R}_M + \mathcal{R}_F). \qquad (\text{D.19}) $$

535 Dividing by $\mathcal{R}_M^2 \mathcal{R}_F^2$ yields:

$$ \left( \frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M} \right)^2 + 1 > 2 \left( \frac{1}{\mathcal{R}_F} + \frac{1}{\mathcal{R}_M} \right). $$

536 Taking the inverse and multiplying by 4 yields:

$$ \frac{2}{\frac{1}{\mathcal{R}_F} + \frac{1}{\mathcal{R}_M}} > \frac{4}{\left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1}, \qquad (\text{D.20}) $$

537 that is a lower-bound to the harmonic mean of $\mathcal{R}_F$ and $\mathcal{R}_M$. Since $\mathcal{R}_F, \mathcal{R}_M > 1$, we have $0 < \frac{1}{\mathcal{R}_F}, \frac{1}{\mathcal{R}_M} < 1$, so that $-1 < \frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M} < 1$ and $1 \le \left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1 < 2$. It follows that $2 < \frac{4}{\left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1} \le 4$ so that (D.20) imposes that the harmonic mean is larger than 2, and is necessarily satisfied when the harmonic mean is larger than 4 (Fig. D.2).
<figure><img src="image_8.png" /><figcaption>**Fig. 7** Simulations of female density over time ($F$). Four cases are considered: no SIT (black curve), SIT without residual fertility (blue curves), SIT with 1% cost-free residual fertility (orange curves), and SIT with 1% costly residual fertility (purple curves). The dashed curves were obtained using the release rate $\sigma = 500 \text{ ind.ha}^{-1}\text{day}^{-1}$, and the solid curves were obtained using the release rate $\sigma = 5000 \text{ ind.ha}^{-1}\text{day}^{-1}$.</figcaption></figure>

the parameter space. An ANOVA was then performed to obtain the decomposition of variance; we employed a linear model with two-way interactions between the output and the parameters. Finally, Sobol sensitivity indices (SIs) and total sensitivity indices (TSIs) were calculated for each parameter as follows: the SI was the ratio between the sum of squares of the parameter's main effect and the total sum of squares; the TSI was the ratio between the sum of squares of the parameter main effect plus its interactions and the total sum of squares.

The parameter with the greatest influence on the percent reduction in female density is the female mortality rate $\mu_F$; its TSI is greater than 50% (Fig. 8). Two parameters related to species biology, female competition for oviposition sites $\beta$ and the emergence rate $r$, also have a significant influence. Finally, parameters directly related to SIT -the residual fertility rate ($\delta$ or $\epsilon$), the release rate $\sigma$, and the cost of sterilization $\eta$- also have a major influence. The other parameters, namely the mortality rates of wild males and sterilized males ($\mu_M$ and $\mu_S$, respectively) and the mating half-saturation constant $k$, have a negligible influence compared to the others.

# 4 Discussion

We investigated the impact of residual fertility on the efficiency of releasing sterilized males as part of SIT deployment. As hypothesized, we found that the relative degree of sterilization has a significant impact on SIT effectiveness. In addition, when there is a fitness cost associated with irradiation, pest control can be achieved even at higher rates of residual fertility. Our findings highlight the impacts of residual fertility and
<figure><img src="image_12.png" /><figcaption>Figure D.2: Contour plot illustrating the existence of $\sigma(F^*)$ numerator roots as a function of the reproduction numbers $\mathcal{R}_M$ and $\mathcal{R}_F$. Three zones are defined: negative real roots (dark blue zone), no real roots (blue-green zone) and positive real roots (light green zone). The two curves represent the harmonic mean of $\mathcal{R}_M$ and $\mathcal{R}_F$ equal to 2 (dotted black curve) and equal to 4 (black curve).</figcaption></figure>

In this case there are two roots. The product of the roots has the sign of $\frac{\mathcal{R}_F}{\mathcal{R}_M}$, which is positive. So the roots have the same sign.

The sum of the roots is:

$$ -\frac{\mathcal{R}_M + \mathcal{R}_F - \mathcal{R}_M \mathcal{R}_F}{\mathcal{R}_M} \qquad (D.21) $$

Thus the sum of the roots is positive and therefore both roots are positive if and only if:

$$ \begin{align*} \mathcal{R}_F + \mathcal{R}_M < \mathcal{R}_M \mathcal{R}_F &\iff \frac{1}{\mathcal{R}_M} + \frac{1}{\mathcal{R}_F} < 1, \\ &\iff \frac{2}{\frac{1}{\mathcal{R}_M} + \frac{1}{\mathcal{R}_F}} > 2. \end{align*} $$

Hence, the harmonic mean of $\mathcal{R}_F$ and $\mathcal{R}_M$ must be larger than 2, which we have shown to be a consequence of equation (D.20). Hence, (D.19) or (D.20) suffice for the numerator to have two positive roots if $\mathcal{R}_F, \mathcal{R}_M > 1$ (Fig. D.2).

# E Parameter values

## E.1 Mortality rates

Different values for *Ceratitis capitata* lifespans were collected from the literature. The mortality rate was estimated as the inverse of the lifespan. Male and females lifespans and corresponding mortality rates are summarized in Table E.1.
<figure><img src="image_9.png" /><figcaption>**Fig. 8** Total sensitivity index per parameter, separated into main effect (black bar, SI) and two-way interactions (gray bar). Sensitivity analysis performed to examine the parameters that most influenced reductions in female density. A: results for the cost-free residual fertility submodel (5) and B: results for the costly residual fertility submodel (6).</figcaption></figure>

emphasize the need to better understand the sterilization-and-release process. Indeed, irradiated males that remain fertile are a greater threat to SIT effectiveness if they have the same phenotypic and behavioral traits as wild males (as in the cost-free residual fertility submodel) than if they are more similar to sterilized males (as in the costly residual fertility submodel). Our work emphasizes the capital importance of clarifying the mechanisms that underpin residual fertility and of understanding how males escape sterilization, including the impacts of the different SIT steps (mass breeding, irradiation, transport, and release). Do irradiated males become less fit while still possessing some viable sperm? Or do some males escape the sterility-inducing gamma rays and retain all their fitness? In short, our results illustrate the importance of considering several factors: the release rate of irradiated males required to reduce population density in a timely manner as well as the residual fertility rate and any associated fitness costs.

In our modeling work, irradiated males experienced fitness costs at two levels: first, they had a higher mortality rate than did wild males and, second, they had a lower mating rate (parameter η). The latter can result from reduced attractiveness, flight capacity, and dispersal ability (Guerfali et al., 2011) or from lower competitiveness (Dyck et al., 2005). The dose of radiation needed to induce total sterility in *C. capitata* causes a significant drop in male performance (Robinson et al., 2002), so SIT programs rarely aim for 100% sterility (Parker and Mehta, 2007). They also often utilize aromatherapy: ginger root oil is known to improve the attractiveness of sterilized males (Morelli et al., 2013). Indeed, aromatherapy has been confirmed to play an essential role in increasing SIT success with *C. capitata* (Dumont and Oliva, 2023). In this species, the attractiveness of sterilized males is crucial in reproduction: males gather in leks (Whittier et al., 1992), to court, attract, and finally mate with females (Shelly and McInnis, 2016). According to our sensitivity analysis, the fitness
<table><caption>Table E.1: *Ceratitis capitata* male and female lifespans based on the literature, with corresponding mortality rates.</caption><thead><tr><th>References</th><th>Female lifespan (days)</th><th>Male lifespan (days)</th><th>μ<sub>F</sub> (day<sup>-1</sup>)</th><th>μ<sub>M</sub> (day<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Shoukry and Hafez (1979)</td><td>[25.5-31]</td><td>[25-36.5]</td><td>[0.032-0.039]</td><td>[0.027-0.040]</td></tr><tr><td>Carey (1984)</td><td>[39-46.4]</td><td>[40.5-47.9]</td><td>[0.022-0.026]</td><td>[0.021-0.025]</td></tr><tr><td>Vargas et al. (2000)</td><td>54.8</td><td>71.6</td><td>0.018</td><td>0.014</td></tr><tr><td>Papadopoulos et al. (2002)</td><td>21.1</td><td>29.5</td><td>0.047</td><td>0.034</td></tr><tr><td>Pieterse et al. (2020)</td><td>[12-17.5]<sup>1</sup></td><td>[17.6-35.5]<sup>2</sup></td><td>[0.057-0.083]<sup>1</sup></td><td>[0.036-0.057]<sup>2</sup></td></tr></tbody></table>

## E.2 Emergence rate

<table><caption>Table E.2: *C. capitata* development data extracted from Pieterse *et al.* (2020)</caption><thead><tr><th>Fruit type</th><th>Eggs per females (24h)</th><th>Egg hatch in fruit (24h)</th><th>Pupal eclosion (24h)</th></tr></thead><tbody><tr><td>Nectarine</td><td>25.75 [22.25-29.25]</td><td>97% [96%-98%]</td><td>94.7% [89.9%-99.5%]</td></tr></tbody></table>

## E.3 Competition coefficient between females

The competition coefficient between females $\beta$ represents oviposition competition. This parameter was estimated from the model without control by SIT at equilibrium. From equation (12), which allows us to find infestation equilibria of the model, we isolated $\beta$:

$$X(M(F))C(F) = \frac{1}{\mathcal{R}_F},$$

$$\frac{\frac{p\mu_F}{(1-p)\mu_M} F}{k + \frac{p\mu_F}{(1-p)\mu_M} F} \frac{1}{1 + \beta F} = \frac{1}{\mathcal{R}_F},$$

$$1 + \beta F = \frac{\mathcal{R}_F \frac{p\mu_F}{(1-p)\mu_M} F}{k + \frac{p\mu_F}{(1-p)\mu_M} F},$$

$$\beta = \frac{1}{F} \left( \frac{\mathcal{R}_F \frac{p\mu_F}{(1-p)\mu_M} F}{k + \frac{p\mu_F}{(1-p)\mu_M} F} - 1 \right).$$

We based our estimate of females density $F$ on several mass trapping studies (Demirel and Akyol, 2017; Hafsi et al., 2020b; Leza et al., 2008; Martinez-Ferrer et al., 2012), in which about 1,000 females per day and per hectare were captured, assuming a balanced sex-ratio. We assumed that these traps captured most individuals and gave an approximation of *C. capitata* equilibrium density without SIT. Thus, taking $F = 1000 \text{ ind.ha}^{-1}$ and the parameter values listed in the Table 1, we obtain $\beta = 0.24 (\text{ind.ha}^{-1})^{-1}$.

<footnote n="1">¹ Data extracted on nectarine and plum to collect extremum for females.</footnote>

<footnote n="2">² Data extracted on apple and plum to collect extremum for males.</footnote>
costs associated with irradiation ($\eta$) had a significant influence on reductions in pop-
ulation density, which highlights the importance of determining the traits that are
affected by the sterilization process and of quantifying their impacts on male mating
capacity. In contrast, there was no major influence of the mortality rate of sterilized
males, which is linked to the radiation dose. This point illustrates the complexity of
the trade-off between the radiation dose and fitness and, subsequently, the compet-
itiveness of irradiated males. Other authors have addressed this trade-off and have
suggested that SIT effectiveness could be optimized by using lower doses than those
applied to achieve high sterilization rates, noting that any increase in residual fertility
is more than offset by the increased competitiveness of the released insects (Parker and
Mehta, 2007). In our study, the sensitivity analysis showed that the residual fertility
rate ($\delta$ or $\epsilon$) has a similar, if not slightly greater, impact on SIT effectiveness relative
to the cost of sterilization (1 - $\eta$), which suggests that increased residual fertility rates
could be mitigated by the fact that irradiated males maintain competitiveness. Other
species-specific parameters played a significant role in reducing population density:
they were either linked to density-dependence processes (e.g., competition strength)
or population growth (e.g., fertility and emergence rate). These results underscore
the importance of having both a qualitative and quantitative understanding of the
demographic processes driving pest population dynamics.

In the agricultural settings, the objective of SIT programs is to reduce fruit dam-
age and, therefore, pest population levels, below an acceptable threshold for economic
damage (Dyck et al., 2021). This study confirms that SIT effectiveness largely depends
on the residual fertility rate (Dyck et al., 2005; Parker and Mehta, 2007; Dumont and
Oliva, 2023): the lower the residual fertility rate, the more effective the level of control.
As highlighted in the introduction, most theoretical studies examine SIT effectiveness
from an eradication perspective and thus focus on reaching a stable pest-free equi-
librium (Bliman et al., 2019; Aronna and Dumont, 2020). Similarly to Aronna and
Dumont (2020), we found that, in the costly residual fertility submodel, the thresh-
old allowing population eradication is $1/R_F$ (15), which is less than 0.5%. It is even
lower for the cost-free residual fertility submodel. Hence, population eradication is
only possible for very low residual fertility rates. However, even if eradication is theo-
retically impossible, the same is not true for population control. Indeed, in the South
African SIT program targeting *C. capitata*, the objective was to drive medfly popula-
tions below an economic threshold and thus create an internationally recognized area
of low pest prevalence (Reyes and Hendrichs, 2017). Such can be achieved when the
pest-free equilibrium is stable, as well as when the pest-free equilibrium is unstable
but the equilibrium associated with low pest density is stable. Thus, we show that
when the residual fertility threshold $\frac{1}{R_F} + \frac{qF_F^*\beta}{R_F}$ (17) is higher than the eradication
threshold, it is theoretically possible to control the population. We found that, up to
this threshold equal to 10% of residual fertility with the biological parameters used,
the population can be reduced by $1 - q = 90\%$ of the equilibrium associated with
uncontrolled pest density. This 90% reduction would indeed be useful since it would
bring the number of males caught (in 50 traps across 1 hectare) from 982, the uncon-
trolled density at equilibrium, down to 98; this figure is below the economic threshold
of approximately 3 males per trap per day for *C. capitata* (Hafsi et al., 2020). Thus,
Funding sources

568

The authors acknowledge the financial support of the CeraTIS - Corse project, within the framework of ECOPHYTO 2019, "Leviers territoriaux pour réduire l'utilisation et les risques liés aux produits phytopharmaceutiques (2020 - 2023)". M. Courtois's PhD is funded by INRAE.

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efficiently deploying SIT does not necessarily require the eradication of the species
from the area of interest because the persistence of a low-density population will result
in negligible economic losses.

For a residual fertility threshold of 1%, which is the typical threshold for *C. capitata*, the IAEA recommends releasing about 4,000 individuals per week per hectare (Reyes and Hendrichs, 2017). Our results suggest that, if residual fertility is 1% and associated with fitness costs, releasing around 400 sterilized males per day per hectare would be sufficient to control the population (i.e., to obtain at least a 90% reduction; in Fig. 6). This figure corresponds to 2,800 males per week per hectare. Consequently, our model is able to estimate release rates of the same order of magnitude as those recommended for *C. capitata*. According to our model, if residual fertility is 1% and unassociated with fitness costs, it would be necessary to release more than 8,000 males per week per hectare to achieve a 90% reduction in the population. This figure is more than twice what the IAEA recommends. Therefore, release recommendations should account for both residual fertility and the fitness of irradiated males.

Here, we explored questions around residual fertility, which means considering
the possibility of introducing fertile irradiated males into the environment, a situa-
tion that raises ethical and safety issues (Oliva et al., 2021). Given current IAEA
recommendations, the risk that residual fertility might compromise pest population
control efforts is fairly low. However, there may be concerns around introducing non-
endemic medfly strains into a wild population (Lynch and Thomas, 2000; Benedict
et al., 2018). Indeed, fertile irradiated males can reproduce and transfer their genes,
potentially introducing new alleles and new phenotypes into the target population.
Moreover, the strains used for SIT releases might be different from wild populations
in many regards. First, strains may be genetically selected for specific characteristics,
such as traits that facilitate genetic sexing (Robinson, 2002; Ramírez-Santos et al.,
2016). Second, mass rearing across many generations has probably led to adaptations
to laboratory conditions (Hoffmann and Ross, 2018). Third, exposure to radiation can
lead to numerous unknown mutations. Residual fertility is thus a concern in SIT pro-
grams in the sense that it makes self-replication possible, such that sterilized insects
are comparable to other biocontrol agents (Kapranas et al., 2022). Indeed, crosses
between introduced and local biocontrol agents are possible and can generate hybrids
with non-controlled traits that potentially increase the risk of invasion (Turgeon et al.,
2011). SIT is generally comparable to inundation biological control (mass introduction
of macroorganisms; (Eilenberg et al., 2001)) because (i) the released species is already
established in the target area (i.e., it is the target species) and (ii) sterilized insects
are mass released. In inundation or classical biological control, the main non-target
effect of concern is that the introduced biocontrol agents could attack species other
than the target pest (Lynch and Thomas, 2000). SIT is exploits intra-specific competi-
tion between sterilized and wild males and is thus highly specific to the target species
(Abram et al., 2021). Residual fertility might thus lead to two different non-target
effects: (i) when SIT is used preventively against a species that is not present in the
target area, it might promote that species' establishment (Kapranas et al., 2022) and
(ii) it might allow gene flow between released individuals and wild individuals. To our
knowledge, no gene flows has been observed following the release of sterilized insects,
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but this question has also not been studied. Further research might be necessary for
quantitative risk assessment (David et al., 2013; Kapranas et al., 2022).

The model’s cases of bistability show that, with low release rates, it is possible to quickly control a population if its initial density is not too high (i.e., below the unstable equilibrium). This is why SIT is particularly effective at low population densities. In order to achieve effective population control, the idea is either to start releases early, before the pest’s strong invasion phase, or to use other control methods in tandem to reduce densities beforehand. The most obvious method is prophylaxis, namely all the measures designed to prevent a pest’s arrival on the crop or to greatly reduce its degree of outbreak, thus limiting the size of the pest population even before SIT is implemented (Benedict, 2021). For fruit flies such as *C. capitata*, crushing fallen fruits and plowing the soil over the winter to expose the pupae to moisture, frost, and predators can be effective prophylactic measures. SIT is often part of AW-IPM programs, where it is used in combination with pesticides (Vreysen et al., 2006). A concern is that cases of resistance in *C. capitata* have been reported (Magaña et al., 2007), and although helpful in reducing population densities beforehand pesticides negatively affect the released sterilized males. Thus, the current goal is to reduce pesticide use. Consequently, to control pests without using pesticides, SIT could be combined with trapping. However, this option is subject to debate. Indeed, some researchers claim that, for *C. capitata*, concomitant mass trapping within a limited area does not require an increase in release efforts (Duarte et al., 2022). More specifically, if the traps preferentially attract males (Leza et al., 2008), regardless of whether they are wild or sterilized, the proportion of wild and sterilized males will be maintained within the population, preserving SIT effectiveness. However, it also seems ineffective to release sterilized males that are doomed to be captured. In conclusion, identifying effective pest population control measures remains and important area of theoretical research. The results can provide valuable insights into the expected effectiveness of various trap deployment strategies.

## Data accessibility and reproductibility

All analyses were performed using the Python programming language (version 3.11.0). The complete Python script, specifying the library versions used, is available at the following link: https://gitlab.com/marinecourtoism2/sit_residual_fertility.

## Statements and Declarations

### Conflict of interest

The authors declare that they have no competing interests that are directly or indirectly related to this work.
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# Appendix A Existence of equilibria

The infestation equilibria are the solutions of $G(F) = X(M(F))C(F) = \frac{1}{\mathcal{R}_F}$ (12) for female density, from which male density is deduced using (11). The equilibria thus correspond to the intersections between the curves $X(M(F))$ and $\frac{1}{\mathcal{R}_F C(F)}$. The first is concave, and the second is convex; there can be 2, 1, or 0 infestation equilibria. To study the derivative of $G(F)$ (12), we first note that

$$ \frac{d}{dF} \left[ \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) C(F) \right] = \frac{d}{dF} (X(M(F))C(F)). $$

Developing the derivative in the parentheses, we arrive at:

$$ \frac{d}{dF} \left[ \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) C(F) \right] = \frac{d}{dF} \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) C(F) + \left( X(M(F)) - \frac{1}{\mathcal{R}_F C(F)} \right) \frac{d}{dF} C(F). $$

Merging the two equations and noting that at equilibrium $X(M(F^*)) = \frac{1}{\mathcal{R}_F C(F^*)}$, we obtain

$$ \frac{d}{dF} (X(M(F)C(F))) = \left( \frac{d}{dF} (X(M(F))) - \frac{d}{dF} \left( \frac{1}{\mathcal{R}_F C(F)} \right) \right) C(F). $$

As $C(F) > 0$, the sign of the derivative of $X(M(F))C(F)$ depends on the sign of the difference between the derivatives of $X(M(F))$ and $\frac{1}{\mathcal{R}_F C(F)}$. The difference is nonnegative for the first equilibrium, called $F_1^*$ (when it exists), as $X(M(F)) < \frac{1}{\mathcal{R}_F C(F)}$ for $F < F_1^*$; the reverse is true for $F_2^*$ (Fig. A1). Under conditions of concavity, this non-negativity turns positive, and under conditions of convexity, this non-positivity turns negative (Fig. A1).

Thus, when $\frac{d}{dF} X(M(F)) > \frac{d}{dF} \left( \frac{1}{\mathcal{R}_F C(F)} \right)$, we have $\frac{dG}{dF}(F^*) > 0$ as $X(M(F)C(F) = G(F)$, corresponding to $F_1^*$; and when $\frac{d}{dF} X(M(F)) < \frac{d}{dF} \left( \frac{1}{\mathcal{R}_F C(F)} \right)$, we have $\frac{dG}{dF}(F^*) < 0$, corresponding to $F_2^*$.

We can summarize as follows:

(i) If the curves intersect at two positive points (Fig. A1A): there are two solutions $F_2^* > F_1^* > 0$, such that $\frac{dG}{dF}(F_1^*) > 0$ and $\frac{dG}{dF}(F_2^*) < 0$. We then have two infestation equilibria $(F_1^*, M_1^*)$ and $(F_2^*, M_2^*)$.

(ii) If the curves intersect at single positive point (Fig. A1B), there is only one solution, such that $\frac{dG}{dF} < 0$. This solution is named $F_2^*$ and corresponds to the infestation equilibrium $(F_2^*, M_2^*)$.

(iii) If the curves do not intersect, there is no solution and thus no infestation equilibrium.

# Appendix B Calculating the determinant

In this section, we describe the calculations used to obtain the determinant of the Jacobian matrix in section 2.2.3. As a reminder, the Jacobian matrix $J_{(F^*, M^*)}$ is
<figure><img src="image_10.png" /><figcaption>**Fig. A1** Female density at the infestation equilibria of the model (1), which are the intersections between the concave curve ($X(M(F))$) (solid gray line) and the convex curve $\frac{1}{R_F C(F)}$ (dashed gray line). The curves were manually generated. In case A, there are two infestation equilibria $F_1^*$ (green point) and $F_2^*$ (blue point). In case B, there is only one infestation equilibrium: $F_2^*$ (blue point).</figcaption></figure>

expressed as follows:

$$J_{(F^*, M^*)} = \begin{pmatrix} -r(1-p)X(M^*)C(F^*) + r(1-p)X(M^*)(C'(F^*)F^* + C(F^*)) & r(1-p)C(F^*)X'(M^*)F^* \\ rpX(M^*)(C'(F^*)F^* + C(F^*)) & -\mu_M + rpC(F^*)X'(M^*)F^* \end{pmatrix}$$

The determinant is calculated as follows:

$$ \begin{aligned} Det(J(F^*, M^*)) &= r(1-p)X(M^*)C'(F^*)F^*(-\mu_M + rpC(F^*)X'(M^*)F^*) \\ &\quad - r(1-p)C(F^*)X'(M^*)F^*rpX(M^*)(C'(F^*)F^* + C(F^*)) \\ &= r(1-p)F^*[X(M^*)C'(F^*)(-\mu_M + rpC(F^*)X'(M^*)F^*) \\ &\quad - C(F^*)X'(M^*)rpX(M^*)(C'(F^*)F^* + C(F^*))], \\ &= r(1-p)F^*(-\mu_M X(M^*)C'(F^*) - C(F^*)X'(M^*)rpX(M^*)C(F^*)], \\ &= r(1-p)F^*[-\mu_M X(M^*)C'(F^*) - C(F^*)X'(M^*)\frac{p\mu_F}{(1-p)}], \end{aligned} $$

as $X(M^*)C(F^*) = \frac{\mu_F}{r(1-p)}$ (10). In addition, according to (11), $M'(F^*) = \frac{p\mu_F}{(1-p)\mu_M}$, so the determinant can be written as follows:

$$ \begin{aligned} Det(J(F^*, M^*)) &= -r(1-p)\mu_M F^* \left[ (X(M^*)C'(F^*) + \frac{p\mu_F}{(1-p)\mu_M})C(F^*)X'(M^*) \right], \\ &= -r(1-p)\mu_M F^* [X(M^*)C'(F^*) + M'(F^*)C(F^*)X'(M^*)], \\ &= -r(1-p)\mu_M F^* \frac{dG}{dF}(F^*), \end{aligned} $$

where $X(M^*)C'(F^*) + X'(M^*)M'(F^*)C(F^*) = \frac{dG}{dF}(F^*)$.

Thus, the sign of the determinant is equal to the sign of the slope of $G(F)$ at the equilibrium.
## Appendix C Expressing $\sigma(F^*)$

In this section, we describe the calculations leading to the explicit expression of $\sigma(F^*)$ (13). In equation (12), we substitute $X(S, M, \epsilon)$ via explicit expression (2) and obtain

$$ (12) \iff \frac{M(F^*)+\epsilon\eta S^*}{k+M(F^*)+\eta S^*} \cdot C(F^*) = \frac{1}{\mathcal{R}_F}. $$

Substituting $S^* = \frac{(1-\delta)\sigma}{\mu_S}$ and $M(F^*) = \frac{p\mu_F}{(1-p)\mu_M}F^* + \frac{\delta\sigma}{\mu_M}$ from equation (11), we then obtain

$$ \iff \frac{\frac{p\mu_F}{(1-p)\mu_M}F^* + \frac{\delta\sigma}{\mu_M} + \frac{\epsilon\eta(1-\delta)\sigma}{\mu_S}}{k + \frac{p\mu_F}{(1-p)\mu_M}F^* + \frac{\delta\sigma}{\mu_M} + \eta\frac{(1-\delta)\sigma}{\mu_S}} \cdot C(F^*) = \frac{1}{\mathcal{R}_F}, $$

$$ \iff \frac{\mathcal{R}_F p \mu_F}{(1-p)\mu_M} F^* + \frac{\mathcal{R}_F \delta \sigma}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta) \sigma}{\mu_S} = \frac{1}{C(F^*)} \left( k + \frac{p \mu_F}{(1-p) \mu_M} F^* + \frac{\delta \sigma}{\mu_M} + \eta \frac{(1-\delta) \sigma}{\mu_S} \right), $$

$$ \iff \frac{\mathcal{R}_F \delta \sigma}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta) \sigma}{\mu_S} - \frac{1}{C(F^*)} \frac{\delta \sigma}{\mu_M} - \frac{1}{C(F^*)} \left( \eta \frac{(1-\delta) \sigma}{\mu_S} \right) = \frac{1}{C(F^*)} \left( k + \frac{p \mu_F}{(1-p) \mu_M} F^* \right) - \frac{\mathcal{R}_F p \mu_F}{(1-p) \mu_M} F^*, $$

$$ \iff \sigma \left( \frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \frac{1}{C(F^*)} \frac{\delta}{\mu_M} - \frac{1}{C(F^*)} \left( \eta \frac{(1-\delta)}{\mu_S} \right) \right) = \frac{1}{C(F^*)} \left( k + \frac{p \mu_F}{(1-p) \mu_M} F^* \right) - \frac{\mathcal{R}_F p \mu_F}{(1-p) \mu_M} F^*, $$

$$ \iff \sigma(F^*) = \frac{\left( \frac{1}{C(F^*)} \right) \left( k + \frac{p\mu_F}{(1-p)\mu_M} F^* \right) - \frac{\mathcal{R}_F p\mu_F}{(1-p)\mu_M} F^*}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \frac{1}{C(F^*)} \frac{\delta}{\mu_M} - \frac{1}{C(F^*)} \left( \eta \frac{(1-\delta)}{\mu_S} \right)}. $$

By explicitly $C(F^*)$ using expression (3), we arrive at

$$ \sigma(F^*) = \frac{(1+\beta F^*) \left( k + \frac{p\mu_F}{(1-p)\mu_M} F^* \right) - \frac{\mathcal{R}_F p\mu_F}{(1-p)\mu_M} F^*}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - (1+\beta F^*) \frac{\delta}{\mu_M} - (1+\beta F^*) \left( \eta \frac{(1-\delta)}{\mu_S} \right)}, \\ = \frac{\frac{\beta p \mu_F}{(1-p)\mu_M} (F^*)^2 + \left( \frac{p\mu_F}{(1-p)\mu_M} \left( \frac{1-\mathcal{R}_F}{(1-p)\mu_M} \right) + \beta k \right) F^* + k}{\frac{\mathcal{R}_F \delta}{\mu_M} + \frac{\mathcal{R}_F \epsilon \eta (1-\delta)}{\mu_S} - \frac{\delta(1+\beta F^*)}{\mu_M} - \frac{\eta(1-\delta)(1+\beta F^*)}{\mu_S}}. $$

This expression corresponds to equation (13) in section 2.2.4. By replacing $C(F^*)$ in the same way with the expressions proposed in section 2.1.1 in $\sigma(F^*)$ (13), we obtain bifurcation diagrams with shapes similar to those described in section 2.2.4.

## Appendix D Studying the numerator of $\sigma(F^*)$

The numerator of $\sigma(F^*)$ is expressed as follows:

$$ \frac{\beta p \mu_F}{(1-p)\mu_M} (F^*)^2 + \left( \frac{p \mu_F (1 - \mathcal{R}_F)}{(1-p)\mu_M} + \beta k \right) F^* + k, $$

The expression can be rewritten as

$$ \frac{k}{\mathcal{R}_F} \left( \mathcal{R}_M (\beta F^*)^2 + (\mathcal{R}_M + \mathcal{R}_F - \mathcal{R}_M \mathcal{R}_F) (\beta F^*) + \mathcal{R}_F \right). $$
It does not depend on residual fertility rates, unlike the denominator. The discriminant of the $\beta F^*$ second order polynomial is

$$ \begin{aligned} \Delta &= (\mathcal{R}_M + \mathcal{R}_F - \mathcal{R}_M \mathcal{R}_F)^2 - 4 \mathcal{R}_F \mathcal{R}_M, \\ &= \mathcal{R}_M^2 + \mathcal{R}_F^2 + \mathcal{R}_F^2 \mathcal{R}_M^2 + 2 \mathcal{R}_F \mathcal{R}_M - 2(\mathcal{R}_M + \mathcal{R}_F) \mathcal{R}_M \mathcal{R}_F - 4 \mathcal{R}_F \mathcal{R}_M. \end{aligned} $$

Thus, the discriminant is positive if and only if

$$ (\mathcal{R}_M - \mathcal{R}_F)^2 + \mathcal{R}_M^2 \mathcal{R}_F^2 > 2 \mathcal{R}_M \mathcal{R}_F (\mathcal{R}_M + \mathcal{R}_F). \quad (\text{D1}) $$

Dividing by $\mathcal{R}_M^2 \mathcal{R}_F^2$ yields

$$ \left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1 > 2\left(\frac{1}{\mathcal{R}_F} + \frac{1}{\mathcal{R}_M}\right). $$

Taking the inverse and multiplying by 4 yields

$$ \frac{2}{\frac{1}{\mathcal{R}_F} + \frac{1}{\mathcal{R}_M}} > \frac{4}{\left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1}, \quad (\text{D2}) $$

which is a lower bound for the harmonic mean of $\mathcal{R}_F$ and $\mathcal{R}_M$. Since $\mathcal{R}_F, \mathcal{R}_M > 1$, we have $0 < \frac{1}{\mathcal{R}_F}, \frac{1}{\mathcal{R}_M} < 1$, such that $-1 < \frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M} < 1$ and $1 \le \left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1 < 2$. It follows that $2 < \frac{4}{\left(\frac{1}{\mathcal{R}_F} - \frac{1}{\mathcal{R}_M}\right)^2 + 1} \le 4$, meaning that (D2) requires the harmonic mean to be larger than 2; this requirement is necessarily satisfied when the harmonic mean is larger than 4 (Fig. D2).

In this case, there are two roots. The product of the roots has the sign of $\frac{\mathcal{R}_F}{\mathcal{R}_M}$, which is positive. Thus, the roots have the same sign. The sum of the roots is

$$ -\frac{\mathcal{R}_M + \mathcal{R}_F - \mathcal{R}_M \mathcal{R}_F}{\mathcal{R}_M}. \quad (\text{D3}) $$

Consequently, the sum of the roots is positive and therefore both roots are positive if and only if

$$ \begin{aligned} \mathcal{R}_F + \mathcal{R}_M < \mathcal{R}_M \mathcal{R}_F &\iff \frac{1}{\mathcal{R}_M} + \frac{1}{\mathcal{R}_F} < 1, \\ &\iff \frac{2}{\frac{1}{\mathcal{R}_M} + \frac{1}{\mathcal{R}_F}} > 2. \end{aligned} $$

Therefore, the harmonic mean of $\mathcal{R}_F$ and $\mathcal{R}_M$ must be larger than 2, which we have shown to be a consequence of equation (D2). It follows that (D1) or (D2) suffices for the numerator to have two positive roots if $\mathcal{R}_F, \mathcal{R}_M > 1$ (Fig. D2).
<figure><img src="image_11.png" /><figcaption>**Fig. D2** Contour plot illustrating the $\sigma(F^*)$ numerator roots as a function of the reproduction numbers $\mathcal{R}_M$ and $\mathcal{R}_F$. Three zones are defined: one containing the negative real roots (dark blue), one in which there are no real roots (blue-green) and one containing the positive real roots (light green). The two curves represent the harmonic means of $\mathcal{R}_M$ and $\mathcal{R}_F$, equal to 2 (dashed black curve) and equal to 4 (solid black curve), respectively.</figcaption></figure>

# Appendix E Parameter values

## E.1 Mortality rates

Different values for *Ceratitis capitata* lifespans were collected from the literature. The mortality rate was defined as the inverse of the lifespan. The lifespans of females and males and the corresponding mortality rates, are provided in Table E1.

<table><caption>**Table E1** Female and male lifespan in *Ceratitis capitata* based on values in the literature; the mortality rates ($\mu_F$ and $\mu_M$) were estimated using the inverse of lifespan.</caption><thead><tr><th>References</th><th>Female lifespan (days)</th><th>Male lifespan (days)</th><th>μ<sub>F</sub> (day<sup>-1</sup>)</th><th>μ<sub>M</sub> (day<sup>-1</sup>)</th></tr></thead><tbody><tr><td>Shoukry and Hafez (1979)</td><td>[25.5-31]</td><td>[25-36.5]</td><td>[0.032-0.039]</td><td>[0.027-0.040]</td></tr><tr><td>Carey (1984)</td><td>[39-46.4]</td><td>[40.5-47.9]</td><td>[0.022-0.026]</td><td>[0.021-0.025]</td></tr><tr><td>Vargas et al. (2000)</td><td>54.8</td><td>71.6</td><td>0.018</td><td>0.014</td></tr><tr><td>Papadopoulos et al. (2002)</td><td>21.1</td><td>29.5</td><td>0.047</td><td>0.034</td></tr><tr><td>Pieterse et al. (2020)</td><td>[12-17.5]<sup>1</sup></td><td>[17.6-35.5]<sup>2</sup></td><td>[0.057-0.083]<sup>1</sup></td><td>[0.036-0.057]<sup>2</sup></td></tr></tbody></table>

<footnote n="1">¹ Data from research on nectarines and plums that provided extremum for females.</footnote>

<footnote n="2">² Data from research on apples and plums that provided extremum for males.</footnote>
## E.2 Emergence rate

The emergence rate *r* represents the mean number of eggs that produce adults per female per day. To estimate this parameter, data (see Table E2) were taken from Pieterse et al. (2020). Only values associated with nectarines were used because the emergence rate varies greatly among fruit types, and our aim was to obtain a rough estimate for the purposes of model calibration. On average, females lay 25.75 eggs a day, of which 97% hatch. The larvae then become pupae, of which 94.7% reach adulthood. So, $r = 25.75 \times 0.97 \times 0.947 = 23.7$ day⁻¹, with values ranging between 19.2 and 28.5 day⁻¹.

<table><caption>**Table E2** Development data for *C. capitata* taken from Pieterse et al. (2020)</caption><thead><tr><th>Fruit type</th><th>Eggs per females (24h)</th><th>Hatching success (24h)</th><th>Pupal eclosion success (24h)</th></tr></thead><tbody><tr><td>Nectarine</td><td>25.75 [22.25-29.25]</td><td>97% [96%-98%]</td><td>94.7% [89.9%-99.5%]</td></tr></tbody></table>

## E.3 Competition among females for oviposition sites

The coefficient *β* represents the strength of female competition for oviposition sites. This parameter was estimated from the model without SIT-based population control at equilibrium. From equation (12), which allows us to find the infestation equilibria of the model, we isolated *β* as follows:

$$ 
\begin{aligned}
X(M(F))C(F) &= \frac{1}{\mathcal{R}_F}, \\
\frac{\frac{p\mu_F}{(1-p)\mu_M}F}{k + \frac{p\mu_F}{(1-p)\mu_M}F} \frac{1}{1 + \beta F} &= \frac{1}{\mathcal{R}_F}, \\
1 + \beta F &= \frac{\mathcal{R}_F \frac{p\mu_F}{(1-p)\mu_M}F}{k + \frac{p\mu_F}{(1-p)\mu_M}F}, \\
\beta &= \frac{1}{F} \left( \frac{\mathcal{R}_F \frac{p\mu_F}{(1-p)\mu_M}F}{k + \frac{p\mu_F}{(1-p)\mu_M}F} - 1 \right).
\end{aligned}
$$

We based our estimate of female density *F* on several mass-trapping studies (Demirel and Akyol, 2017; Hafsi et al., 2020; Leza et al., 2008; Martinez-Ferrer et al., 2012), in which about 1,000 females per day per hectare were captured; we assumed the sex ratio was balanced. We also assumed that these traps captured most of the individuals present, which thus leads to an estimate of the equilibrium density of *C. capitata* in the absence of SIT. Thus, using *F* = 1000 ind.ha⁻¹ and the parameter values listed in the Table 1, we found *β* = 0.24 (ind.ha⁻¹)⁻¹.

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