# Polyandry: A Threat or An opportunity for the Sterile Insect Technique?

Marine A Courtois, Louise van Oudenhove, Suzanne Touzeau, Frédéric Grognard, Ludovic Mailleret

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Marine A Courtois, Louise van Oudenhove, Suzanne Touzeau, Frédéric Grognard, Ludovic Mailleret. Polyandry: A Threat or An opportunity for the Sterile Insect Technique?. 2025. hal-05208104

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# Polyandry: A Threat or An opportunity for the Sterile Insect Technique?

Marine A. Courtois<sup>a,*</sup>, Louise van Oudenhove<sup>a</sup>, Suzanne Touzeau<sup>a,b</sup>, Frédéric Grognard<sup>b</sup>, Ludovic Mailleret<sup>a,b</sup>

**Abstract.** The sterile insect technique (SIT) is a pest control strategy based on the mass release of sterilized males to disrupt natural reproduction and suppress wild populations. However, its effectiveness can be challenged by biological factors such as female multiple mating and sperm use bias. While multiple mating is widespread among many insect species, the mechanisms governing sperm use remain poorly understood. In this study, we develop and analyze a compartmental mathematical model based on differential equations to investigate the overall impact of multiple mating on SIT efficiency. We further analyze the effect of sperm use biases with an agent-based model, calibrated on *Drosophila suzukii*, allowing to explore different scenarios: preferential use of first vs last sperm, of fertile vs sterile sperm, and mixed sperm use. Our results highlight how multiple mating and sperm use biases influence SIT effectiveness. In the longer term, multiple mating is disadvantageous as it requires additional releases of sterilized males to control the pest population. However, in the shorter term, it can be beneficial by disrupting further female reproductive output by "defertilizing" females mated with wild males. This study provides new information on how the way sperm is processed after mating can impact sterile insect control strategies, highlighting the limited influence of these biological processes depending on the release efforts that can be deployed.

**Keywords:** ABM; Crop protection; Modeling; Population dynamics; Pest management; Polyandry; Sperm use bias; *Drosophila suzukii*.

## 1 Introduction

The sterile insect technique (SIT) is an autocidal pest control method: it uses the pest species to control its own population by regular inundative releases of sterilized males (Dyck et al., 2021). Insects of the target species are mass-reared, sexed (when possible), and then sterilized before being released in large numbers. Once released, sterilized males compete with wild males for mates, reducing the proportion of females that produce viable offspring. This leads progressively to a decline of the pest population size. Originally conceptualized by the entomologist E. F. Knipling in the 1950s (Knipling, 1955), SIT was first used to eradicate the New World screwworm fly (*Cochliomyia hominivorax*) (Knipling, 1959a). Today, SIT offers a sustainable alternative or complementary approach to chemical control methods in various pest management programs (Table 1). Regarding human health, SIT has been applied to control disease vectors such as mosquitoes (Dyck et al., 2021). In agriculture, it has been used against pests such as the melon fly *Bactrocera cucurbitae* and the Mediterranean fruit fly *Ceratitis capitata* (Enkerlin et al., 2017). Recently, SIT programs have been extended to new invasive pests, including *Drosophila suzukii*, commonly known as the Spotted Wing Drosophila (SWD). *Drosophila suzukii* is one of the most damaging pests in soft fruit crops, particularly affecting berries, cherries, and sometimes grapes and peaches (Hauser, 2011; Walsh et al., 2011). Unlike other *Drosophila* species, *D. suzukii* targets healthy, non-rotting fruits, causing severe economic losses due to larval feeding and oviposition wounds, which favour pathogen introduction (Haye et al., 2016; Rossi Stacconi et al., 2013). Due to its broad host

<footnote><sup>a</sup>Université Côte d'Azur, INRAE, CNRS, ISA, France</footnote>

<footnote><sup>b</sup>Université Côte d'Azur, Inria, INRAE, MACBES, France</footnote>

<footnote>*Corresponding author, marine.courtois@inrae.fr</footnote>
range and aggressive damaging of healthy fruits, *D. suzukii* represents a major agricultural challenge worldwide. Current control strategies heavily rely on chemical methods, which poses environmental and resistance-related challenges, thereby encouraging the exploration of alternative approaches such as SIT (Haye *et al.*, 2016).

The successful implementation of SIT depends on identifying key biological and technical factors that influence its outcomes. Among these, the mating behavior of the target species, particularly polyandry, has been highlighted as an influential aspect. Historically, SIT programs have targeted species where females are predominantly monogamous, i.e., monoandrous, mating only once in their lifetime (Knipling, 1955). Knipling (1955) proposed that one principle to consider when determining if SIT is appropriate for a species was: “Females must normally mate only once.” However, increasing attention has been given to polyandrous species, in which females mate with multiple males over their reproductive lifetime. The timing and frequency of such matings are often constrained by refractory rates (i.e., intervals between matings). In such species (Table 1), multiple mating can significantly shape population dynamics because females mating with multiple males may reduce the chances that sterilized males successfully prevent reproduction. Various techniques have been developed to mitigate re-mating in polyandrous species, such as dietary modifications or semiochemical treatments that enhance the ability of sterilized males to suppress further mating (Gavriel *et al.*, 2009; Morelli *et al.*, 2013). For example, in *Ceratitis capitata*, a polyandrous species, sterilized males treated with ginger root oil inhibit female re-mating more effectively than untreated males (Shelly *et al.*, 2004a,b).

While polyandry could be perceived as a challenge for SIT, its impact may largely depend on factors such as mat-
ing order, sperm storage, and sperm competition. Early theoretical studies reconsidered the necessity of monogamous
mating systems for SIT success (Knipling, 1959a,b), emphasizing that the key determinant is the competition between
wild and sterile sperm, rather than the number of matings per female (von Borstel and Knipling, 1960). More recent
works support this view, showing that as long as sterilized male sperm remains competitive and mating occurs ran-
domly, polyandry is not inherently incompatible with SIT, although its effectiveness can be compromised when wild
male sperm is preferentially used by females for fertilization (Dyck et al., 2021). The impact of re-mating on SIT
efficiency hence depends on sperm selection biases, among other factors such as male fertility influence on re-mating
probability (Pérez-Staples et al., 2013; Vreysen et al., 2007). These processes can either dilute or bypass the sterilizing
effect of the initial mating.

In the context of *D. suzukii*, multiple mating behavior and sperm use biases are particularly relevant. The females
of this species can mate with multiple males, being able to produce offspring from up to five mates (Puppato et al., 2023). Additionally, SWD females possess two spermathecae, which may enable differential sperm storage and use (Avanesyan et al., 2017). The mechanisms driving sperm selection and use in SWD remain poorly understood, studies on related species provide valuable insights. For instance, in *Drosophila melanogaster*, sperm storage structures can influence sperm use: some genotypes are associated with a preferential use of first-mating sperm due to anatomical features limiting storage of later sperm (Hopkins et al., 2020), while other studies report last-male precedence, with the most recent male siring over 80% of offspring (Bloch Qazi et al., 2003). These contrasting findings highlight the complexity of post-copulatory processes and sperm use. Reflecting this complexity, mixed sperm use strategies, where females store and use sperm from multiple males simultaneously, have also been proposed (Bangham et al., 2003). Sterilization processes may further influence SIT outcomes by reducing sperm quantity or altering ejaculate composition. For example, irradiated mosquitoes produce significantly fewer sperm than un-irradiated males (Helinski and Knols, 2009). While no equivalent data exist for SWD, lower sperm numbers in sterilized males could reduce the likelihood of successful sperm use. Similar effects have been observed in other insect species (such as corn earworm and boll weevil), where sterilization-induced sperm reduction can lower sterilized male competitiveness by promoting female re-mating with wild males (Carpenter et al., 1987; Haynes and Mitchell, 1977). Additionally, studies suggest that mating duration correlates positively with male ejaculate investment (Lanouette et al., 2017), and that longer copulations increase the likelihood of that sperm being used in polyandrous contexts (Bretman et al., 2009). Further, mating plugs, such as those observed in *Drosophila hibisci*, may limit the access of subsequent males' sperm to the female reproductive tract, thereby favoring first-mating sperm (Polak et al., 2001).

This study aims to evaluate the impact of multiple mating, in particular sperm use biases on SIT outcomes using
mathematical modeling to explore different scenarios and hypotheses. Various modeling studies have already con-
tributed to advance knowledge on SIT effectiveness by investigating key factors such as release strategies (Cai et al.,
2014), residual fertility in sterilized males released (Aronna and Dumont, 2020; Barclay, 1982; Courtois et al., 2025),
sterilized male competitiveness (Parker and Mehta, 2007) or female mate preferences (Chargui et al., 2018; Vreysen
et al., 2006). Interestingly, a modeling study focusing specifically on the case of two successive matings showed that in
some cases, re-mating may enhance SIT outcomes, depending on female reproductive parameters (Dumont and Oliva,
2024). We adopt a two-scale modeling approach to study the influence of polyandry and sperm-use biases on pest pop-
ulation dynamics under SIT. We first develop a compartmental model based on ordinary differential equations, which provides a global understanding of how re-mating affects qualitative results and population-level outcomes. We then complement this with an agent-based model (ABM), calibrated on SWD, that enables us to investigate more detailed scenarios of sperm use associated with multiple mating events. In particular, we explore various sperm use strategies, including preferential use of first sperm, preferential use of last sperm, mixed sperm use, and female preference for fertile or sterile sperm. This dual modeling framework helps evaluate the robustness of SIT under different biological assumptions and sheds light on the role of female post-mating processes in shaping control outcomes.

This article is structured as follows: Section 2 presents the compartmental model, which explores the global dynamics of re-mating. Section 3 introduces the agent-based model for analyzing finer-scale sperm use biases and their implications for SIT. Finally, Section 4 discusses the impact of re-mating and sperm use biases on population dynamics, the complementarity of the two approaches, their implications for optimizing SIT, and perspectives for further refining this pest control method.
<table><caption>**Table 1** Summary of species targeted by sterile insect technique (SIT) programs: **Location** represents where SIT has been implemented; **Polyandry?** indicates whether the species is polyandrous (multiple matings) or not (single mating per female); **Date** indicates the date or approximate time frame for the initial study on the implementation of the sterile insect technique for the species; **Reference** cites key studies.</caption><thead><tr><th>Species</th><th>Location</th><th>Polyandry?</th><th>Date</th><th>Reference</th></tr></thead><tbody><tr><td><strong>Livestock pests</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Cochliomyia hominivorax (New World screwworm)</td><td>USA, Mexico, Central America, Libya</td><td>No</td><td>1950s</td><td>(Bushland and Hopkins, 1951; Wyss, 2000)</td></tr><tr><td>Chrysomyia bezziana (Old World screwworm)</td><td>Papua New Guinea, Australia</td><td>No</td><td>1989</td><td>(Hendrichs and Robinson, 2009; Spradbery et al., 1989)</td></tr><tr><td><strong>Agricultural pests: Fruit flies (Tephritidae)</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Bactrocera dorsalis (Oriental fruit fly)</td><td>Thailand, China, East Africa</td><td>Yes</td><td>1960s</td><td>(Dyck et al., 2021; Prastiti et al., 2021)</td></tr><tr><td>Ceratitis capitata (Mediterranean fruit fly)</td><td>Worldwide (Spain, Croatia, Argentina, Brazil, Israel etc.)</td><td>Yes</td><td>1970s</td><td>(Dumont and Oliva, 2024; Dyck et al., 2021)</td></tr><tr><td>Anastrepha ludens (Mexican fruit fly)</td><td>Mexico, USA</td><td>Yes</td><td>1980s</td><td>(Perez-Staples et al., 2021)</td></tr><tr><td>Anastrepha obliqua (West Indian fruit fly)</td><td>Mexico, Central America</td><td>Yes</td><td>1980s</td><td>(Orozco-Dávila et al., 2017)</td></tr><tr><td>Bactrocera tryoni (Queensland fruit fly)</td><td>Australia</td><td>Yes</td><td>1980s</td><td>(Shadmany et al., 2022)</td></tr><tr><td>Zeugodacus cucurbitae (Melon fruit fly)</td><td>USA, Japon, Inde, Africa</td><td>Yes</td><td>1980s</td><td>(Kuba et al., 2020)</td></tr><tr><td>Bactrocera oleae (Olive fruit fly)</td><td>Mediterranean region, North America, Israel</td><td>Yes</td><td>2000s</td><td>(Dyck et al., 2021; Gerofotis et al., 2015)</td></tr><tr><td><strong>Agricultural pests: Other insects</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Pectinophora gossypiella (Pink bollworm)</td><td>Mexico, USA</td><td>Yes</td><td>1960s</td><td>(Tabashnik et al., 2021; Torres-Vila et al., 2004)</td></tr><tr><td>Eldana saccharina (African sugarcane borer)</td><td>South Africa</td><td>No</td><td>1970s</td><td>(Malinga, 2024; Walton and Conlong, 2016)</td></tr><tr><td>Lyantria dispar (Gypsy moth)</td><td>USA</td><td>No</td><td>1975</td><td>(Torres-Vila et al., 2004; USDA Forest Service, 2023)</td></tr><tr><td>Delia antiqua (Onion maggot)</td><td>USA, Europe, Israel, Netherlands</td><td>No</td><td>1981</td><td>(Dyck et al., 2021; Fortier et al., 2024)</td></tr><tr><td>Anthonomus grandis (Boll weevil)</td><td>Mexico, Brazil, USA</td><td>Yes</td><td>1990s</td><td>(Mayer and Brazzel, 1963; USDA APHIS, 1997)</td></tr><tr><td>Cactoblastis cactorum (Cactus moth)</td><td>USA, Mexico</td><td>Yes</td><td>1990s</td><td>(Hight et al., 2005)</td></tr><tr><td>Cydia pomonella (Codling moth)</td><td>Canada</td><td>No</td><td>1990s</td><td>(Dyck et al., 2021; Tyson et al., 2008)</td></tr><tr><td>Cylas formicarius (Sweetpotato weevil)</td><td>Japan</td><td>Yes</td><td>2000s</td><td>(Arnqvist and Nilsson, 2000; Himuro et al., 2022b)</td></tr><tr><td>Ectomyelois ceratontiae (Carob moth)</td><td>Tunisia, Iran</td><td>Yes</td><td>2000s</td><td>(Soutbaf et al., 2023)</td></tr><tr><td>Euscepes posifasciatus (Sweetpotato weevil)</td><td>Japan</td><td>Yes</td><td>2000s</td><td>(Himuro et al., 2022a; Ikegawa and Himuro, 2017)</td></tr><tr><td>Helicoverpa armigera (Cotton bollworm)</td><td>China</td><td>Yes</td><td>2000s</td><td>(Lu et al., 2002; Teng and Zhang, 2009)</td></tr><tr><td>Lobesia botrana (European grapevine moth)</td><td>Chile</td><td>No</td><td>2000s</td><td>(Simmons et al., 2021; Torres-Vila et al., 2004)</td></tr><tr><td>Liriomyza trifoli (Leafminer)</td><td>USA</td><td>Yes</td><td>2000s</td><td>(Kaspi and Parrella, 2003, 2006, 2008)</td></tr><tr><td>Thaumatotibia leucotreta (False codling moth)</td><td>South Africa</td><td>Yes</td><td>2000s</td><td>(Azrag et al., 2021; Githae et al., 2024)</td></tr><tr><td>Aethina tunida (Small hive beetle)</td><td>USA, Australia</td><td>Yes</td><td>2010s</td><td>(Neumann et al., 2015; Papach et al., 2023)</td></tr><tr><td>Amyelois transitella (Navel orangeworm)</td><td>USA</td><td>No</td><td>2010s</td><td>(Parra-Pedrazzoli and Leal, 2006; Reger et al., 2021)</td></tr><tr><td>Halyomorpha halys (Brown marmorated stink bug)</td><td>USA, Europe</td><td>Yes</td><td>2010s</td><td>(Leskey et al., 2013; Suckling et al., 2019)</td></tr><tr><td>Drosophila suzukii (Spotted-wing drosophila)</td><td>Europe, North America</td><td>Yes</td><td>2020s</td><td>(Puppato et al., 2023)</td></tr><tr><td>Spodoptera frugiperda (Fall armyworm)</td><td>Americas, Africa, Asia</td><td>Yes</td><td>2020s</td><td>(Jiang et al., 2023)</td></tr><tr><td><strong>Disease vectors</strong></td><td></td><td></td><td></td><td></td></tr><tr><td>Aedes aegypti (Dengue mosquito)</td><td>Brazil, Thailand, China, USA, Australia, Mexico, Europe, Mauritius</td><td>Yes</td><td>1960s</td><td>(Degner and Harrington, 2016; Lees et al., 2015)</td></tr><tr><td>Glossina spp. (Tsetse fly)</td><td>Sub-Saharan Africa, Burkina Faso</td><td>No</td><td>1970s</td><td>(Dyck et al., 2021; Maudlin, 1999; Mirieri et al., 2024)</td></tr><tr><td>Anopheles arabiensis (Malaria mosquito)</td><td>Sub-Saharan Africa</td><td>Yes</td><td>2000s</td><td>(Helinski et al., 2008a,b)</td></tr><tr><td>Aedes albopictus (Asian tiger mosquito)</td><td>Europe, Asia, Americas</td><td>Yes</td><td>2010s</td><td>(Balatsos et al., 2024; Yamada et al., 2024)</td></tr></tbody></table>
## 2 Compartmental Modeling: Impact of Multiple Mating on SIT Effectiveness

### 2.1 Model description

The model inspired by Anguelov et al. (2017), describes the population dynamics of the following compartments: immature stage L (including all immature sub-stages: eggs, larvae, and pupae), sexually mature females that are yet unmated or available for re-mating $F_U$, fertilized females $F_F$ (females mated with wild males), infertile females $F_I$ (females mated with sterilized males), wild males M, and sterilized males S released as part of the sterile insect technique deployment (Fig. 1). The system is defined as follows:

$$ 
\left\{
\begin{aligned}
\dot{L} &= \omega(1 - \frac{L}{K})F_F - (\mu_L + \nu)L, \\
\dot{M} &= \nu p L - \mu_M M, \\
\dot{F}_U &= \nu(1 - p)L + \tau_F F_F + \tau_I F_I - (\mu_F + \chi)F_U, \\
\dot{F}_I &= \chi \frac{\eta S}{M + \eta S} F_U - (\tau_I + \mu_F)F_I, \\
\dot{F}_F &= \chi \frac{M}{M + \eta S} F_U - (\tau_F + \mu_F)F_F, \\
\dot{S} &= -\mu_S S + \sigma.
\end{aligned}
\right.
\tag{1}
$$

with the dot notation representing the time derivative.

The dynamics of larvae L are affected by their mortality $\mu_L$, the larval hatching rate $\nu$, and the egg-laying rate of fertilized females ($F_F$) $\omega$. Larval competition is represented by a logistic-like competition function through the load-carrying capacity K, i.e. the maximum number of juveniles that can be contained in the fruits. Resources for oviposition and larval development are assumed to be finite and constant. The dynamics of males M are driven by their mortality $\mu_M$ and the emergence of new males, which is dependent on the larval hatching rate $\nu$ and sex ratio $p$. The dynamics of unmated and available females $F_U$ depend on their mortality $\mu_F$, the emergence of unmated females (with larval hatching rate $\nu$ and sex ratio $p$) and the rate at which females mate $\chi$. The dynamics of $F_U$ are also influenced by females that become available again, with a refractory rate $\tau_F$ for fertilized females $F_F$ and a refractory rate $\tau_I$ for infertile females. The refractory rate is defined as the inverse of time required for a mated female to become available for mating again. The dynamics of fertilized females $F_F$ (respectively infertile females $F_I$) are affected by their mortality $\mu_F$, the proportion of unmated females $F_U$ that mate with a wild male M (respectively a sterilized male S) and the refractory rate $\tau_F$ (respectively $\tau_I$) of females that become available again. We assume here that males
are not limiting so that successful matings only depends on the ratio between fertile males and attractive sterilized males, with parameter $\eta$ accounting for sterilized males competitiveness. Finally, the dynamics of sterilized males $S$ are affected by their mortality $\mu_S$ and the release rate $\sigma$.

<figure><img src="image_3.png" /><figcaption>**Fig 1** Flow diagram of the population dynamics model (1). The compartments (and color code) correspond to larvae $L$ (red), wild males $M$ (green), sterilized males $S$ (pink), unmated females $F_U$ (purple), fertilized females $F_F$ (blue) and infertile females $F_I$ (orange). Solid arrows correspond to flows, dotted arrows to matings. Sterilized males are released at rate $\sigma$. Reproduction is represented by the egg-laying rate $\omega$, the larvae hatching rate $\nu$, the proportion of males among offspring $p$ and the proportion of successfull matings $\frac{M}{M+\eta S}$. Finally, all compartments are affected by specific mortality rates $\mu$.</figcaption></figure>

The parameter values used, calibrated for *Drosophila suzukii*, are listed in Table 2. Details on the extraction and calculation of parameter values are provided in Appendix A. As said in the introduction, *D. suzukii* is known to be polyandrous (Krüger et al. (2019); Lanouette et al. (2017); Puppato et al. (2023), Table 1). Male availability is not a limiting factor, nor is their mating capacity. Wild and irradiated males of this species have been reported to possibly achieve more than six matings within a 24-hour period (Lanouette et al., 2017). Females of the species *D. suzukii* typically mate shortly after their emergence, once they reach sexual maturity (Revadi et al., 2015). Moreover, the high density of males promotes rapid interactions and immediate mating.

Therefore, we assume in our model that females remain in the unmated compartment ($F_U$) for a very short time, reflecting this pest's ability to quickly colonize environments through efficient reproduction (Burrack et al., 2013). Based on this assumption, we perform a slow-fast approximation, as described in Appendix B. This approximation
allows us to reduce the system (1) to the following model:

$$
\left\{
\begin{aligned}
\dot{L} &= \omega \left(1 - \frac{L}{K}\right) F_F - (\mu_L + \nu)L, \\
\dot{M} &= \nu p L - \mu_M M, \\
\dot{F}_I &= \nu(1-p) \frac{\eta S}{M+\eta S} L + \tau_F \frac{\eta S}{M+\eta S} F_F - (\mu_F + \tau_I \frac{M}{M+\eta S}) F_I, \\
\dot{F}_F &= \nu(1-p) \frac{M}{M+\eta S} L + \tau_I \frac{M}{M+\eta S} F_I - (\mu_F + \tau_F \frac{\eta S}{M+\eta S}) F_F, \\
\dot{S} &= -\mu_S S + \sigma.
\end{aligned}
\right.
\tag{2}
$$

<table><caption>**Table 2** Model parameters estimated for *Drosophila suzukii*</caption><thead><tr><th>Parameter/Description</th><th>Value</th><th>Unit</th><th>Reference</th></tr></thead><tbody><tr><td>μ<sub>L</sub> (Larvae mortality rate)</td><td>0.037</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>μ<sub>F</sub> (Female mortality rate)</td><td>0.012</td><td>day<sup>-1</sup></td><td>(Papanastasiou et al., 2021)</td></tr><tr><td>μ<sub>M</sub> (Male mortality rate)</td><td>0.013</td><td>day<sup>-1</sup></td><td>(Papanastasiou et al., 2021)</td></tr><tr><td>μ<sub>S</sub> (Sterilized male mortality rate)</td><td>0.054</td><td>day<sup>-1</sup></td><td>(Lanouette et al., 2017)</td></tr><tr><td>ω (Egg-laying rate)</td><td>6</td><td>eggs per female per day</td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>ν (Larvae hatching rate)</td><td>0.08</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>p (Sex ratio)</td><td>0.50</td><td>-</td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>χ (Mating rate)</td><td>-</td><td>day<sup>-1</sup></td><td>Assumed high<sup>[1]</sup></td></tr><tr><td>η (Male competitiveness)</td><td>0.60</td><td>-</td><td>(Lanouette et al., 2020)</td></tr><tr><td>K (Carrying capacity)</td><td>36,000</td><td>larvae per tunnel</td><td>Calculated<sup>[2]</sup></td></tr><tr><td>σ (Sterilized male release rate)</td><td>-</td><td>ind.day<sup>-1</sup></td><td>-</td></tr><tr><td>τ<sub>F</sub> (Refractory rate of fertilized females)</td><td>0.12</td><td>day<sup>-1</sup></td><td>(Krüger et al., 2019)</td></tr><tr><td>τ<sub>I</sub> (Refractory rate of infertile females)</td><td>0.14</td><td>day<sup>-1</sup></td><td>(Krüger et al., 2019)</td></tr></tbody></table>

## 2.2 Analysis of equilibria

The $\dot{S}$ equation is independent and $S$ tends to its equilibrium value $S^* = \frac{\sigma}{\mu_S}$. Equilibrium values for other variables are obtained for $S = S^*$ (see Appendix D). There are between one and three equilibria for model (2). Among these equilibria, the pest-free equilibrium, denoted as $E_0^*$, always exists and represents a case where the pest population is completely eradicated. This pest-free equilibrium $E_0^*$ is shown to be locally asymptotically stable as soon as $\sigma > 0$, as

<footnote n="1">¹ Refer to Appendix B</footnote>

<footnote n="2">² Estimation based on larval density calculations from *D. suzukii* for a strawberry tunnel of approximately 500 $m^2$ (more details in Appendix A).</footnote>
demonstrated in Appendix C.

In addition to this, infestation equilibria, which correspond to cases where the pest population persists, are obtained from the solutions of the equation $\zeta(L^*) = \sigma$ (Appendix D):

$$ \zeta(L^*) = \frac{\gamma\mu_S(\mu_F + \tau_I)}{\eta(\mu_F + \tau_F)} \left( -\frac{\mathcal{R}_0}{K} L^{*2} + (\mathcal{R}_0 - 1)L^* \right), \quad \text{with } \gamma = \frac{\nu p}{\mu_M}. \qquad (3) $$

In this expression, the basic reproduction number, is defined as: $\mathcal{R}_0 = \frac{\omega(1-p)\nu}{\mu_F(\mu_L+\nu)}$, and represents the average number of viable female offspring produced by a single pest female.

The parabola described by $\zeta(L^*)$ allows us to identify the conditions that lead to pest population eradication or persistence by analyzing how larval density at equilibrium $L^*$ responds to variations in the release intensity $\sigma$. In order to determine the stability of this infestation equilibria, a numerical analysis was conducted on the eigenvalues of the Jacobian matrix associated with the system at these equilibria (Appendix D).

We define the tipping point $\bar{\sigma}$ as the critical release rate of sterilized males beyond which the pest-free equilibrium becomes the only stable state of the system. This threshold, referred to as the *eradication threshold*, corresponds to the minimum number of sterilized males $\sigma$ that must be released to ensure effective control of pest populations in agricultural systems. Mathematically, it corresponds to the summit of parabola (3), and is thus given by:

$$ \bar{\sigma} = \frac{K\gamma\mu_S(\mathcal{R}_0 - 1)^2(\mu_F + \tau_I)}{4\eta\mathcal{R}_0(\mu_F + \tau_F)}. \qquad (4) $$

Releasing a rate of sterilized males beyond this threshold drives the system into the stability region of the pest-free equilibrium (Fig. 2), theoretically ensuring successful pest control.

## Summary: equilibria and local stability

(i) If $\mathcal{R}_0 \le 1$ or $\sigma > \bar{\sigma}$ (Fig. 2), the pest-free equilibrium is the unique equilibrium of the system (2) and is locally asymptotically stable.

(ii) If $\mathcal{R}_0 > 1$ and $\sigma < \bar{\sigma}$ (Fig. 2), the system (2) admit three equilibria: the pest-free equilibrium which is locally asymptotically stable, the infestation equilibrium $E_1^*$ such that $L^* < \frac{K}{2}(1 - \frac{1}{\mathcal{R}_0})$ which is unstable, and, the infestation equilibrium $E_2^*$ such that $L^* > \frac{K}{2}(1 - \frac{1}{\mathcal{R}_0})$ which is locally asymptotically stable.
<figure><img src="image_4.png" /><figcaption>**Fig 2** Bifurcation diagram showing the larval density at equilibrium $L^*$ as a function of the sterilized male release rate $\sigma$. The equilibria of system (2) correspond to the intersection points between the function $\zeta(L)$ (8) (blue, *First* scenario = Without MM, when $\tau_F = \tau_I = 0$, or orange, *Last* scenario, when $\tau_F$ and $\tau_I > 0$) and the $\sigma$ constant. Stability is represented by solid lines, and unstable equilibria are shown by dashed lines. The *eradication thresholds* for each case are also shown (blue or orange dotted lines). The diagram was generated by setting the parameters to the values listed in Table 2.</figcaption></figure>

## 2.3 Impact of Multiple Mating

### 2.3.1 Definition of Scenarios

We define two distinct scenarios for multiple mating (MM):

**The Last Scenario** In this scenario, females can re-mate and sperm from the last mating takes precedence over earlier sperm, so the female status (fertilized or infertile) and the production of offspring are determined by the last male the female mated with. This is the default scenario in the reduced model (2) with $\tau_F > 0$ and $\tau_I > 0$, where fertilized females ($F_F$) continue to lay eggs until they mate with a sterilized male and transition into the infertile female compartment (orange in Fig. 2).

**The First Scenario (= Without MM):** In this scenario, the production of offspring is determined exclusively by the sperm of the first male the female mates with, regardless of any subsequent matings. This scenario also represents the absence of multiple mating and corresponds to model (2), where parameters $\tau_I = \tau_F = 0$. It ensures that females, once fertilized, remain in that compartment ($F_F$) and continue to lay eggs throughout their lifespan (blue in Fig. 2).
### 2.3.2 Asymptotic behavior

The asymptotic behavior is quite similar for both the *First* and *Last* scenarios (Fig. 2). Indeed, three distinct cases can be identified based on the initial infestation level and the sterilized male release effort: (1) a high initial infestation with a small release rate, (2) a low initial infestation with a small release rate, and (3) a sufficiently high release rate. These three cases are illustrated in Fig. 3. The simulations reveal that, for a high initial infestation level of pests at the start of SIT treatment, when the release rate $\sigma$ is below the eradication thresholds for the *First* and *Last* scenarios (Fig. 2, case (1)), the larval population is reduced compared to the “Without Release” scenario (WR) but remains at a relatively high level (Fig. 3A). At intermediate release rates between eradication thresholds, population control is achieved without multiple mating, but maintained when multiple mating is present. For release rates $\sigma$ exceeding the eradication thresholds, the larval populations are effectively controlled in both the *First* and *Last* scenarios (Fig. 3A, case (3)). For a low initial infestation level of pests, the larval population is successfully controlled across all three ranges of release rates $\sigma$ (Fig. 3B), which corresponds to cases (2) and (3).

In the long term, the pest population seems easier to control in the *First* scenario (= Without MM) rather than in the *Last* scenario (Fig. 3). This observation is supported by two arguments: (1) The locally asymptotically stable infestation equilibrium is slightly higher in the *Last* case (Fig. 2). (2) The eradication threshold for the *Last* scenario is higher than for the *First* scenario (Fig. 2). A higher sterilized male release rate $\sigma$ will be required to control a population where females mate multiple times and use the last sperm (*Last*). When the release rate $\sigma$ is between the eradication thresholds of the *First* and *Last* scenarios ($\sigma = 450,000$, Fig. 2), the larval population is eradicated under the *First* scenario, whereas it persists at a controlled level under the *Last* scenario (Fig. 3A).

### 2.3.3 Transient dynamics

During the transient regime, i.e., the short term, it can be observed that the population decreases much more quickly at the beginning of the simulation in the *Last* scenario (Fig. 3A). Indeed, with fertilized females and larvae as initial conditions and by examining the model equations (2), in the *Last* scenario, we observe that fertilized females ($F_F$) can transition directly into the infertile female compartment ($F_I$) upon encountering sterilized males, due to the large number of sterilized males released from the initial time. In contrast, in the *First* case, fertilized females $F_F$ cannot directly become “infertile”; there is a delay. These females will produce larvae *L* and produce larvae from the beginning of the simulation.
<figure><img src="image_5.png" /><figcaption>**Fig 3** Simulations of larval density *L* over time for different initial infestation levels and sterilized male release rates (σ). (A) shows a high initial infestation level (initial conditions: *L* = 30,000, *M* = 100,000, *F*<sub>I</sub> = 0, *F*<sub>F</sub> = 100,000, *S* = σ/μ<sub>S</sub>), while (B) shows a low initial infestation level (initial conditions: *L* = 100, *M* = 300, *F*<sub>I</sub> = 0, *F*<sub>F</sub> = 300, *S* = σ/μ<sub>S</sub>) at the beginning of the SIT treatment. The black curves represent no sterilized male releases (σ = 0). Blue curves show the *First* sperm use scenario (τ<sub>I</sub> = τ<sub>F</sub> = 0), and orange curves show the *Last* scenario (τ<sub>I</sub> > 0 and τ<sub>F</sub> > 0). Thick curves correspond to σ = 300,000 (below the eradication threshold in Fig. 2), medium curves to σ = 450,000 (between the eradication thresholds in Fig. 2), and thin curves to σ = 600,000 (above the eradication threshold in Fig. 2).</figcaption></figure>

To further explore the impact of re-mating dynamics on control efficiency over time, we computed the differ-
ence in larval density reduction proportion between the *Last* and *First* scenarios, relative to a no-SIT baseline (*WR*)
($\frac{L_{First} - L_{Last}}{L_{WR}}$), as a function of time and sterilized male release rate $\sigma$ (Fig. 4). We assumed that the infestation starts
in a tunnel with the arrival of fertilized flies. This visual comparison highlights the transient advantages of the *Last*
scenario, especially during the initial deployment period. When release rates are moderate, the *Last* scenario achieves
a sharper initial decline in larval density due to the immediate sterilization of fertilized females, whereas in the *First*
scenario, a delay is introduced by the time required for offspring maturation and secondary mating events. However,
this advantage tends to fade over time, and in both scenarios, sufficiently high release rates lead to effective long-
term suppression, minimizing differences. Conversely, at low release rates, SIT impact is limited in both cases, and
differences remain marginal. Interestingly, for a narrow range of intermediate release intensities, the *First* scenario
becomes more effective in the very long term, likely due to its slower yet more stable impact on population dynamics.
This heatmap thus helps clarify how mating dynamics interact with release intensity to influence SIT efficiency across
different timescales.

Thus, in the short term (e.g., a timeframe of about 100 days, corresponding to the strawberry harvest period),
multiple mating can be advantageous, as long as τ<sub>F</sub> is not close to zero, meaning that not only females mated with
sterilized males re-mate.
<figure><img src="image_6.png" /><figcaption>**Fig 4** Heatmap showing the difference in larval density reduction between the *Last* and *First* scenarios, relative to a no-SIT situation (WR), across time (days, horizontal axis) and sterilized male release rates $\sigma$ (vertical axis). Positive values (orange tones) indicate greater reduction in the *Last* scenario; negative values (blue tones) indicate better performance of the *First* scenario. Simulations were run over 3000 days under a high-infestation setting (initial conditions: $L = 30,000$, $M = 100,000$, $F_I = 0$, $F_F = 100,000$). Parameters as in Table 2.</figcaption></figure>

## 2.4 Impact of refractory rates

The *eradication threshold* (4) depends on the refractory rates $\tau_F$ and $\tau_I$. These rates correspond to the inverse of the time during which females are unable to mate again. When the refractory rates are equal whatever mating just occurred ($\tau_F = \tau_I$) the *eradication threshold* is the same as without re-mating ($\tau_F = \tau_I = 0$). However, when females re-mate faster after a sterilized male than after a wild male ($\tau_I > \tau_F$), as illustrated in Figure 2, the *eradication threshold* increases, and vice versa when they re-mate slower.

The impact of re-mating in the *Last* scenario can therefore be examined according to the difference between refractory rates according to the nature of the last male the female mated with. So three cases emerge (see Fig. 5A):

* When $\tau_I = \tau_F$, females re-mate at the same rate regardless of whether they were previously mated with a wild or a sterilized male. In this case, in the long term, re-mating has a neutral effect on the control strategy, as function $\zeta(L^*)$ (3) is then independent of the refractory rates.

* When $\tau_I > \tau_F$, females previously mated with sterilized males re-mate faster than when mated with wild males. This scenario is the most detrimental to control efforts, as it increases the chances that females will be fertilized by wild males, thereby raising the *eradication threshold*.

* Conversely, when $\tau_F > \tau_I$, females previously mated with wild males re-mate faster than when mated with ster-
<figure><img src="image_7.png" /><figcaption>**Fig 5** Heatmaps of the *eradication threshold* (4) as a function of (A) refractory rates $\tau_F$ (x-axis) and $\tau_I$ (y-axis) and (B) the competitiveness of sterilized males $\eta$ (x-axis) and the refractory rate of females mated with sterilized males $\tau_I$ (y-axis). Default parameter values are represented by white crosses on heatmpas. (A) The parameter $\eta$ is fixed at 0.6, as listed in Table 2, and the white dashed line represents the scenario of equal re-mating ($\tau_F = \tau_I$). The top-left region corresponds to $\tau_F < \tau_I$, while the bottom-right region corresponds to $\tau_F > \tau_I$. (B) The parameter $\tau_F$ is fixed at 0.12, as detailed in Table 2. The eradication threshold was capped at one million; values exceeding this limit are included in the yellow area of the heatmap.</figcaption></figure>

ilized males. This scenario tends to facilitate population suppression since wild males have a higher probability of being replaced in subsequent matings. Biologically this case seems less realistic.

However, the two refractory rates ($\tau_I$ and $\tau_F$) do not exert the same influence on the *eradication threshold* (Fig.5A).

When $\tau_F > \tau_I$, the *eradication threshold* is relatively insensitive to changes in $\tau_F$ and, to a lesser extent, in $\tau_I$.

In contrast, when $\tau_I > \tau_F$, the *eradication threshold* becomes highly sensitive to even small increases in $\tau_I$ and $\tau_F$, especially when the re-mating rates are small, which can cause the required release ratio of sterilized males to reach impractical levels, rendering the population uncontrollable. So re-mating rates tend to increase the *eradication threshold* but there are interactions between the two parameters.

The *eradication threshold* depends not only on the refractory rates but also on the competitiveness $\eta$ of sterilized males (Fig.5B). Recall that when $\eta = 0$, sterilized males are not competitive at all, whereas when $\eta = 1$, sterilized males are as competitive as wild males. Importantly, the interaction between the re-mating rate and the competitiveness of sterilized males is not simply linear. While the *eradication threshold* is highly sensitive to variations in $\tau_I$, it is much less affected by changes in $\eta$, provided $\eta$ remains sufficiently high. As expected, the most advantageous case for
control, where the *eradication threshold* is the lowest, occurs when sterilized males are as competitive as wild males ($\eta = 1$) and when females previously mated with a sterilized male re-mate as little as possible ($\tau_I$ close to 0, as they remain unavailable and “sterilized”) (Fig. 5B). We observe that the interaction between sterilized male competitiveness $\eta$ and the refractory rate $\tau_I$ influences the *eradication threshold*. Specifically, to return to biological values, when competitiveness exceeds the fixed value of 0.6, variations in $\tau_I$ have only a minor effect on the *eradication threshold* (Fig. 5B). In contrast, for lower values of competitiveness, $\tau_I$ plays a more significant role. However, even with low re-mating rates, $\eta$ must still be sufficiently high to prevent the *eradication threshold* from reaching impractical levels that would require unattainable release ratios of sterilized males.

# 3 Agent-based model: Impact of sperm use bias on SIT effectiveness

The previous section highlighted that female polyandry has a significant impact on the release efforts required for efficient SIT deployment. Furthermore, a question still remains unsolved: how are the population dynamics affected by multiple mating in cases of complex biases in the use of sperm beyond the extreme cases of using only the sperm of the *First* or *Last* mating? To answer this question within a pertinent timescale and in an agricultural context with non-perennial crops, we developed an agent-based model to explore how potential sperm use biases impact pest control with SIT, focusing on proxies of SIT efficiency on a short timescale.

## 3.1 Agent-based model description

As part of this study, an agent-based model was developed to simulate the reproductive dynamics of *D. suzukii* females under SIT deployment context. The model, implemented in Python using the Mesa package (Masad and Kazil, 2015), operates in discrete time steps (one step corresponds to one day) and assumes a spatially well-mixed population. A brief description of the model is provided in the text, but the full ODD-based (Overview, Design concepts, and Details) description is provided in Appendix E to ensure clarity and reproducibility. In this model, we maintain similar assumptions to those established in the previous compartmental model, but we explore more scenarios of sperm use bias.
<figure><img src="image_8.png" /><figcaption>**Fig 6** Flowchart describing the steps of the agent-based model. The top black rectangle outlines the general model framework, beginning with the initialization phase: (1) input of data (biological data linked to the life cycle, release quantity), (2) creation of counters to track population densities, (3) implementation of the scheduler, (4) creation of agents at the initial time step (four types: sterilized males, wild males, larvae, and females, and (5) parametrization of the data collector to specify the data to be recorded at each step. The subsequent steps, executed at each time step, include: advancing the simulation time, releasing sterilized males (creating new sterile male agents), performing agent-specific steps for each category, collecting data, and repeating the cycle with the next time step. The bottom rectangle illustrates, in algorithmic diagrams, the specific steps for each type of agent: sterilized males (pink), wild males (green), larvae (red), and females (purple). These diagrams summarize the sequence of decisions and events for each agent category. The agent-based model is described in further detail in Section 3.</figcaption></figure>
The model includes four types of agents (Fig. 6): wild males, sterilized males, females, and larvae (a compartment encompassing eggs, larvae, and pupae). Each agent is characterized by a set of biological traits such as longevity and mortality. Additionally, larvae have an emergence time, while females are also defined by their fecundity (number of eggs laid), their post-mating refractory time, and their ability to store and use sperm from multiple mates.

A central feature of the model is indeed the explicit representation of polyandrous female behavior. Females may mate multiple times during their lifetime, accumulating sperm from different males. The use of this sperm follows one of six predefined scenarios:

* **First = Without MM**: exclusive use of the sperm from the first mate, equivalent to a case without multiple mating;

* **Last**: exclusive use of the sperm from the last mate;

* **Mixed**: sperm use according to the proportion of fertile and sterilized sperm stored;

* **Preference W** (W for **Wild**): complete preference for fertile sperm, if available;

* **Preference I** (I for **Intermediate**): relative preference for fertile sperm over sterilized sperm. In this study: 80% preference for fertile sperm over sterilized sperm, when both are stored;

* **Preference S**: complete preference for sterilized sperm, if available.

These scenarios directly influence the probability that a given egg is fertilized by a fertile versus a sterilized male, and thus affect population growth.

The model also allows a flexible configuration of sterilized male release strategies, including the timing and intensity of releases, to investigate their combined effect with female re-mating behavior.

## 3.2 Results

Simulation results highlight how different sperm use biases influence larval density and SIT success. The number of larvae was chosen as the indicator of SIT success because they are responsible for a large proportion of fruit damage. Figure 7 illustrates larval population dynamics and cumulated population reduction, compared to a situation without SIT (WR), under different sperm use biases (colors) and sterilized male release intensities (A–C panels).
Simulation results, obtained for low infestation levels, show a strong short-term efficacy of SIT, with no pattern
differences between sperm use biases at the beginning of the season (Fig. 7). Increasing the number of sterilized
males released consistently enhances SIT effectiveness, leading to high larval population reduction. Even in the least
favorable scenario for SIT, where females prefer fertile sperm when stored ("Preference W"), a substantial increase in
release effort allows for nearly complete larval suppression.

A noticeable difference appears between the *First* and *Last* scenarios (σ between 10,000 and 20,000 in Fig. 7). The *Last* scenario, where females use sperm from their latest mate, proves more effective in reducing larval density than the *First* scenario. However, this difference diminishes as the number of released sterilized males increases, confirming trends observed in the compartmental model (Fig. 4).

The *First* and *Preference I* scenarios yield very similar results, which is interesting given that the former represents
a situation with no re-mating, while the latter allows re-mating but with a slight preference for fertile sperm—a more
biologically realistic setting. The *Mixed* scenario leads to intermediate larval reduction levels between the *First* and
*Last* scenarios.

Overall, at sufficiently high release rates, differences between sperm use strategies become less pronounced, with
SIT achieving successful larval population reduction across all tested scenarios (see Fig. 8).

In addition, simulation results, obtained for high infestation levels, with initial conditions including fertilized fe-
males and larvae, reproducing an early-season infestation scenario in a tunnel where mated flies and larvae are already
present in some fruits, reveal a dynamic inversion over time between sperm use scenarios, consistent with findings from
the compartmental model (see Sect. 2.3.3). In particular, the *Last* scenario shows a faster initial decline in larval density
compared to *First* (= Without MM), but may lead to less effective suppression in the long term if release rates are not
sufficiently high. This temporal shift in effectiveness is visible as in the compartmental model (details in Appendix F),
reinforcing the robustness of this pattern across modeling approaches. However, this result remains dependent on the
initial conditions and thus on the simulated starting context: for example, if the simulation starts only with larvae and
no adults, the dynamics with and without re-mating are equivalent, with no delay effect.
<figure><img src="image_9.png" /><figcaption>**Fig 7** Dynamics of larval densities (A-B-C) and percentage reduction in areas under the curves (D-E-F) for different sperm bias and three sterilized male release rates σ: (A-D) σ = 10,000, (B-E) σ = 15,000, (C-F) σ = 20,000.</figcaption></figure>

(A-B-C) Larval densities over time for different sperm use scenarios: *WR* (black) represents the reference case Without Release (no sterile insect technique applied), *First* (dark blue) corresponds to females using sperm from the first male they mated with, *Last* (orange) represents females using sperm from the last male they mated with, *Mixed* (pink) assumes a proportional use of fertile sperm linked to the proportion of eggs laid and *Preference* reflects female preference for a specific sperm category, with three subcases: *Preference W* (gray) indicates total preference for fertile sperm, *Preference I* (light blue) indicates intermediate preference for fertile sperm, and *Preference S* (yellow) indicates total preference for sterile sperm.

(D-E-F) Histograms showing the percentage reduction in areas under the curves for the sperm use scenarios described in A-B-C, relative to the reference scenario *WR* without release, over a 100-day period. The error bars in these graphs represents the 95% confidence interval (CI 95%). Simulations were conducted under the same initial conditions: 1000 wild males, 1000 females, and 0 larvae. Results are based on 100 simulations performed with the agent-based model described in Section 3.
<figure><img src="image_10.png" /><figcaption>**Fig 8** Graph showing the percentage reduction in areas under the curve (AUC) over a 100-day period, for different sperm use scenarios, relative to the reference scenario *WR* without release (no sterile insect technique applied) as a function of sterile male released rates. Simulations were conducted under the same initial conditions: 1000 wild males, 1000 females, and 0 larvae. Each point is the average over 100 simulations performed with the agent-based model described in Section 3.</figcaption></figure>

**4 Discussion**

Our results highlight how multiple mating and sperm use biases influence SIT outcomes. In our study, the compartmental model offers a rapid and intuitive understanding of global dynamics, whereas the agent-based model captures more nuanced biological mechanisms and stochastic variability. Used together, they provide a more comprehensive framework for evaluating the impact of sperm use biases on SIT outcomes. Regarding *Drosophila suzukii*, in the long term, polyandry is disadvantageous as it requires additional sterilized male releases to control pest populations, making SIT more costly to implement. Indeed, the existence of re-mating behavior raises the *eradication threshold*, as females previously mated with sterilized males re-mate more, requiring more sterilized males for effective population control.

In contrast, polyandry can be beneficial in the short term, such as during a single non-perennial fruit season, by further reducing pest populations. However, the difference between sperm use biases remains limited. These results depend on the parameters, particularly the refractory rates and sterilized male competitiveness, which influence the release effort required.

Intuitively, if only females that previously mated with a sterilized male re-mate, we can expect that re-mating would complicate the effectiveness of SIT. However, if refractory rates are similar regardless of the female previous mate (wild or sterilized male), multiple mating can be beneficial in the short term. Indeed, at the onset of an infestation in a crop
tunnel, when fertilized females and larvae are already present, the transition between reproductive statuses occurs more rapidly when females re-mate, whereas without re-mating, a full generation is needed before a noticeable shift in the proportion of offspring sired by sterilized males occurs. These results highlight the importance of reproductive dynamics in SIT efficiency: effectiveness is not determined solely by the quantity of released males but also by their ability to engage in mating. Our findings align with previous studies, such as Barclay (1984), which has explored the implications of multiple mating on SIT. Barclay (1984) presented a model assuming strict female monogamy and compared it to a scenario where females frequently mate. He found that the system is easier to control under monogamy, particularly when mating propensity is low, although eradication takes longer. Furthermore, stability does not seem to be affected by mating frequency in his model. Additionally, Dyck et al. (2021) discussed how multiple mating complicates control but does not disqualify a pest species as a candidate for SIT. While this review covers various studies on SIT, it does not highlight cases where multiple mating is explicitly beneficial. Instead, it suggests that while multiple mating may introduce complications, it does not render SIT ineffective. Our study provides a more unexpected perspective: in the short term, multiple mating can actively contribute to the success of SIT, rather than merely being a neutral or complicating factor. In addition, we demonstrate the existence of an interaction between release strategy and the species reproductive mode (Fig. 4).

Our simulations of the agent-based model reveal that the impact of different sperm use biases varies depending on the release effort of sterilized males. Quantitatively, over a range of release rates (from 10,000 to 60,000 sterilized males released) within an arbitrarily defined season, the larval population is reduced by 43.7% to 97.6% when females preferentially use fertile sperm, the most unfavorable scenario, whereas it is reduced by 79.3% to 99.7% when females utilize the last sperm, a more favorable scenario. This highlights a strong short-term effectiveness of pest reduction through increased release efforts, regardless of the sperm use bias tested (see Fig. 8). However, scenarios where females favor fertile sperm or without multiple mating tend to reduce the short-term effectiveness of SIT. Contrary to our initial hypothesis, which assumed that the sperm use bias would have a strong impact on the ability of SIT to suppress populations, and to what is suggested in the literature (Pérez-Staples et al., 2013), the quantitative difference in SIT effectiveness depending on sperm use bias remains limited.

The biases in sperm use were arbitrarily defined in our simulations but may correspond to natural variability in sperm storage and use. *D. suzukii* females possess two spermathecae (Avanesyan et al., 2017) and can produce offspring
from up to five different males (Puppato et al., 2023). Some data suggest a slight preference for the sperm from the first mating (Chen et al., 2022), while others indicate that mating duration is positively correlated with male ejaculate investment (Lanouette et al., 2017), influencing sperm use probability in polyandrous contexts (Bretman et al., 2009). If sterilized males transfer less sperm due to shorter matings, females are more likely to preferentially use fertile sperm simply because it is more abundant. The first mating duration has been observed to be shorter than the second one (Chen et al., 2022), and irradiation is known to alter the quantity of sperm transferred to females (Pérez-Staples et al., 2013). Additionally, we cannot exclude the possibility of a more active mechanism, where females might “recognize” and react to the mating status of their partner, as suggested by research on *Wolbachia*-infected males.

In fact, multiple mating and sperm use biases are not only relevant to the success of SIT but also impact other pest control strategies, such as those involving *Wolbachia*. These factors directly influence reproduction dynamics, bacterial spread, and the efficiency of population replacement or suppression methods. For instance, *Wolbachia* infection alters female receptivity to mating and re-mating frequency, particularly after mating with an infected male (He et al., 2018; Osorio et al., 2023). He et al. (2018) showed that females mated with infected males exhibited lower refractory rate, potentially enhancing *Wolbachia*-based population replacement strategies. Conversely, Osorio et al. (2023) found that females, after mating with infected males, showed an increased tendency to re-mate, potentially undermining population replacement efforts if re-mating occurs with uninfected males. These findings emphasize the need for a deeper understanding of mating dynamics to optimize pest population control strategies effectively.

To conclude, our findings highlight the critical role of female re-mating dynamics in shaping the effectiveness of SIT strategies. In particular, rapid re-mating following copulation with a sterilized male, combined with biased sperm use in favor of fertile males, can drastically increase the number of sterilized males required, sometimes beyond operational feasibility. While improving sterilized male competitiveness remains essential, better understanding and integrating the biological determinants of re-mating (e.g., refractory period variability, sperm selection mechanisms) could open new avenues to optimize SIT deployment. In this context, moving beyond eradication to short-term population suppression strategies, particularly in seasonal crops, offers a pragmatic path forward. Future models should also account for damage caused by sterile oviposition, which remains a critical factor when evaluating the environmental impact of female mating behavior on SIT efficiency.
## Code and data Availability

All analyses were performed using the Python programming language (version 3.9.12). The complete script coded in Python, specifying the versions of the libraries used, is available on the following link:

https://gitlab.com/marinecourtois/sit_remating_abm.

## Acknowledgments

The authors are 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. We would like to thank the CTIFL for sharing biological data cited in this work. We also extend our gratitude to Crésus Kounoudji and Taha Belkhayate for their contributions to this topic during their Master's internships. Finally, we sincerely thank the GenoToul platform for providing access to the computing cluster, which enabled us to run the agent-based model simulations developed in this paper.
# Appendix A: Description of Biological Parameters and Data Sources

## A.1 Details on the compartmental model parameters

The biological parameters used to model *Drosophila suzukii* were estimated from several studies and datasets and reported in Table 2.

**Estimation of the larval mortality rate** We assume that the survival probability of immature sub-stages (eggs, larvae and pupae) can be approximated by an exponential decay function:

$$ \phi(t) = e^{-\mu_L t}, $$

where $\mu_L$ is a constant mortality rate, and $t$ is the development time (in days). We aim to estimate the overall survival probability from egg to adult emergence.

To do so, we used stage-specific survival probabilities extracted from the study by Emiljanowicz et al. (2014), where the authors provide the proportion of individuals alive at age $x$ that survive through each developmental interval. Specifically, the proportion of individuals surviving the egg stage is 0.868. The larval stage is divided into three substages, with survival proportions of 0.913, 0.952, and 0.900 respectively. Finally, the survival proportion during the pupal stage is 0.944.

To compute the total survival probability from egg to adult, we multiply the proportions across all sub-stages:

$$ \phi = 0.868 \times 0.913 \times 0.952 \times 0.900 \times 0.944 = 0.640. $$

The total development time from egg to adult is approximately 12 days, so mortality rate $\mu_L$ can be deduced as follows:

$$ \mu_L = -\frac{\ln(\phi)}{t} = -\frac{\ln(0.640)}{12} = 0.037. $$

This estimation provides a biologically grounded value for $\mu_L$ used in the population model (2).

**Estimation of the adult mortality rate** Female ($\mu_F$) and wild male ($\mu_M$) mortality rates were estimated from figure 4 of Papanastasiou et al. (2021), focusing on the summer morph in the mainland condition. Using the R package
digitize (Poisot, 2011), we extracted the minimum, maximum, median, first quartile (Q1), and third quartile (Q3) values from the box plots for both females and males.

For females, the extracted longevity values were: minimum = 22.09, Q1 = 68.51, median = 92.82, Q3 = 101.66, and maximum = 143.86. For males, the corresponding values were: minimum = 29.83, Q1 = 60.77, median = 76.24, Q3 = 89.50, and maximum = 129.28.

To estimate the mean longevity, we applied the formula proposed by Wan et al. (2014), which is appropriate when the sample size is between 15 and 70 (here sample size is equal to 50 for females and for males) and when the minimum, maximum, median, Q1, and Q3 are available. The formula used is:

$$Mean = \frac{\min + 2 \times Q_1 + 2 \times \text{median} + 2 \times Q_3 + \max}{8}$$

This yields estimated mean longevities of 86.5 days for females and 76.5 days for males. Mortality rates were then calculated as the inverse of the estimated longevities, leading to $\mu_F = 0.012$ and $\mu_M = 0.013$.

**Estimation of egg-laying rate, larval hatching rate and sex ratio** The daily egg-laying rate ($\omega$) was taken from Emiljanowicz et al. (2014), which reported a mean production of $5.7 \pm 0.24$ eggs per day, rounded up to 6 in the parameter set. From egg to adult, individuals develop in 13 days, leading to a larval hatching rate $\nu$ of $\frac{1}{13} = 0.08$ days$^{-1}$. The sex ratio ($p$) was considered equal to 0.5, as indicated by Emiljanowicz et al. (2014).

**Estimation of the sterilized male competitiveness** The competitiveness of sterilized males ($\eta$) was calculated based on data from Lanouette et al. (2020), which indicated that irradiated males achieved a mating success rate of 37.5%, compared to 62.5% for wild males. Parameter eta in model (2) represents the relative mating success of irradiated males compared to wild males, so $\eta = \frac{37.5}{62.5} = 0.60$.

**Estimation of the carrying capacity ($K$)** To estimate the carrying capacity $K$, we used data on strawberry production and larval infestation (source: Occitanie Chamber of Agriculture, https://occitanie.chambre-agriculture.fr). Over the course of a year, corresponding to approximately 100 days of production, the daily yield is about 0.036 kg/m², or 18 kg of strawberries per day in a 500 m² tunnel. Given that a strawberry weighs approximately 20 grams, this corresponds to around 900 strawberries produced daily in the tunnel.
According to experiments conducted by the CTIFL (Centre Technique Interprofessionnel des Fruits et des Légumes, personal communication), each strawberry contains on average 20 larvae, based on fruit sampling and larval counts [min = 0, max = 265]. Assuming that only ripe fruits can host larvae (i.e., fruits that are likely to be harvested), this results in approximately 18,000 larvae per day across the 500 m² production area.

Several studies have shown that *D. suzukii* females strongly prefer laying eggs in ripe fruits, which are typically the fruits harvested on the same day, and thus at the end of their maturation process. This oviposition preference is well documented by Karageorgi et al. (2017), who demonstrated that *D. suzukii* evolved to target healthy, intact, ripening fruits, unlike most other Drosophila species that prefer decaying fruits. Although females can occasionally lay eggs in unripe or overripe fruits when no better options are available, such behavior remains marginal. Kienzle et al. (2020) confirmed that even under conditions of limited resource availability, females still predominantly choose ripe, non-fermented fruits. These findings highlight a marked, though flexible, preference for ripe fruits as oviposition sites.

In our model, we assume that the resource (i.e., the number of ripe fruits available for oviposition) is constant over time. Since harvest takes place every two days, we consider that, at any given time, the resource available to the flies corresponds to twice the daily production.

Thus, the carrying capacity *K*, defined as the maximum number of larvae that can be supported by the available resource, is estimated to be:

$$K = 2 \times \text{daily number of strawberries} \times \text{average number of larvae per fruit} = 2 \times 900 \times 20 = 36,000.$$

$$K = 2 \times 900 \times 20 = 36,000 \text{larvae per tunnel per day.}$$

**Estimation of the refractory rates** Finally, the refractory rates for “fertilized” and “sterilized” females ($\tau_F$ and $\tau_I$) were estimated from Table 1 of Krüger et al. (2019), which reports the average refractory periods depending on the type of male with which the females previously mated. Specifically, the average refractory period was 8.25 days for females that mated with wild males and 7.40 days for females that mated with sterilized males. The refractory rates were obtained by taking the inverse of these average values, resulting in $\tau_F \approx 0.12 \text{ days}^{-1}$ and $\tau_I \approx 0.14 \text{ days}^{-1}$.
## A.2 Details on the agent-based model parameters

All data extracted from the literature to calibrate the agent-based model are listed in Table 3 and detailed in the previous section. In addition, the table specifies the type of distribution used (uniform $U$, Bernoulli $B$) or if a fixed value was applied for each parameter.

<table><caption>**Table 3** Range of values used to estimate parameters for the agent-based model 3 on *Drosophila suzukii*</caption><thead><tr><th>Parameter/Description</th><th>Value or Range</th><th>Unit</th><th>Reference</th></tr></thead><tbody><tr><td>Sterilized male release rate</td><td>[0 - 60,000]</td><td>ind.days<sup>-1</sup>.ha<sup>-1</sup></td><td>-</td></tr><tr><td>Sterilized male lifespan</td><td>U(1, 36)</td><td>day</td><td>(Lanouette et al., 2017)</td></tr><tr><td>Sterilization cost</td><td>0.40</td><td>-</td><td>(Lanouette et al., 2020)</td></tr><tr><td>Wild male lifespan</td><td>U(31, 129)</td><td>day</td><td>(Papanastasiou et al., 2021)</td></tr><tr><td>Average number of eggs laid</td><td>6</td><td>eggs per female per day</td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Mortality rate of eggs</td><td>0.13</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Mortality rate of larvae stage L1 (1-2 days)</td><td>0.087</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Mortality rate of larvae stage L2 (2-3 days)</td><td>0.048</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Mortality rate of larvae stage L3 (3-6 days)</td><td>0.10</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Mortality rate of pupae</td><td>0.056</td><td>day<sup>-1</sup></td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Larval emergence</td><td>U(10, 20)</td><td>day</td><td>(Emiljanowicz et al., 2014)<sup>[1]</sup></td></tr><tr><td>Sex ratio</td><td>B(1, 0.5)</td><td>-</td><td>(Emiljanowicz et al., 2014)</td></tr><tr><td>Carrying capacity</td><td>36,000</td><td>larvae</td><td>Calculated<sup>[2]</sup></td></tr><tr><td>Female lifespan</td><td>U(21, 144)</td><td>day</td><td>(Papanastasiou et al., 2021)</td></tr><tr><td>Refractory period for fertilized females</td><td>U(3, 13)</td><td>day</td><td>(Krüger et al., 2019)</td></tr><tr><td>Refractory period for infertile females</td><td>U(4, 11)</td><td>day</td><td>(Krüger et al., 2019)</td></tr></tbody></table>

$$ \text{Eggs per female} = \text{Average number of eggs laid} \times \left( 1 - \frac{\text{larvae population size}}{\text{Carrying capacity}} \right). $$

## Appendix B: Model reduction

We adopt a slow-fast approximation approach used in ecology (Auger et al., 2008) to reduce the system dimension. We assume that mating occurs very rapidly when females are sexually mature and available (Revadi et al., 2015), implying

<footnote n="1">¹ In the literature, larval emergence occurs at 12.8 ± 0.2 days (Emiljanowicz et al., 2014). However, we chose a broader range to account for environmental variability.</footnote>

<footnote n="2">² Carrying capacity is set to 36,000 larvae for a strawberry tunnel of 500 m². Larval growth follows a logistic model to account for density-dependent competition. The mean number of eggs laid per female agent per day is adjusted, to represent the fact that not all larvae will be able to develop, as follows:</footnote>
that $\chi$ is much larger than the other parameters. We define $X$ such as $\chi = \frac{X}{\epsilon}$, with $0 < \epsilon \ll 1$. Model (1) becomes:

$$ 
\left\{
\begin{aligned}
\dot{L} &= \omega(1 - \frac{L}{K})F_F - (\mu_L + \nu)L, \\
\dot{M} &= \nu p L - \mu_M M, \\
\dot{F}_U &= \nu(1-p)L + \tau_F F_F + \tau_I F_I - \mu_F F_U - \frac{X}{\epsilon}F_U, \\
\dot{F}_I &= \frac{X}{\epsilon}\frac{\eta S}{M+\eta S}F_U - (\tau_I + \mu_F)F_I, \\
\dot{F}_F &= \frac{M}{\epsilon(M+\eta S)}F_U - (\tau_F + \mu_F)F_F, \\
\dot{S} &= -\mu_S S + \sigma.
\end{aligned}
\right.
$$

We also introduce a new variable:

$$ G_U = \chi F_U = \frac{X}{\epsilon} F_U. $$

This change has the advantage that all fast terms involving $\chi F_U$ now appear as $G_U$, simplifying the separation of time scales. The original model becomes:

$$ 
\left\{
\begin{aligned}
\dot{L} &= \omega \left(1 - \frac{L}{K}\right) F_F - (\mu_L + \nu)L, \\
\dot{M} &= \nu p L - \mu_M M, \\
\dot{G}_U &= \frac{X}{\epsilon} \left[\nu(1-p)L + \tau_F F_F + \tau_I F_I - \mu_F \frac{\epsilon}{X} G_U\right] - \frac{X}{\epsilon} G_U, \\
\dot{F}_I &= \frac{\eta S}{M + \eta S} G_U - (\tau_I + \mu_F) F_I, \\
\dot{F}_F &= \frac{M}{M + \eta S} G_U - (\tau_F + \mu_F) F_F, \\
\dot{S} &= -\mu_S S + \sigma.
\end{aligned}
\right.
$$

Introducing the fast time-scale $\tilde{t} = \frac{t}{\epsilon}$ and defining $L', M', F'_U, F'_I, F'_F, S'$ as the derivatives with respect to $\tilde{t}$, we obtain:
$$
\left\{
\begin{aligned}
L' &= \epsilon \left[ \omega \left( 1 - \frac{L}{K} \right) F_F - (\mu_L + \nu)L \right], \\
M' &= \epsilon(\nu p L - \mu_M M), \\
G'_U &= X \left[ \nu(1-p)L + \tau_F F_F + \tau_I F_I - G_U - \mu_F \frac{\epsilon}{X} G_U \right], \\
F'_I &= \epsilon \left( \frac{\eta S}{M + \eta S} G_U - (\tau_I + \mu_F) F_I \right), \\
F'_F &= \epsilon \left( \frac{M}{M + \eta S} G_U - (\tau_F + \mu_F) F_F \right), \\
S' &= \epsilon(-\mu_S S + \sigma).
\end{aligned}
\right.
$$

In this fast time-scale, letting $\epsilon$ tend to 0, we obtain:

$$
\left\{
\begin{aligned}
L' &= 0, \\
M' &= 0, \\
G'_{U} &= X [\nu(1-p)L + \tau_{F}F_{F} + \tau_{I}F_{I} - G_{U}], \\
F'_{I} &= 0, \\
F'_{F} &= 0, \\
S' &= 0.
\end{aligned}
\right.
$$

This system implies that $L, M, F_I, F_F$, and $S$ are constant in the fast time scale, while $G_U$ converges exponentially fast to:

$$G_U = \nu(1-p)L + \tau_F F_F + \tau_I F_I.$$

By replacing this quasi-equilibrium expression back into the slow system (for $\dot{F}_I$ and $\dot{F}_F$), we obtain the reduced model:

$$
\left\{
\begin{aligned}
\dot{L} &= \omega \left(1 - \frac{L}{K}\right) F_F - (\mu_L + \nu)L, \\
\dot{M} &= \nu p L - \mu_M M, \\
\dot{F}_I &= \nu(1-p) \frac{\eta S}{M+\eta S} L + \tau_F \frac{\eta S}{M+\eta S} F_F - (\mu_F + \tau_I \frac{M}{M+\eta S}) F_I, \\
\dot{F}_F &= \nu(1-p) \frac{M}{M+\eta S} L + \tau_I \frac{M}{M+\eta S} F_I - (\mu_F + \tau_F \frac{\eta S}{M+\eta S}) F_F, \\
\dot{S} &= -\mu_S S + \sigma.
\end{aligned}
\right.
$$
This reduced model captures the slow dynamics under the assumption that mating occurs almost instantaneously after females become available.

# Appendix C: Analysis of the pest-free equilibrium $E_0^*$

## C.1 Equilibrium $E_0^*$

The pest-free equilibrium of system (2) is given by:

$$E_0^* = (L^*, M^*, F_I^*, F_F^*, S^*) = (0, 0, 0, 0, \frac{\sigma}{\mu_S}).$$

## C.2 Stability analysis of $E_0^*$

We analyze the local stability of the pest-free equilibrium $E_0^*$ by evaluating the Jacobian matrix of the $(L, M, F_I, F_F)$ subsystem of the reduced model (2) at $S = S^*$. The general Jacobian matrix is given by:

$$J = \begin{pmatrix} -\frac{\omega}{K} F_F - (\mu_L + \nu) & 0 & 0 & \omega(1 - \frac{L}{K}) \\ \nu p & -\mu_M & 0 & 0 \\ \nu(1-p) \frac{\eta_S}{M+\eta_S} & -\frac{\eta_S}{(M+\eta_S)^2} (\nu(1-p)L - \tau_F F_F - \tau_I F_I) & -(\mu_F + \tau_I \frac{M}{M+\eta_S}) & \tau_F \frac{\eta_S}{M+\eta_S} \\ \nu(1-p) \frac{M}{M+\eta_S} & \frac{\eta_S}{(M+\eta_S)^2} (\nu(1-p)L + \tau_F F_F + \tau_I F_I) & \tau_F \frac{M}{M+\eta_S} & -(\mu_F + \tau_F \frac{\eta_S}{M+\eta_S}) \end{pmatrix}. \quad (5)$$

At the equilibrium $E_0^*$, where $L = M = F_I = F_F = 0$, the Jacobian matrix becomes:

$$J(E_0^*) = \begin{pmatrix} -(\mu_L + \nu) & 0 & 0 & \omega \\ \nu p & -\mu_M & 0 & 0 \\ \nu(1-p) & 0 & -\mu_F & \tau_F \\ 0 & 0 & 0 & -(\mu_F + \tau_F) \end{pmatrix}$$

This matrix has a block triangular structure, which allows us to identify its eigenvalues directly from the eigenvalues of each block on the diagonal:

* The lower-right block $-(\mu_F + \tau_F)$ (isolated by the black vertical line) is an eigenvalue.

* The upper-left 3×3 block is lower-triangular and yields eigenvalues $-(\mu_L + \nu)$, $-\mu_M$ and $-\mu_F$.

**Conclusion:** Since all eigenvalues of $J(E_0^*)$ have strictly negative real parts, the pest-free equilibrium $E_0^*$ is locally asymptotically stable. This conclusion holds regardless of the value of $S^*$.
# Appendix D: Analysis of infestation equilibria

## D.1 Equilibria of the subsystem

For the $(L, M, F_I, F_F)$ subsystem at $S = S^*$, the equilibria of system (2) are the solutions of the following equations.

We introduce the notation $\gamma = \frac{\nu p}{\mu_M}$ to simplify expressions:

$$ \Leftrightarrow \begin{cases} F_F^* = \frac{(\mu_L + \nu)}{\omega(1 - \frac{L^*}{K})} L^*, \\ M^* = \frac{\nu p}{\mu_M} L^* = \gamma L^*, \\ F_I^* = \left( \frac{\nu(1-p)}{\mu_F} - \frac{(\mu_L + \nu)}{\omega(1 - \frac{L^*}{K})} \right) L^*, \\ \nu(1-p) \frac{\eta S^*}{M^* + \eta S^*} L^* + \tau_F \frac{\eta S^*}{M^* + \eta S^*} F_F^* - (\mu_F + \tau_I \frac{M^*}{M^* + \eta S^*}) F_I^* = 0. \end{cases} \quad (6) $$

with $F_I^*$ obtained from $\dot{F}_I + \dot{F}_F = 0$. Substituting $F_I^*$ and $F_F^*$ into the last equation, we obtain:

$$ \nu(1-p) \frac{\eta S^*}{M^* + \eta S^*} L^* + \tau_F \frac{\eta S^*}{M^* + \eta S^*} \frac{(\mu_L + \nu)}{\omega(1 - \frac{L^*}{K})} L^* - \left( \frac{\mu_F(M^* + \eta S^*) + \tau_I M}{M^* + \eta S^*} \right) \left( \frac{\nu(1-p)}{\mu_F} - \frac{(\mu_L + \nu)}{\omega(1 - \frac{L^*}{K})} \right) L^* = 0. $$

Replacing $M^*$ by its expression in (6) and simplifying, we arrive at an equation that solely depends on the density of larvae $L$:

$$ -\frac{\nu(1-p)\omega(\mu_F + \tau_I)\gamma}{K\eta\mu_F(\mu_L + \nu)(\mu_F + \tau_F)}(L^*)^2 + \frac{(\mu_F + \tau_I)(\nu(1-p)\omega - \mu_F(\mu_L + \nu))\gamma}{\eta\mu_F(\mu_L + \nu)(\tau_F + \mu_F)}L^* = S^*. $$

We define the basic reproduction number $R_0 = \frac{\omega(1-p)\nu}{\mu_F(\mu_L + \nu)}$, and substitute it, along with $S^* = \frac{\sigma}{\mu_S}$, into the equation above to obtain:

$$ -\frac{\gamma\mu_S(\mu_F + \tau_I)R_0}{K\eta(\mu_F + \tau_F)}(L^*)^2 + \frac{\gamma\mu_S(\mu_F + \tau_I)(R_0 - 1)}{\eta(\mu_F + \tau_F)}L^* = \sigma. \quad (7) $$

Thus, $L^*$ is given by the intersection between the straight line $y = \sigma$ and the parabola $y = \zeta(L^*)$ 2, where:

$$ \zeta(L^*) = \frac{\gamma\mu_S(\mu_F + \tau_I)}{\eta(\mu_F + \tau_F)} \left( -\frac{R_0}{K} (L^*)^2 + (R_0 - 1)L^* \right). \quad (8) $$

When $\sigma = 0$, the roots of (8) are easy to deduce: $L^* = 0$ and $L^* = K(1 - \frac{1}{R_0})$.
## D.2 Stability analysis of infestation equilibria

**Case** $R_0 \le 1$ The parabola $\zeta(L^*)$ admits a zero root and a non positive root. In this case, since the highest-degree term is negative, the equation $\zeta(L^*) = \sigma$ has no solution, and there is no infestation equilibrium.

**Case** $R_0 > 1$ The parabola $\zeta(L^*)$ admits a zero root and a positive root. In this case, the equation $\zeta(L^*) = \sigma$ may have 2 or 0 solutions: $E_1^*$ and $E_2^*$, which correspond to the points of intersection between the straight line $y = \sigma$ and the parabola $\zeta(L^*)$.

The summit of the parabola is reached for $L^*$ in the middle of the two roots $\zeta(L)$, so that, when $E_1^*$ and $E_2^*$ exist, $L^*(E_1^*) < \frac{K}{2}(1 - \frac{1}{R_0}) < L^*(E_2^*)$. Also numerical exploration of the Jacobian matrix (5) of the $(L, M, F_F, F_I)$ subsystem at $S = S^*$ of model (2) shows that $E_1^*$ is always unstable and $E_2^*$ asymptotically stable. This analysis was performed for a range of release rates $\sigma$ to explore the bifurcation structure of the system.

# Appendix E: Agent-based model description with ODD protocol

The model is described using aspects of the Overview, Design concepts, Details (ODD) protocol (Grimm et al., 2020) and was implemented using Python 3.9.12 with the Mesa package (Masad and Kazil, 2015), specifically designed for agent-based models. Numerical simulations were calibrated on the species *D. suzukii*.

## E.1 Overview

### E.1.1 Purpose

The model aims to simulate the reproductive dynamics of *D. suzukii* females, which can mate multiple times and store sperm in a spermatheca, i.e. a specialized organ in female that stores sperm after mating. Different sperm biases are tested to evaluate their influence on population dynamics when sterilized males are released. Although not the primary objective of this study, the model also serves as a framework to explore release scenarios, by assessing how different release rates affect population dynamics.

### E.1.2 Entities, State Variables, and Scales

The model consists of four types of agents: sterilized males *S*, wild males *M*, larvae *L* (including all immature stages: eggs, larvae and pupae) and females *F*.
The model operates on discrete time steps, tracking population dynamics over multiple days, and is spatially implicit.

### E.1.3 Process Overview and Scheduling

At each time step, the time counter is incremented. When sterilized males are scheduled for release, they are added to the population. Subsequently, agents execute their category-specific behaviors, such as mortality, mating, and egg-laying. Agents are processed randomly at each time step to prevent ordering biases. Finally, data collection is performed to record population statistics and agent attributes. Figure 6 illustrates the process for a given time step for each agent type.

## E.2 Design Concepts

### E.2.1 Interactions

In our model, available females can mate at any time, as one of our key assumptions is that males are not a limiting factor. The model is not spatially explicit, meaning that individuals are well-mixed, without any spatial structure. As a consequence, mating events are assumed to be random and depend solely on the proportion of wild and sterilized males in the population.

### E.2.2 Sensing

The model is based on biological principles governing *D. suzukii* reproduction, particularly multiple mating and sperm bias. It incorporates different levels of sperm sensing, which influence how females use stored sperm for fertilization.

The level of sensing determines the sperm use biases studied as follows:

* **No sensing (Mixed):** Females do not distinguish between sperm sources.

- **Mixed:** Egg-laying is proportional to the fertile sperm content of the spermatheca. A scenario in which only a single sperm from a single ejaculate was chosen and used in the spermatheca was tested (results not shown), and the results of this scenario nearly overlapped with those of the *Mixed* scenario.

* **Limited sensing (First, Last):** Females exhibit a simple rule-based selection.

- **First:** Females use sperm from their first mating.
- *Last*: Females use sperm from their last mating.

* **Strong sensing (Preference):** Females actively favor specific sperm types.

- *Preference*: Total fertile sperm preference (*Preference W*), intermediate fertile sperm preference (*Preference I*), or total sterile sperm preference (*Preference S*).

### E.2.3 Stochasticity

Key life history traits (e.g., lifespan, mating intervals, emergence times) are modeled using uniform and Poisson distributions with parameters based on published data.

### E.2.4 Observation

Data collection tracks population size by agent type, female mating history and offspring production.

## E.3 Details

### E.3.1 Initialization

The model is initialized with user-defined population sizes, number of sterilized males released and sperm use biases (First, Last, Mixed, Preference W, Preference I or Preference S).

The initialization phase also includes the setup of counters for tracking the population of each agent category, a time counter, and a release counter. A scheduler is created to manage the activation of agents, ensuring that all agents are activated in a random order at each time step. Subsequently, the initial population of agents is created, and a data collector is configured to record global metrics (e.g., total population, number of agents in each category, number of releases).

### E.3.2 Input Data

The model does not require external datasets but relies on empirical parameter values from published studies. All parameters are listed in Table 3.
### E.3.3 Initialization of New Agents

Each agent is characterized by a unique identifier (a number), an attribute of category (S, W, L, or F), its creation date (`creation_date`) (i.e., the time step $t$ at which the agent is created in the model), and its death date (`death_date`).

Adult agents (S, W, and F) are assigned a death date as follows:

* If the agent (S, W, or F) is created at time $t = 0$, its lifespan is drawn from a uniform distribution between minimum and maximum lifespan found in the literature, an initial age is randomly assigned between 0 and its lifespan, and its death date is adjusted accordingly based on the lifespan and initial age.

* If the agent (S, W, or F) is created at time $t > 0$, its lifespan is drawn from a uniform distribution between minimum and maximum lifespan found in the literature, and its death date is determined at creation by adding this lifespan to the creation date.

The emergence date of larvae (L) is calculated by adding a randomly drawn development duration to its creation date. However, it will only emerge if it survives all sub-stages. A mortality probability, depending on the sub-stage (eggs, larvae, pupae), is applied at each time step (see Table 3).

Females (F) have additional characteristics: their spermatheca, the total number of eggs laid (`egg_laid`) and the time remaining until their next mating (`time_until_next_mating`). The number of eggs laid and the time until next mating are initialized to zero. The spermatheca content corresponds to the number of matings with wild males (`fertile Matings`) and sterilized males (`sterile Matings`), as well as the details of their first (`first_mating`) and last mating (`last_mating`) events. The former are initialized to 0, the latter to "None". New females are hence considered to be virgin and ready to mate.

Upon creation, the counter for the agent category in the model is incremented. The initially created agents perform their steps at time 0, so the structure of the female population (`fertilized` vs. `infertile`) depends on the structure of the male population (number of wild and sterilized males at the initial time).

### E.3.4 Submodels

At each time step $t$,
**Sterilized Male Release** New sterilized males are added to the population, simulating the release of sterilized males as part of SIT.

**Mortality** At each time step, the survival status of all agents is evaluated by comparing the current time to their death date. If $t \ge \text{death_date}$, the agent is removed from the scheduler, and the corresponding category counter is decremented.

**Emergence** Larvae undergo an additional check: if $t \ge \text{emergence_date}$, they transition to adulthood, becoming either a wild male or a female according to a Bernoulli distribution with a 0.5 probability for each sex. A new agent (male or female) is then created and integrated into the model.

**Mating** Females assess their availability for mating based on the value of `time_until_next_mating`. If they are available (`time_until_next_mating = 0`), they mate with either a wild or a sterilized male, depending on the proportion of each male type in the population. The spermatheca is updated according to the mating outcome. If both `fertile Matings` and `sterile matings` are 0, then `first_mating` is set to 1 if the mating is with a wild male, and 0 otherwise. If either `fertile matings` or `sterile matings` is greater than 0, then `last_mating` is set to 1 if the mating is with a wild male, and 0 otherwise. The value of `first matings` or `sterile mating` is increased by 1, depending on the male the female mated with. The time until the next mating is updated to reflect the refractory rate associated with the male the female mated with. If the female is not available (`time_until_next_mating > 0`), then `time_until_next_mating` is decremented by 1.

**Sperm Use** Upon mating, sperm is used according to the selected sperm use bias (`sperm_selection`), as described in section E.2.2. The use of sperm refers to testing whether fertile sperm is used or not, depending on the scenario being explored.

* In the *First* scenario, the first mating determines the sperm use,

* whereas in the *Last* scenario, it is the last mating.

* In the *Mixed* scenario, egg-laying is proportional to the amount of fertile sperm stored.
* In the “Preference” scenarios, when there is a complete preference for fertile (Preference W) or sterile sperm (Preference S), we simply check if the preferred sperm is stored, and if so, egg-laying will occur or not. When the preference is intermediate (Preference I), the preference rate decreases the probability of using fertile sperm to fertilize the eggs.

**Egg Laying** During the egg-laying step, females that have stored fertile sperm (fertilematings > 0) may use it to fertilize eggs. The number of eggs laid (egg_laid) per time step is drawn from a Poisson distribution, where the mean depends on a logistic growth function that limits larval production as density approaches the carrying capacity. The egg-laying behavior depends on the implemented sperm use bias. In the *First* and *Last* scenarios, females lay eggs only if the sperm from their first (first_mating == 1) or last (last_mating == 1) mating, respectively, came from a fertile male. In contrast, in the *Mixed* scenario, the number of eggs laid is proportional to the proportion of fertile sperm stored, computed as the ratio between fertile and total mating. This proportion is then applied to scale the expected number of eggs. This allows for partial fertility when both sterile and fertile sperm are present. The scenarios of preference incorporate probabilistic egg-laying decisions based on the relative amounts of fertile and sterile sperm stored, modulated by a preference parameter $\tilde{\eta}$. When $0 < \tilde{\eta} < 1$, corresponding to the *Preference I* scenario, the probability of using fertile sperm is given by:

$$P = \frac{\text{fertile\_matings}}{\text{fertile\_matings} + \tilde{\eta} \times \text{sterile\_matings}}$$

When $\tilde{\eta} = 0$ (Preference W), females exclusively use fertile sperm when available. Conversely, when $\tilde{\eta} = 1$ (Preference S), females use sterile sperm by default if available; otherwise, if only fertile sperm is stored, they proceed to lay eggs, with the number of eggs drawn from a Poisson distribution as previously described.

In all cases, each fertilized egg results in the creation of a new larval agent, which is added to the simulation.

## Appendix F: Comparison Between the Compartmental Model and the Agent-Based Model

This appendix compares the larval dynamics obtained from the compartmental model (Eq. (2), Fig. 10A-B) and the agent-based model (ABM) (Sect. 3, Fig. 10C-D) across different sterilized male release rates ($\sigma$) and sperm use scenarios (described in Sect. 2.3.1). The goal is to evaluate the consistency between the two approaches and to highlight
both their convergences and divergences in simulating larval population dynamics. The simulated release rates $\sigma$ in the two subsequent scenarios (high infestation and low infestation) differ substantially, as they are adjusted to the initial conditions, to illustrate different contexts such as population suppression or maintenance.

**High infestation** These simulations are conducted under the same initial conditions as in Fig. 3A, allowing for direct comparison. To start with only fertilized females ($F_F$ in the compartmental model) in the ABM, the females present in the system at the initial time have fertile sperm stored in their spermatheca. As observed with the compartmental model, when release rates are below the eradication thresholds for the *First* and *Last* scenarios, larval suppression is initially more rapid and pronounced in the *Last* scenario (which includes multiple mating). However, over the long term, this suppression proves less effective than in the *First* scenario, thus confirming the reversal in trend already noted.

When the release rate is sufficiently high, the *Last* scenario still leads to a faster suppression of the larval population, although eradication is achieved in all cases. The intermediate case observed in the compartmental model—where the population is maintained only in the *Last* scenario for intermediate $\sigma$, is not reproduced with the agent-based model.
<figure><img src="image_11.png" /><figcaption>**Fig 9** Simulations of larval density *L* over time for different initial infestation levels and sterilized male release rates (σ). This figure shows results for a high initial infestation level (initial conditions: *L* = 30,000, *M* = 100,000, *F*<sub>I</sub> = 0, *F*<sub>F</sub> = 100,000, *S* = σ/μ<sub>S</sub>). Black curves represent the scenario without releases (σ = 0). Blue curves show the *First* sperm use scenario (τ<sub>I</sub> = τ<sub>F</sub> = 0), and orange curves show the *Last* scenario (τ<sub>I</sub> > 0, τ<sub>F</sub> > 0). Solid curves correspond to σ = 300,000 (below the eradication threshold in Fig. 2), dash-dot curves to σ = 450,000 (between thresholds), and dotted curves to σ = 600,000 (above the eradication threshold). Each curve represents the mean of 10 stochastic simulations, and the shaded area around each line shows the range between the minimum and maximum larval densities across replicates.</figcaption></figure>

**Low infestation** Under the “without release” scenario (WR, Fig. 10), both models yield similar larval dynamics, with populations gradually stabilizing around the carrying capacity *K*. However, the ABM displays slightly slower convergence.

For varying release rates tested in Fig. 10, the overall trends and curve shapes remain consistent across models. The compartmental model generally reaches equilibrium about twice as quickly. At σ = 10,000, the compartmental model shows faster convergence in the *Last* scenario compared to the *First*, while this distinction is not observed in the agent-based model.

At higher release rates (σ = 25,000), the divergence becomes more evident. In the compartmental model, the larval population still tends to stabilize near *K* for both sperm use scenarios. In contrast, in the agent-based model, the population stabilizes much more slowly in the *First* scenario, and it fails to persist altogether in the *Last* scenario. Consequently, the relative reductions in larval populations differ considerably between the two approaches.
At very high release rates ($\sigma = 50,000$), neither model maintains the larval population, regardless of the sperm use scenario, indicating successful suppression in all cases.

Despite differences in convergence speeds and long-term behavior, both models consistently rank sperm use scenarios in the same order: *Last* yields more effective short-term suppression than *First*, as previously discussed in Sect. 2. This consistency reinforces the robustness of qualitative insights while showcasing the complementarity of both approaches.

<figure><img src="image_12.png" /><figcaption>**Fig 10** Comparison of larval dynamics and percentage reduction in areas under the curves between the compartmental model (A-B) (2) and the agent-based model (described in Sect. 3, C-D) under different sterilized male release rates ($\sigma$) and sperm use scenarios. Simulations were initialized with 1000 wild males, 1000 females, and 0 larvae. (A-C) Larval density over time. Black curves represent the scenario without sterilized male releases ($\sigma = 0$). Blue curves correspond to the *First* sperm use scenario, and orange curves to the *Last* scenario. Solid, dash-dot, and dotted lines indicate $\sigma = 10,000$, $\sigma = 25,000$, and $\sigma = 50,000$, respectively. For the agent-based model, results are based on 20 simulations, with the main (thicker) curve representing the average of these simulations. (B-D) Percentage reduction in the area under the curve relative to the reference scenario (WR, no release), for the compartmental model (B) and the agent-based model (D).</figcaption></figure>

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