# A sex- and stage-structured population dynamics model for pest control using the sterile insect technique

Crésus Kounoudji, Suzanne Touzeau, Frédéric Grognard, Louise van Oudenhove, Ludovic Mailleret

► To cite this version:

Crésus Kounoudji, Suzanne Touzeau, Frédéric Grognard, Louise van Oudenhove, Ludovic Mailleret. A sex- and stage-structured population dynamics model for pest control using the sterile insect technique. MPDEE 2023 - Mathematical Population Dynamics, Ecology and Evolution, Apr 2023, Marseille, France. hal-04144139

HAL Id: hal-04144139

https://hal.inrae.fr/hal-04144139

Submitted on 8 Aug 2023

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# A sex- and stage-structured model for pest control using the sterile insect technique

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C. Kounoudji, S. Touzeau<sup>1,2</sup>, F. Grognard<sup>2</sup>, L. van Oudenhove<sup>1</sup>, L. Mailleret<sup>1,2*</sup>

1. M2P2, ISA (INRAE, CNRS, Univ. Côte d'Azur)

2. MACBES (Inria, INRAE, CNRS Univ. Côte d'Azur)

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# Fruit flies

* widespread polyphagous dipteran insects that lay their eggs in fruits

* *Ceratitis capitata*: citrus, stone fruits...

* *Drosophila suzukii*: berries, cherries...

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* after hatching, maggots develop inside fruits, entailing massive damage

* make them unfit for consumption

* may cause early fruit drop

* create entry points for diseases

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# Fruit flies control

* control of fruit flies has long relied on chemicals

* sustainability and health issues

* development of resistant flies

* more stringent state regulations (EU)

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* alternative eco-friendly control means include

* crop sanitation, mass trapping

* biological control through natural enemies

* taking advantage of sexual mode of reproduction through the sterile insect technique

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# Sterile Insect Technique (SIT)

* flood agricultural plots with factory produced sterilized males

* prevent matings between *wild males* and *wild females*

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Where is Waldo?

SIT is like a football game with so many people on the ground
you simply cannot find your teammates ¹
Outline

* SIT model in an agricultural context relevant to fruit flies

* access to reliable estimates of crop damage caused by the larvae

* analyze influence of sterile male introductions on model dynamics and damage levels

* study if and how sterile male pattern of introduction can be optimized
# Fruit flies life cycle

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pupae develop

credit to noun project for art
# Model diagram

* 4 stages: eggs/larvae *L*, unmated females *V*, males *M*, mated females *F*¹

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# Mating model

* frequency dependent mating probability¹

* # males *M* abundant
    females *V* mated at rate *v*<sub>F</sub>

* # males *M* limiting
    mating proba. *γM/V*, *V* mating rate *γM/V* *v*<sub>F</sub>

* overall mating rate per unmated females *V*

<figure><img src="image_19.png" /></figure>

$$v_F \min \left( \frac{\gamma M}{V}, 1 \right)$$
# Population dynamics model

$$ 
\left\{
\begin{aligned}
\dot{L} &= r \left(1 - \frac{L}{K}\right) F - v_L L - \mu_L L \\
\dot{M} &= p v_L L - \mu_M M \\
\dot{V} &= (1 - p) v_L L - v_F \min\left(\frac{\gamma M}{V}, 1\right) V - \mu_F V \\
\dot{F} &= v_F \min\left(\frac{\gamma M}{V}, 1\right) V - \mu_F F
\end{aligned}
\right.
$$

* in an agricultural context, the insect pest settles in crops at high densities s.t.

$$ \eta_0 = \frac{r (1 - p) v_L v_F}{\mu_F (\mu_F + v_F) (\mu_L + v_L)} > 1 $$

* in that case, the positive equilibrium of the saturated submodel is GAS for the full model (Anguelov et alii, 2017)

* thus the min(.) necessarily saturates to 1 after some transient times
# Reduced model

* in what follows, we therefore concentrate on the simpler form

$$ 
\begin{cases}
\dot{L} = r \left(1 - \frac{L}{K}\right) F - v_L L - \mu_L L \\
\dot{M} = p v_L L - \mu_M M \\
\dot{V} = (1 - p) v_L L - v_F V - \mu_F V \\
\dot{F} = v_F V - \mu_F F
\end{cases}
$$

* assuming that the basic reproduction number

$$ \eta_0 = \frac{r (1 - p) v_L v_F}{\mu_F (\mu_F + v_F) (\mu_L + v_L)} > 1 $$
# Model diagram (with sterile males)

* 5th stage: sterile males $M_S$ (= constant for now)

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$M_S$ divert a part of unmated females V to mated-with-sterile females
# Model with sterile males

* only a proportion $\frac{M}{M+M_s}$ of matings yield egg-laying females

$$ 
\left\{ 
\begin{aligned}
\dot{L} &= r \left(1 - \frac{L}{K}\right) F - v_L L - \mu_L L \\
\dot{M} &= p v_L L - \mu_M M \\
\dot{V} &= (1 - p) v_L L - v_F V - \mu_F V \\
\dot{F} &= v_F \frac{M}{M+M_s} V - \mu_F F
\end{aligned}
\right. 
$$
# Analysis: equilibria

* equilibria are solutions of

$$F = \frac{v_L + \mu_L}{r(1 - \frac{L}{K})} L \quad (i)$$

$$M = \frac{p v_L}{\mu_M} L \quad (ii)$$

$$V = \frac{(1-p)v_L}{v_F + \mu_F} L \quad (iii)$$

$$F = \frac{v_F}{\mu_F} \frac{M}{M+M_s} V \quad (iv)$$

* so that (0,0,0,0) is always an equilibrium

* and, using (i), (ii) and (iii) in (iv), other equilibria must verify

$$\frac{v_L + \mu_L}{r \left(1 - \frac{L}{K}\right)} = \frac{v_F}{\mu_F} \frac{\frac{p v_L}{\mu_M} L}{\frac{p v_L}{\mu_M} L + M_s} \frac{(1 - p) v_L}{v_F + \mu_F}$$
# Analysis: equilibria

* rearranging, other equilibria must verify

$$1 = \eta_0 \left(1 - \frac{L}{K}\right) \frac{\frac{pv_L}{\mu_M} L}{\frac{pv_L}{\mu_M} L + M_s} \Leftrightarrow M_s = \frac{pv_L}{\mu_M} L \left(\eta_0 - 1 - \frac{\eta_0}{K} L\right)$$

* RHS term is a concave parabola in L, with roots: 0, and: $K \left(1 - \frac{1}{\eta_0}\right) > 0$

* if $M_s$ larger than max of parabola: no equilibrium other than 0

* if $M_s$ smaller than max of parabola, there exists two positive equilibria with $0 < L_1^* < L_2^* < K$

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# Analysis: stability

* Jacobian matrix is 4D 😤

$$J = \begin{pmatrix}
-\frac{r}{K}F - \mu_L - v_L & 0 & 0 & r\left(1 - \frac{L}{K}\right) \\
pv_L & -\mu_M & 0 & 0 \\
(1-p)v_L & 0 & -(v_F + \mu_F) & 0 \\
0 & v_F \frac{M_s}{(M+M_s)^2} V & v_F \frac{M}{(M+M_s)} & -\mu_F
\end{pmatrix}$$

* but with non-negative off-diagonal elements at equilibria 😊

$$J^* = \begin{pmatrix}
\bullet & 0 & 0 & + \\
+ & \bullet & 0 & 0 \\
+ & 0 & \bullet & 0 \\
0 & + & + & \bullet
\end{pmatrix}$$
# Analysis: stability

* 0 equilibrium is always LAS, thanks to a nice block-triangular structure

$$J_0^* = \begin{pmatrix} -(\mu_L + v_L) & 0 & 0 & r \\ pv_L & -\mu_M & 0 & 0 \\ (1-p)v_L & 0 & -(v_F + \mu_F) & 0 \\ 0 & 0 & 0 & -\mu_F \end{pmatrix}$$

* for positive equilibria built on $L_1^*$ and $L_2^*$

* strong clues for fold bifurcation at

$$M_s = \overline{M_s} = \frac{pv_L}{4\mu_M} \frac{(\eta_0 - 1)^2}{\eta_0} K$$

* so that, given 0 is always LAS, $E_1^*$ would be a saddle and $E_2^*$ would be LAS

Equilibria computation

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# Analysis: stability of $E_i^*$

* use the special structure of the Jacobian: a Metzler matrix

$$J = \begin{pmatrix} \vdots & \geq 0 \\ \geq 0 & \vdots \end{pmatrix}$$

* Bowong's lemma¹

Let $J$ be a Meztler matrix that can be decomposed into blocks as

$$J = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$$

Then $J$ is stable if and only if $A$ and $D - CA^{-1}B$ are stable Metzler matrices
# Analysis: stability of $E_i^*$

* express $J$ in function of $L^*$

$$ J = \begin{pmatrix} \frac{-\eta_0(\mu_L + v_L) \frac{pv_L}{\mu_M} L^*}{\frac{pv_L}{\mu_M} L^* + M_s} & 0 & 0 & r \left(1 - \frac{L^*}{K}\right) \\ pv_L & -\mu_M & 0 & 0 \\ (1-p)v_L & 0 & -(\mu_F + v_F) & 0 \\ 0 & \frac{\eta_0 \mu_F (\mu_L + v_L) M_s L^*}{r \left(\frac{pv_L}{\mu_M} L^* + M_s\right)^2} & \frac{v_F \frac{pv_L}{\mu_M} L^*}{\frac{pv_L}{\mu_M} L^* + M_s} & -\mu_F \end{pmatrix} $$

* so that

$$ A = \begin{pmatrix} \frac{-\eta_0(\mu_L + v_L) \frac{pv_L}{\mu_M} L^*}{\frac{pv_L}{\mu_M} L^* + M_s} & 0 \\ pv_L & -\mu_M \end{pmatrix} \quad \text{is Metzler stable} $$
• and

$$B = \begin{pmatrix} 0 & r \left(1 - \frac{L^*}{K}\right) \\ 0 & 0 \end{pmatrix}, \quad C = \begin{pmatrix} (1-p)v_L & 0 \\ 0 & \frac{\eta_0 \mu_F (\mu_L + v_L) M_s L^*}{r \left(\frac{p v_L}{\mu_M} L^* + M_s\right)^2} \end{pmatrix}, \quad \text{and} \quad D = \begin{pmatrix} -(\mu_F + v_F) & 0 \\ \frac{v_F \frac{p v_L}{\mu_M} L^*}{\frac{p v_L}{\mu_M} L^* + M_s} & -\mu_F \end{pmatrix}$$
# Stability of $E_i^*$

* further computations show

$$D - CA^{-1}B = \begin{pmatrix} -(\mu_F + v_F) & \frac{\mu_F(\mu_F + v_F)(1 - \frac{L^*}{K})(\frac{pv_L}{\mu_M}L^* + M_s)}{v_F \frac{pv_L}{\mu_M}L^*} \\ \frac{v_F \frac{pv_L}{\mu_M}L^*}{\frac{pv_L}{\mu_M}L^* + M_s} & \mu_F \left( \frac{M_s(1 - \frac{L^*}{K})}{\frac{pv_L}{\mu_M}L^* + M_s} - 1 \right) \end{pmatrix} \quad \text{is Metzler}$$

* and

$$\det(D - CA^{-1}B) = -\mu_F(\mu_F + v_F) \left[ \left(1 - \frac{L^*}{K}\right) \left(1 + \frac{M_s}{\frac{pv_L}{\mu_M}L^* + M_s}\right) - 1 \right] \\ = -\frac{\mu_F(\mu_F + v_F)}{\eta_0} \left(\eta_0 - 1 - \frac{2\eta_0L^*}{K}\right)$$

given that $M_s = \frac{pv_L}{\mu_M}L^* (\eta_0 - 1 - \frac{\eta_0L^*}{K})$ at equilibrium $E_i^*$
# Stability of $E_i^*$

* and this is it: the slope of the parabola at $L_i^*$ is

$$ \frac{dM_s}{dL} = \frac{pv_L}{\mu_M} \left( \eta_0 - 1 - \frac{2\eta_0L_i^*}{K} \right) $$

slope sign at $L^*$ sets $det(D - CA^{-1}B)$ sign

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* at $E_2^*$, $\frac{dM_s}{dL} < 0$ which implies: $det(D - CA^{-1}B) > 0$ and $tr(D - CA^{-1}B) < 0$
  from Bowong's lemma, $J(E_2^*)$ is thus stable and $E_2^*$ is LAS

* at $E_1^*$, $\frac{dM_s}{dL} > 0$ which implies: $det(D - CA^{-1}B) < 0$
  from Bowong's lemma, $E_1^*$ is unstable
Bifurcation diagram

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• thanks to the cooperativity of the model

■ when $M_S > \bar{M}_S$, 0 equilibrium is GAS

■ when $M_S < \bar{M}_S$ trajectories converge to either 0 or $E_2^*$
$M_S < \overline{M_S}$ : bi-stability

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$M_S > \overline{M_S} : 0$ is GAS

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# Mated females invade

* 0 is still GAS, but...

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# Mated females invade

* but GAS is not always enough

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# Larvae vs. population size

* total population not a very good proxy for larvae population / crop damage

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# Model with pulsed $M_S$ introductions

* same equations as before, but $M_S$ is dynamic

$$ \begin{cases} \dot{M}_S = -\mu_M M_S & \forall t \in (kT, (k+1)T) \\ M_S(kT^+) = M_S(kT) + \sigma T & \end{cases} $$

* classical trick to compare different introduction regimes for given introduction rate $\sigma$¹

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# Numerical experiments: pulses

* which introduction strategy works best: late introductions situation

<figure><img src="image_44.png" /><figcaption>max. Larvae at end of program</figcaption></figure>
# Numerical experiments: pulses

* which introduction strategy works best: early introduction situations

<figure><img src="image_45.png" /><figcaption>max. Larvae at end of program</figcaption></figure>
# Conclusion

* sex- and stage- structured model of Anguelov *et alii* in a SIT context

* quite thorough mathematical analysis

* Metzler matrices and cooperativity tools

* showed importance of stage-structure consideration

* dynamics are very different depending on initial condition

* larvae density (damage) poorly correlates with total population size

* introduction strategy

* timing is the essence

* early, and not late introductions

* if not possible, small and frequent introductions perform best by far

* SIT most efficient in a preventive context
# Perspectives

* quantify basins of attraction in the bistable cases

* account for multiple female matings in the model

* provide mathematical grounds for the results on *T* / introduction strategies

* address complementary questions of biological interest, e.g.

* what happens if *sterile* males are not *that sterile*?

Marine Courtois will give insights on this topic wednesday at 11 AM
Thank you

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Acknowledgements

* Ecophyto Ceratis Corse

* ANR Suzukiiss:me

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Where is
Waldo?

ÉCOPHYTO

RÉDUIRE ET AMÉLIORER
L'UTILISATION DES PHYTOS

anr®