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Feedback stabilization for a spatial-dependent Sterile Insect Technique model with Allee Effect

Kala Agbo Bidi, Luís Almeida, Jean-Michel Coron

TL;DR

Problem addressed: achieving extinction of spatially distributed pest populations using SIT with an Allee effect. Approach: formulate a two-dimensional reaction-diffusion SIT model, prove well-posedness, and design a global, positive state-feedback control via backstepping that stabilizes the extinction equilibrium. Key results: existence/uniqueness of weak solutions in L1, global exponential extinction under a suitable reproduction parameter, and an explicit feedback law with Lyapunov-based justification; numerical simulations in 2D validate rapid suppression and provide insights on diffusion and localized control. Significance: demonstrates robust, space-aware SIT strategies capable of long-term pest suppression with potentially reduced release demands and feasibility for field deployment using localized measurements.

Abstract

This work focuses on feedback control strategies for applying the sterile insect technique (SIT) to eliminate pest populations. The presentation is centered on the case of mosquito populations, but most of the results can be extended to other species by adapting the model and selecting appropriate parameter values to describe the reproduction and movement dynamics of the species under consideration. In our study, we address the spatial distribution of the population in a two dimensional bounded domain by extending the temporal SIT model analyzed in [2], thereby obtaining a reaction-diffusion SIT model. After the analysis of the existence and the uniqueness of the solution of this problem, we construct a feedback law that globally asymptotically stabilizes the extinction equilibrium thus yielding a robust strategy to keep the pest population at very low levels in the long term.

Feedback stabilization for a spatial-dependent Sterile Insect Technique model with Allee Effect

TL;DR

Problem addressed: achieving extinction of spatially distributed pest populations using SIT with an Allee effect. Approach: formulate a two-dimensional reaction-diffusion SIT model, prove well-posedness, and design a global, positive state-feedback control via backstepping that stabilizes the extinction equilibrium. Key results: existence/uniqueness of weak solutions in L1, global exponential extinction under a suitable reproduction parameter, and an explicit feedback law with Lyapunov-based justification; numerical simulations in 2D validate rapid suppression and provide insights on diffusion and localized control. Significance: demonstrates robust, space-aware SIT strategies capable of long-term pest suppression with potentially reduced release demands and feasibility for field deployment using localized measurements.

Abstract

This work focuses on feedback control strategies for applying the sterile insect technique (SIT) to eliminate pest populations. The presentation is centered on the case of mosquito populations, but most of the results can be extended to other species by adapting the model and selecting appropriate parameter values to describe the reproduction and movement dynamics of the species under consideration. In our study, we address the spatial distribution of the population in a two dimensional bounded domain by extending the temporal SIT model analyzed in [2], thereby obtaining a reaction-diffusion SIT model. After the analysis of the existence and the uniqueness of the solution of this problem, we construct a feedback law that globally asymptotically stabilizes the extinction equilibrium thus yielding a robust strategy to keep the pest population at very low levels in the long term.

Paper Structure

This paper contains 18 sections, 10 theorems, 246 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. ODE Model for Mosquito Population with Allee Effect
  3. Reaction diffusion model for mosquito populations
  4. Life cycle model for a mosquito population in a bounded domain
  5. Well-posedness of the Cauchy problem
  6. Asymptotic stability of the life cycle model
  7. Reaction diffusion model for SIT
  8. Well-posedness of the closed-loop system
  9. Design of state-feedback controllers
  10. Numerical simulation in 2D
  11. Conclusion and perspectives
  12. Stability properties of the ODE dynamical systems
  13. Proof of Theorem \ref{['Steadystates']}
  14. Proof of Theorem \ref{['stablityofstates']}
  15. Proof of Theorem \ref{['stablityofsitode']}
  16. ...and 3 more sections

Key Result

Theorem 2.1

$\space$

Figures (7)

  • Figure 1: Numerical simulation of the function $K$ for $\zeta = 500$, $\Lambda_1 = 2\times 10^5$, $\Lambda_2 = 1.5\times 10^5$, $\Lambda_3 = 1\times 10^5$. The domain $\Omega = [0,\ell] \times[0,\ell]$ and $\ell= 5$ km which we discretize with $N_x = 50=N_y$ and $dx=dy=0.1$. $\sigma_1=\sigma_2=\sigma_3 = 1$, $\mu_1 = 2.5$, $\mu_2 = 1.5=\xi_2=\xi_3$, $\mu_3 = 4=\xi_1$.
  • Figure 2: Fecundated egg density at the time $t=10$ days when applying the backstepping feedback law \ref{['eq:backcontr1PDE']}.
  • Figure 3: Fecundated egg density at the time $t=200$ days when applying the backstepping feedback law \ref{['eq:backcontr1PDE']}. We remark that it is very close to zero everywhere in the domain as also shown in Figure \ref{['fig:L1normE']}
  • Figure 4: Evolution of $t \mapsto \int_\Omega E(t)$, representing the total number of fertile eggs across the entire domain.
  • Figure 5: Evolution of $t \mapsto \ln\left(\int_\Omega u(t)\right)$, representing the total number of sterile mosquitoes released across the entire domain.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • Definition 3.1
  • Remark 3.1
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.2
  • proof
  • ...and 11 more