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Agro-ecological control of a pest-host system: optimizing the harvest

Baptiste Maucourt

Abstract

We delve into the interactions between a prey-predator and a vector-borne epidemic system, driven by agro-ecological motivations. This system involves an ODE, two reaction--diffusion PDEs and one reaction--diffusion--advection PDE. It has no complete variational or monotonic structure and features spatially heterogeneous coefficients. Our initial focus is to examine the continuity of a quantity known as ''harvest'', which depends on the time-integral of infected vectors. We analyze its asymptotic behaviour as the domain becomes homogeneous. Then we tackle a non-standard optimal control problem related to the linearized harvest and conduct an analysis to establish the existence, uniqueness, and properties of optimizers. Finally, we refine the location of optimizers under specific initial conditions.

Agro-ecological control of a pest-host system: optimizing the harvest

Abstract

We delve into the interactions between a prey-predator and a vector-borne epidemic system, driven by agro-ecological motivations. This system involves an ODE, two reaction--diffusion PDEs and one reaction--diffusion--advection PDE. It has no complete variational or monotonic structure and features spatially heterogeneous coefficients. Our initial focus is to examine the continuity of a quantity known as ''harvest'', which depends on the time-integral of infected vectors. We analyze its asymptotic behaviour as the domain becomes homogeneous. Then we tackle a non-standard optimal control problem related to the linearized harvest and conduct an analysis to establish the existence, uniqueness, and properties of optimizers. Finally, we refine the location of optimizers under specific initial conditions.
Paper Structure (10 sections, 13 theorems, 131 equations, 5 figures)

This paper contains 10 sections, 13 theorems, 131 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Notations
  4. Main result
  5. Harvest around $\lambda_1(\mathcal{L}_{V_s})=0$
  6. Homogenization of the refuges
  7. Optimizers of the linearized harvest
  8. Computation of $\eta_L$
  9. Schwarz rearrangement
  10. Explicit optimizers in the homogeneous case

Key Result

Theorem 1

When the frequency of the refuge goes to infinity, the harvest converges to the harvest of the homogenized system.

Figures (5)

  • Figure 1: Sub-estimate of the harvest in function of a constant $R$ (with biologically consistent parameters, see girardinmaucourt2023).
  • Figure 2: Harvest and linearized harvest in function of a constant $R$ (with biologically consistent parameters, see girardinmaucourt2023). Explicit formula for $\eta_L$ can be found in Remark 1.
  • Figure 3: Harvest and linearized harvest in function of a constant $R$ (with $\beta_{HV}$ ten times bigger).
  • Figure 4: Harvest and linearized harvest in function of a constant $R$ (with $\beta_{VH}$ ten times bigger).
  • Figure 5: Harvest and linearized harvest in function of a constant $R$ (with $T$ three times bigger).

Theorems & Definitions (23)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • proof
  • Lemma 7
  • ...and 13 more