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Boundary and Interior Control in a Diffusive Lotka-Volterra Model

João Carlos Barreira, Maicon Sonego, Enrique Zuazua

TL;DR

This work analyzes a two-species diffusive Lotka-Volterra system on a smooth domain with both boundary actuators and an interior multiplicative control. It proves two main results: (i) asymptotic controllability to single-species survival states for any parameters via a joint boundary and interior strategy that overcomes barrier effects, and (ii) finite-time controllability to a heterogeneous coexistence state through a two-phase approach that first steers near the target with boundary controls and then uses a localized interior control in a region $\omega$ together with a linearization and Carleman-based null controllability argument. The analysis couples nonlinear dynamics with Carleman estimates and an inverse-mapping theorem to bridge linear and nonlinear behavior, and is complemented by numerical simulations illustrating barrier crossing and minimum-time trajectories. The results have ecological relevance for population management and demonstrate the value of combining boundary actuation with distributed interior controls to achieve precise ecological states under practical constraints.

Abstract

We investigate the controllability of a generalized diffusive Lotka-Volterra competition model for two species, incorporating boundary controls and an interior multiplicative control. Considering a smooth, bounded N-dimensional domain, we analyze ecologically pertinent scenarios characterized by constraints on both the controls and system states. Our results demonstrate how integrated control strategies can effectively overcome the limitations identified in previous studies. We prove two main results: (1) asymptotic controllability to single-species survival steady states under arbitrary system parameters, ensured by a combination of boundary and interior controls which act jointly to stabilize the system; and (2) finite-time controllability to a specific heterogeneous coexistence steady state via a two-phase strategy - first steering the system near the target with boundary control, then activating an interior multiplicative control in a localized region. The strong synergy between the two control mechanisms is crucial in both cases. We also analyze extinction dynamics and homogeneous coexistence, and support our findings with numerical simulations. The work concludes with perspectives for future research.

Boundary and Interior Control in a Diffusive Lotka-Volterra Model

TL;DR

This work analyzes a two-species diffusive Lotka-Volterra system on a smooth domain with both boundary actuators and an interior multiplicative control. It proves two main results: (i) asymptotic controllability to single-species survival states for any parameters via a joint boundary and interior strategy that overcomes barrier effects, and (ii) finite-time controllability to a heterogeneous coexistence state through a two-phase approach that first steers near the target with boundary controls and then uses a localized interior control in a region together with a linearization and Carleman-based null controllability argument. The analysis couples nonlinear dynamics with Carleman estimates and an inverse-mapping theorem to bridge linear and nonlinear behavior, and is complemented by numerical simulations illustrating barrier crossing and minimum-time trajectories. The results have ecological relevance for population management and demonstrate the value of combining boundary actuation with distributed interior controls to achieve precise ecological states under practical constraints.

Abstract

We investigate the controllability of a generalized diffusive Lotka-Volterra competition model for two species, incorporating boundary controls and an interior multiplicative control. Considering a smooth, bounded N-dimensional domain, we analyze ecologically pertinent scenarios characterized by constraints on both the controls and system states. Our results demonstrate how integrated control strategies can effectively overcome the limitations identified in previous studies. We prove two main results: (1) asymptotic controllability to single-species survival steady states under arbitrary system parameters, ensured by a combination of boundary and interior controls which act jointly to stabilize the system; and (2) finite-time controllability to a specific heterogeneous coexistence steady state via a two-phase strategy - first steering the system near the target with boundary control, then activating an interior multiplicative control in a localized region. The strong synergy between the two control mechanisms is crucial in both cases. We also analyze extinction dynamics and homogeneous coexistence, and support our findings with numerical simulations. The work concludes with perspectives for future research.

Paper Structure

This paper contains 21 sections, 11 theorems, 114 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and main results
  3. Biological interpretation and control
  4. Outline
  5. Proof of Theorem \ref{['MT1']}
  6. Preliminaries for the proof of Theorem \ref{['MT3']}
  7. Trajectory control and linear system
  8. Null controllability of linear system
  9. Proof of Theorem \ref{['MT3']}
  10. Homogeneous Coexistence and Extinction
  11. On the target (u^*,v^*)
  12. On the target (0,0)
  13. Numerical Simulations
  14. Crossing barriers
  15. Optimal control - minimum time
  16. ...and 6 more sections

Key Result

Theorem 1.1

There are boundary controls $c_u, c_v\in L^{\infty}(\partial\Omega)$ and an interior control $h\in L^{\infty}(\Omega)$ such that, for every $(u_0,v_0)\in L^{\infty}(\Omega)\times L^{\infty}(\Omega)$, the solution $(u,v)$ of e1 satisfies uniformly in $\Omega$.

Figures (10)

  • Figure 1: Controlled trajectory reaching exactly the target $(u^{**},v^{**})$ at time $T=T_1+\widetilde{T}$.
  • Figure 2: Barriers preventing the trajectories from approaching the target $(a_1/b_1,0)=(1,0)$.
  • Figure 3: Trajectories crossing the barrier under the action of the internal control $\overline{h}$.
  • Figure 4: The target $(u^{**},v^{**})$.
  • Figure 5: Trajectories approaching $u^{**}.$
  • ...and 5 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['MT1']}
  • Remark 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 15 more