Controllability of diffusive Lotka-Volterra strongly competitive systems under boundary constrained controls

TL;DR

This paper studies how to control a competitive ecological system modeled by a Lotka-Volterra reaction-diffusion equation, showing conditions under which species extinction states can be achieved via boundary controls.

Contribution

It provides new theoretical insights into the controllability of competitive reaction-diffusion systems with boundary constraints, including explicit barrier solutions and spectral analysis.

Findings

System can be driven to species extinction states using boundary controls.

Controllability to certain steady states is limited by inter-species competition imbalance.

Numerical simulations support theoretical results and highlight open problems.

Abstract

We investigate the controllability of the competition-diffusion Lotka-Volterra system. Our primary focus is on the one-dimensional setting with Dirichlet boundary controls, interpreted as ecological management policies regulating the density of species at the habitat boundaries and satisfying bilateral constraints. We show that the system can be steered from any initial state to a constant steady state representing the extinction of the less competitive species. In contrast, we prove that controllability toward a steady state where the more competitive species vanishes is generally not achievable when the inter-species competition rates are too unbalanced. This obstruction is due to the existence of barrier solutions, which we explicitly construct based on the spectral properties of the associated reaction-diffusion operators. Our theoretical results are illustrated through numerical simulations and are accompanied by a discussion of open problems and potential directions for future research.