Propagation pattern for a three-species competition system in shifting patchy environments

Abstract

This paper investigates the dynamic behavior of a three-species competition system in shifting patchy environments. By employing Schauder’s fixed point theorem, we first establish the existence of positive traveling wave solutions under the assumption of a pair of generalized super- and sub-solutions. Building on this theoretical foundation, we construct distinct types of super- and sub-solutions to derive three classes of super-critical and critical forced waves, which characterize the transition from the following ecological states to extinction: (i) one native species and two invasive competitors; (ii) two native species and one invasive competitor; (iii) coexistence of all three species. Furthermore, we prove the non-existence of sub-critical forced waves using a contradiction argument. To validate our theoretical findings, we conduct numerical simulations, which illustrate the predicted wave dynamics and species extinction scenarios. The study advances the mathematical theory of nonlinear lattice systems and provides a framework for analyzing ecological transitions under environmental forcing.