Global stability in a three‐species Lotka–Volterra cooperation model with seasonal succession

Abstract

In this paper, we focus on a three‐species Lotka–Volterra cooperation model with seasonal succession. The Floquet multipliers of all nonnegative periodic solutions of such a time‐periodic system are estimated via the stability analysis of equilibria. By Brouwer fixed point theorem and the connecting orbits theorem, it is proved that there admits a unique positive periodic solution under appropriate conditions. Furthermore, sharp global asymptotical stability criteria for extinction and coexistence are established. Compared to the classical three‐species Lotka–Volterra cooperation model, the introduction of seasonal succession may lead to species' extinction. Finally, some numerical examples are given to illustrate the effectiveness of our theoretical results.