Existence and Stability of Stationary States of a Reaction–Diffusion-Advection Model for Two Competing Species

Abstract

It is well known that the research of two species in the Lotka–Volterra competition system could create very interesting dynamics. In our paper, we investigate the global dynamical behavior of a classic Lotka–Volterra competition system by studying the steady states and corresponding stability by mainly employing the methods of monotone dynamical systems theory, Lyapunov–Schmidt reduction and spectral theory and so on. It illustrates that the dynamical behavior substantially relies on certain variable of the maximal growth rate. Furthermore, we obtain that one of the semi-trivial steady state solutions is a global attractor in some special cases. In biology, these results show that both of the species do not coexist and the mutant forces the extinction of resident species under some condition for two similar species system.