Traveling waves of modified Leslie–Gower predator–prey systems

Abstract

This paper focuses on the spreading phenomena within modified Leslie–Gower reaction–diffusion predator–prey systems. Our main objective is to investigate the existence of two distinct types of traveling waves. Specifically, with the aid of the upper and lower solution methods, we establish the existence of traveling waves connecting the prey-present state and the coexistence state or the prey-present state and the prey-free state by constructing different and appropriate Lyapunov functions. Moreover, for traveling wave connecting the prey-present state and the prey-free state, more information about the monotonicity of the wave profile can be obtained by analyzing its asymptotic behavior at negative infinity. Finally, our results are applied to modified Leslie–Gower system with Holling-II type functional response or Lotka–Volterra type functional response, and a novel Lyapunov function is constructed for the latter, which further enhances our results. Meanwhile, some numerical simulations are carried out to support our results.