Abstract
We establish the existence of traveling wave solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator-prey model incorporating a prey refuge. By using the shooting argument, invariant manifold theory, and the Hopf bifurcation theorem, we analyze the dynamic behavior of this model in the three-dimensional phase space. Numerical results are also presented to illustrate the theoretical results.
Highlights
In mathematical biology, one interesting and dominant theme is the dynamic relationship between predators and their prey [1,2,3]
In order to establish the existence of traveling wave solutions of system (5), we assume the system has a solution of the special form H(x, t) = H(x+ct), P(x, t) = P(x+ct), where parameter c(> 0) is the wave speed
We require that the traveling wave solutions H and P are nonnegative and satisfying the following boundary conditions: H (−∞) = r, H (+∞) = H∗, a
Summary
One interesting and dominant theme is the dynamic relationship between predators and their prey [1,2,3]. Dunbar [16] proved the existence of traveling wave solutions of diffusive Lotka-Volterra and used the methods of a shooting argument and a Lyapunov function. Huang et al [23] and Li and Wu [24] used Dunbar’ method to study the existence of traveling solutions of diffusive predator-prey models with Holling type-II and Holling type-III, respectively. In this paper, based on the above discussion, we are interested in the existence of traveling wave solutions of a reaction-diffusion Leslie-Gower-type model incorporating a prey refuge, which is modified from model (1). Remark that the methods we use to prove the existence are similar to these in [16, 23, 24], there are several differences For one thing, it is a different model, a modified Leslie-Gower model incorporating a prey refuge.
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