Spatiotemporal Patterns Induced by Hopf Bifurcations in a Homogeneous Diffusive Predator-Prey System

Meng Lin,Xuguang Yang,Yufeng Wang,Yanyou Chai

doi:10.1155/2019/3907453

Abstract

In this paper, we consider a diffusive predator-prey system where the prey exhibits the herd behavior in terms of the square root of the prey population. The model is supposed to impose on homogeneous Neumann boundary conditions in the bounded spatial domain. By using the abstract Hopf bifurcation theory in infinite dimensional dynamical system, we are capable of proving the existence of both spatial homogeneous and nonhomogeneous periodic solutions driven by Hopf bifurcations bifurcating from the positive constant steady state solutions. Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology.

Highlights

Spatiotemporal pattern formations are one of the core problems arising in biology and ecology
Our results allow for the clearer understanding of the mechanism of the spatiotemporal pattern formations of the predator-prey interactions in ecology
In the model of our concern, the prey is assumed to exhibit herd behavior, so that the predator interacts with the prey along the outer corridor of the herd of prey, and the interaction terms are supposed to use the square root of the prey population rather than the prey population, where the use of the square root properly represents the assumption that the interactions occur along the boundary of the population

Summary

Introduction

Spatiotemporal pattern formations are one of the core problems arising in biology and ecology. In [26], the authors used the same methods to consider a kind of homogeneous diffusive predator-prey system with Holling type-II functional response subject to the homogeneous Neumann boundary conditions They found that, under suitable conditions, for any eigenmode cos(nx/l), the predator-prey system in [26] has 2n Hopf bifurcation points. The remaining parts of this paper are organized as follows: in Section 2, we prove the existence of Hopf bifurcations, and consider the detailed properties of Hopf bifurcations by using center manifold methods and normal form methods; in Section 3, we include numerical simulations to support our theoretical analysis; in Section 4, we end up our discussions by drawing some conclusions

Hopf Bifurcation Analysis

Numerical Simulations

Conclusions

Conflicts of Interest