Pattern Dynamics in a Spatial Predator-Prey System with Allee Effect

Gui-Quan Sun,Li Li,Tao Zhou,Zi-Ke Zhang,Zhen Jin

doi:10.1155/2013/921879

Abstract

We investigate the spatial dynamics of a predator-prey system with Allee effect. By using bifurcation analysis, the exact Turing domain is found in the parameters space. Furthermore, we obtain the amplitude equations and determine the stability of different patterns. In Turing space, it is found that predator-prey systems with Allee effect have rich dynamics. Our results indicate that predator mortality plays an important role in the pattern formation of populations. More specifically, as predator mortality rate increases, coexistence of spotted and stripe patterns, stripe patterns, spotted patterns, and spiral wave emerge successively. The results enrich the finding in the spatial predator-prey systems well.

Highlights

The Allee effect, named after the ecologist Warder Clyde Allee, has been recognized as an important phenomenon of positive density dependence in low-density population [1,2,3,4,5]
We investigated the pattern dynamics of a spatial predatorprey systems with Allee effect
It is found that the predator-prey systems with Allee effect have rich spatial dynamics by performing a series of numerical simulations

Summary

Introduction

The Allee effect, named after the ecologist Warder Clyde Allee, has been recognized as an important phenomenon of positive density dependence in low-density population [1,2,3,4,5]. Long time series of the density of both prey and predator is needed, so it is difficult to analyse their dynamics. There have been a large group of papers on predator-prey systems with Allee effect [8,9,10,11,12,13] These previous works did not take into account the effect of space. Morozov et al found that the temporal population oscillations can exhibit chaotic dynamics even when the distribution of the species in the space was regular [15] They found that the chaos accompanied with patch invasion even though the environments were heterogeneous [16]. It is natural to ask what kind of patterns can be obtained in predator-prey systems with Allee effect by using other initial conditions.

A Predator-Prey System with Allee Effect

Conclusion and Discussion