Pattern Formation in a Reaction-Diffusion Predator-Prey Model with Weak Allee Effect and Delay

Hua Liu,Ming Ma,Weiyuan Ma,Yong Ye,Yumei Wei,Kai Zhang

doi:10.1155/2019/6282958

Abstract

In this paper, we establish a reaction-diffusion predator-prey model with weak Allee effect and delay and analyze the conditions of Turing instability. The effects of Allee effect and delay on pattern formation are discussed by numerical simulation. The results show that pattern formations change with the addition of weak Allee effect and delay. More specifically, as Allee effect constant and delay increases, coexistence of spotted and stripe patterns, stripe patterns, and mixture patterns emerge successively. From an ecological point of view, we find that Allee effect and delay play an important role in spatial invasion of populations.

Highlights

Since the Allee e ect was proposed by Allee [1] in 1931, the predator-prey model with Allee e ect has been studied extensively [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]
Pattern formation in the predator-prey model is an appropriate tool for understanding the basic mechanism of spatiotemporal population dynamics
With the increase of Allee parameters, the length of stripes decreases and some stripes even form a circle. en, we continue to increase the value of the weak Allee parameter like Figure 3 (A 0.1), and we find that pattern formations have changed again

Summary

Introduction

Since the Allee e ect was proposed by Allee [1] in 1931, the predator-prey model with Allee e ect has been studied extensively [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27]. Sen et al [5] established a twoprey one-predator model with Allee e ect, and the e ects of Allee e ect on the dynamics of predator population are discussed. The research on reaction-di usion predator-prey model with Allee e ect is very rich. Wang et al [7] established a reaction-di usion predator-prey model and found the model dynamics exhibits both Allee e ect and di usion controlled pattern formation growth to holes. Ey studied Allee e ect induced instability in a reaction-di usion predator-prey model [4]. Pattern formation in the predator-prey model is an appropriate tool for understanding the basic mechanism of spatiotemporal population dynamics. We nd that there are few studies on delays in reaction-di usion predator-prey model with Allee e ect. C is the death rate of the predator, e and η are coefficients of light attenuation by water and self-shading in the context of plankton mortality, and τ is the searching delay. e weak Allee effect term is u/u + A, where A > 0 is described as a weak Allee effect constant

Turing Instability

Delay-Induced Instability

Amplitude Equations and Pattern Selection

Numerical Simulations