Stability and bifurcation on predator-prey systems with nonlocal prey competition

Shanshan Chen; Jianshe Yu

doi:10.3934/dcds.2018002

Article Contents

2018, Volume 38, Issue 1: 43-62. Doi: 10.3934/dcds.2018002

This issue Previous Article Linking curves, sutured manifolds and the Ambrose conjecture for generic 3-manifolds Next Article Surgery on Herman rings of the standard Blaschke family

Stability and bifurcation on predator-prey systems with nonlocal prey competition

Shanshan Chen^1,2, and
Jianshe Yu^1, ,

1.
School of Mathematics and Information Science, Guangzhou University, Guangzhou, Guangdong, 510006, China
2.
Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, 264209, China

* Corresponding author
* Corresponding author

Received: November 2016

Revised: July 2017

Published: October 2017

The authors are supported by the National Natural Science Foundation of China (Nos. 11471085,11771109)

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Abstract

In this paper, we investigate diffusive predator-prey systems with nonlocal intraspecific competition of prey for resources. We prove the existence and uniqueness of positive steady states when the conversion rate is large. To show the existence of complex spatiotemporal patterns, we consider the Hopf bifurcation for a spatially homogeneous kernel function, by using the conversion rate as the bifurcation parameter. Our results suggest that Hopf bifurcation is more likely to occur with nonlocal competition of prey. Moreover, we find that the steady state can lose the stability when conversion rate passes through some Hopf bifurcation value, and the bifurcating periodic solutions near such bifurcation value can be spatially nonhomogeneous. This phenomenon is different from that for the model without nonlocal competition of prey, where the bifurcating periodic solutions are spatially homogeneous near such bifurcation value.

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Mathematics Subject Classification: Primary:35K57, 35B36;Secondary:45K05.

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Figure 1. The constant steady state loses its stability through Hopf bifurcation, and the solution converges to the bifurcated spatially nonhomogeneous periodic solution. Here initial values: $u(x,0)=0.3+0.1\cos^2\frac{x}{4},v(x,0)=0.2+0.1\cos^2\frac{x}{2},x\in[0,2\pi]$. (Upper): $\gamma=4$; (Lower): $\gamma=9$.

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Figure 2. The constant steady state loses its stability through Hopf bifurcation. (Upper): $\gamma=2.7$, and the solution converges to the bifurcated spatially nonhomogeneous periodic solution. Here initial values: $u(x,0)=0.3+0.1\cos^2\frac{x}{3},v(x,0)=0.2+0.1\cos^2\frac{x}{3},x\in[0,1.5\pi]$. (Lower): $\gamma=6$, and the solution converges to the bifurcated spatially homogeneous periodic solution. Here initial values: $u(x,0)=0.7+0.5\cos^2\frac{x}{3},v(x,0)=0.7+0.5\cos^2\frac{x}{3},x\in[0,1.5\pi]$.

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