Mathematics Faculty Articles

Title

Bifurcation Analysis for a One Predator and Two Prey Model with Prey-Taxis

Document Type

Article

Publication Date

2021

Publication Title

Journal of Biological Systems

Keywords

Competing species, Predator prey, Prey-Taxis, Pattern Formation, Bifurcation

ISSN

1793-6470

Volume

29

Issue/No.

2

First Page

495

Last Page

524

Abstract

This paper concerns spatio-temporal pattern formation in a model for two competing prey populations with a common predator population whose movement is biased by direct prey-taxis mechanisms. By pattern formation, we mean the existence of stable, positive non-constant equilibrium states, or nontrivial stable time-periodic states. The taxis can be either repulsive or attractive and the population interaction dynamics is fairly general. Both types of pattern formation arise as one-parameter bifurcating solution branches from an unstable constant stationary state. In the absence of our taxis mechanism, the coexistence positive steady state, under suitable conditions, is locally asymptotically stable. In the presence of a sufficiently strong repulsive prey defense, pattern formation will develop. However, in the attractive taxis case, the attraction needs to be sufficiently weak for pattern formation to develop. Our method is an application of the Crandall–Rabinowitz and the Hopf bifurcation theories. We establish the existence of both types of branches and develop expressions for determining their stability.

Comments

AMSC: 35K59, 35Q92, 92D25

DOI

10.1142/S0218339021400131

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Peer Reviewed

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