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Dynamics and pattern formation of a diffusive predator–prey model with predator-taxis

School of Science, Nanjing University of Posts and Telecommunications, Nanjing, Jiangsu 210003, P. R. China

School of Mathematics Sciences and Y.Y. Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, Heilongjiang 150001, P. R. China

E-mail Address: jinfengwangmath@163.com

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, and

Junping Shi

Department of Mathematics, College of William and Mary, Williamsburg, Virginia 23187-8795, USA

E-mail Address: jxshix@wm.edu

Corresponding author.

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https://doi.org/10.1142/S0218202518400158Cited by:105 (Source: Crossref)

This article is part of the issue:

Special Issue on Challenges in Modeling and Analysis of Cross-Diffusion Systems; Guest Editors: N. Bellomo, Y. Tao and M. Winkler

Abstract

We propose a new reaction–diffusion predator–prey model system with predator-taxis in which the preys could move in the opposite direction of predator gradient. A similar situation also occurs when susceptible population avoids the infected ones in epidemic spreading. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any predator-taxis sensitivity coefficient are proved. It is also shown that such predator-taxis does not qualitatively affect the existence and stability of coexistence steady state solutions in many cases. For diffusive predator–prey system with diffusion-induced instability, it is shown that the presence of predator-taxis may annihilate the spatial patterns.

Communicated by N. Bellomo, Y. Tao and M. Winkler

Keywords:

AMSC: 35K57, 35K59, 35B45, 35B32, 92D25, 92D30

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