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Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion

Part of: Equations of mathematical physics and other areas of application Partial differential equations Parabolic equations and systems

Published online by Cambridge University Press:  12 August 2020

HAI-YANG JIN  and
ZHI-AN WANG Open the ORCID record for ZHI-AN WANG [Opens in a new window]
HAI-YANG JIN
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou510640, China, email: mahyjin@scut.edu.cn
ZHI-AN WANG
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, email: mawza@polyu.edu.hk
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Abstract

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

Keywords

Predator–prey systemprey-taxisboundednessglobal stabilityLyapunov functional

MSC classification

Primary: 35K51: Initial-boundary value problems for second-order parabolic systems
Secondary: 35A01: Existence problems 35B40: Asymptotic behavior of solutions 35K57: Reaction-diffusion equations 35Q92: PDEs in connection with biology and other natural sciences
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 4 , August 2021 , pp. 652 - 682
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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