Electron. J. Differential Equations, Vol. 2021 (2021), No. 70, pp. 1-31.

Diffusive predator-prey models with fear effect in spatially heterogeneous environment

Shanbing Li, Yanni Xiao, Yaying Dong

This article concerns diffusive predator-prey models incorporating the cost of fear and environmental heterogeneity. Under homogeneous Neumann boundary conditions, we establish the uniform boundedness of global solutions and global stability of the trivial and semi-trivial solutions for the parabolic system. For the corresponding steady-state problem, we obtain sufficient conditions for the existence of positive steady states, and then study the effects of functional responses and the cost of fear on the existence, stability and number of positive steady states. We also discuss the effects of spatial heterogeneity and spatial diffusion on the dynamic behavior and establish asymptotic profiles of positive steady states as the diffusion rate of prey or predator individuals approaches zero or infinity. Our theoretical results suggest that fear plays a very important role in determining the dynamic behavior of the models, and it is necessary to revisit existing predator-prey models by incorporating the cost of fear.

Submitted November 16, 2020. Published August 23, 2021.
Math Subject Classifications: 35B30, 35B32, 92D25.
Key Words: Predator-prey model; fear cost; spatial diffusion; environmental heterogeneity.

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Shanbing Li
School of Mathematics and Statistics
Xidian University
Xi'an 710071, China
email: lishanbing@xidian.edu.cn
Yanni Xiao
School of Mathematics and Statistics
Xi'an Jiaotong University
Xi'an 710049, China
email: yxiao@mail.xjtu.edu.cn
Yaying Dong
School of Science
Xi'an Polytechnic University
Xi'an 710048, China
email: dongyaying@xpu.edu.cn

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