Electronic Journal of Differential Equations, Vol. 2021 (2021), No. 70, pp. 1-31.

Title: Diffusive predator-prey models with fear effect in spatially heterogeneous environment
       
Authors: Shanbing Li (School of Math. and Statistics, Xidian Univ., Xi'an 710071, China)
         Yanni Xiao (Xi'an Jiaotong Univ., Xi'an 710049, China)
         Yaying Dong (Xi'an Polytechnic Univ., Xi'an 710048, China)

Abstract:
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.