Dynamics of a diffusive food chain model with fear effects
Department of Mathematics, National Institute of Technology Raipur, 492010, Raipur, CG, India
Accepted: 28 November 2021
Published online: 17 December 2021
The traditional prey–predator model emphasizes factors showing an instant effect on population decline, but recent studies have highlighted that both direct killing and the stress of being killed can diminish population densities. Some empirical studies validate that predator fear reduces more population density of prey than direct predation. Therefore, in this article, we have proposed and analysed a three-population-based spatial prey–predator model considering double fear. We induce fear mechanisms in two populations, assuming that fear of the specialist predator suppresses the birth rate of prey and that fear for the generalist predator reduces the growth of the specialist predator. We have determined feasible equilibrium points and parametric conditions to prove their stability. Direction and stability of Hopf bifurcation are determined using central manifold theorems. Existence conditions for global positive solutions and Turing instability are derived for the spatial model system. Simulations show that the cost of fear changes the Turing instability to a homogeneous steady state in both one-dimensional and two-dimensional diffusion cases. Our study shows that the low cost of fear stabilizes the temporal system dynamics, whereas the high cost of fear stabilizes the spatial system dynamics. We observe complex dynamics such as chaos, period-doubling and period-half, Hopf bifurcation concerning predator fear. Various patterns are devised to explain the spatial complexities caused by fear and diffusion coefficients. Extensive numerical simulations are performed to justify these claims and understand and interpret the consequence of fear on the system dynamics.
© The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2021