Hopf Bifurcation And Turing Pattern Of A Diffusive R-M Model With Hyperbolic Tangent Functional Response And Fear Factor

In this paper,the dynamic behavior of a diffusive Rosenzweig-MacArthur predator-prey model with hyperbolic tangent functional response and fear factor is investigated.For the local system,we give a detailed classification of equilibria and perform bifurcation analysis.Furthermore,the existence of limit cycles is dis-cussed.Numerical simulations show that bi-stability phenomenon appears in the system without fear factor.This suggests that capturing efficiency of predator is more powerful than fear of predator in stabilizing the system.For the reaction-diffusion system,we consider the local stability of a positive equilibrium,Turing instability of both positive equilibrium and spatially homogeneous periodic orbits,the direction of Hopf bifurcation and the stability of bifurcating periodic solutions,the existence of positive nonconstant steady states.However,in the absence of fear factor,Turing instability of both positive equilibrium and spatially homogeneous periodic orbits does not occur.System only undergoes a supercritical Hopf bifur-cation and the bifurcating periodic solutions are orbitally asymptotically stable.Meanwhile,numerical examples are given to illustrate the corresponding analytical results.