Influence of Fear Effect on Bifurcation Dynamics of Predator-prey System in a Predator-poisoned Environment

In this paper, we propose a predator-prey system with fear effect and predator death due to toxins based on the phenomenon that African predators death due to eating toxin-carrying carcasses. First, the existence and stability of equilibria are proved. Second, the conditions for the occurrence of different types of bifurcations are established by varying parameters. Our results indicate that (1) when the cost of minimum fear is taken as the bifurcation parameter, it can determine the stability and direction of periodic solution of Hopf bifurcation; (2) when the predation rate and the cost of minimum fear are taken as the bifurcation parameters, the system undergoes Bogdanov-Takens (BT) bifurcation of codimension two, and the BT bifurcation will disappear when the system is not affected by toxins. At last, some numerical simulations verify the validity of theoretical results, and we obtain that toxins not only inhibit the survival of predator populations but also affect the density of prey populations. Moreover, only an appropriate cost of minimum fear maintains long-term coexistence of prey and predator populations.