Abstract
In this paper, a class of predator-prey systems with fear effect is investigated, the integrability conditions of the origin and the positive equilibrium are obtained, and the fact that three limit cycles can be bifurcated from the positive equilibrium is proved, so bistable phenomenon can occur for this system.
Highlights
Predator-prey systems which were considered to be among the most important models in ecological systems have been investigated intensively and extensively since they were proposed; there have been many good results for these systems
A class of predator-prey systems with fear effect is investigated, the integrability conditions of the origin and the positive equilibrium are obtained, and the fact that three limit cycles can be bifurcated from the positive equilibrium is proved, so bistable phenomenon can occur for this system
The population dynamics of predator-prey systems with the Holling type II functional response have been studied by many authors; for such a model, the result about the existence of a unique stable limit cycle has been proved
Summary
Predator-prey systems which were considered to be among the most important models in ecological systems have been investigated intensively and extensively since they were proposed; there have been many good results for these systems. The population dynamics of predator-prey systems with the Holling type II functional response have been studied by many authors; for such a model, the result about the existence of a unique stable limit cycle has been proved. The system of Leslie-type predator-prey schemes with a nonmonotonic functional response and Allee effect allows the existence of three limit cycles under certain conditions over the parameters; by Hopf bifurcation [17], the first two cycles. The authors studied the canard phenomenon for predator-prey systems with response functions of Holling types [18] by using this method. For a Gause type predator-prey system with Holling type III functional response and Allee effect on prey, multiple limit cycles were considered in [19] by multiple Hopf bifurcations. Three limit cycles can be bifurcated from the origin; namely, bistable phenomenon can occur for this system; this is not discussed in previous discussion
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