Existence of Multiple Limit Cycles in a Predator-Prey Model with $\arctan(ax)$ as Functional Response

Abstract

We consider a Gause type predator-prey system with functional response given by $θ(x)=\arctan(ax), where $a \gt 0$, and give a counterexample to the criterion given in Attili and Mallak [Commun. Math. Anal. 1:33-40(2006)] for the nonexistence of limit cycles. When this criterion is satisfied, instead this system can have a locally asymptotically stable coexistence equilibrium surrounded by at least two limit cycles.