Bifurcation of periodic solution in a Predator–Prey type of systems with non-monotonic response function and periodic perturbation

•In this paper we consider a system of non-autonomous differential equations: ẋ=x(1−λx−y1αx2+βx+1),ẏ=y(−δ−μy+x1αx2+βx+1),where the dot denotes derivation with respect to time: t.•This system is known in the literature as a predator–prey type of dynamical systems with response function of the type Holling IV, where a time-periodic perturbation has been added.•The analysis is done by extending the system to four dimensional autonomous system of differential equations.•Apart from performing numerical bifurcation analysis using continuation software AUTO, we present also the analysis for the number of equilibria in the system by using geometric argument.•Furthermore, we present an alternative proof for the period of the periodic solution of a periodic vector field.•Exciting bifurcation such as cusp, and Bogdanov–Takens bifurcation for ε= 0 has been observed.•Furthermore, we have observed the occurrence of a Swallowtail Bifurcation for periodic solution.

A Predator–Prey type of dynamical systems with non-monotonic response function and time-periodic perturbation is considered in this paper. We present a proof for the number of equilibria in the unperturbed system at some parts of the parameter space. The perturbed system is a dynamical system defined by a periodic vector field. We present an alternative proof for a classical result on the period of the periodic solution. By using a numerical continuation method AUTO (Doedel et al., 1986 [9]), we present a bifurcation analysis for periodic solution of the perturbed system where we found fold, cusp and Swallowtail bifurcations.