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.