Ratio-dependent predator–prey model of interacting population with delay effect

Abstract

A system of delay differential equation is proposed to account the effect of delay in the predator–prey model of interacting population. In this article, the modified ratio-dependent Bazykin model with delay in predator equation has been considered. The essential mathematical features of the proposed model are analyzed with the help of equilibria, local and global stability analysis, and bifurcation theory. The parametric space under which the system enters into a Hopf-bifurcation has been investigated. Global stability results are obtained by constructing suitable Lyapunov functions. We derive the explicit formulae for determining the stability, direction, and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Using the global Hopf-bifurcation result of Wu (Trans. Am. Math. Soc., 350:4799–4838, 1998) for functional differential equations, the global existence of periodic solutions has been established. Our analytical findings are supported by numerical experiments. Biological implication of the analytical findings are discussed in the conclusion section.