Global analysis of a delayed stage structure prey–predator model with Crowley–Martin type functional response

Abstract

A stage structure prey–predator model that consists of a system of three nonlinear ordinary differential equations in the presence of discrete time delay is proposed and analysed in this paper. The prey population is divided into two categories: immature and mature prey. The predator population depends on mature prey only and that followed by Crowley–Martin type functional response. We analyse positivity, boundedness and existence of equilibrium points. The local and global stability behaviour of the delayed and non-delayed system are also analysed. Considering delay as a bifurcation parameter, the Hopf-bifurcation is also examined for this system. Then we discuss the stability and direction of Hopf-bifurcation using Normal form theory and Centre manifold theory. Numerical simulation is carried out to verify our analytical findings. We observe that, for a set of values of parameters, the bifurcated periodic solution is supercritical, stable with decreasing period and as the time delay increases, interior equilibrium point disappears. Model of this type may be considered to save the immature prey from the predator population and to maintain the prey–predator relation.