Stability and Hopf bifurcation in a delayed predator-prey system with parental care for predators

Abstract

In this paper a stage-structured predator-prey model (stage structure on predators) with two discrete time delays has been discussed. It is assumed that immature predators are raised by their parents in the sense that they cannot catch the prey and their foods are provided by parents. We suppose that the growth is of logistic type. The two discrete time delays occur due to gestation delay and maturation delay. Linear stability analysis for both non delays and as well as with delays reveals that certain thresholds have to be maintained for coexistence. We analyzed the global stability of the interior equilibrium and the boundary equilibrium points by using a suitable Lyapunov function. In addition, the normal form of the Hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these Hopf bifurcations. We present some numerical examples to illustrate our analytical works.