Abstract
We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis.
Highlights
In recent years, much research efforts have been extensively made on interaction and coexistence mechanism of population in prey predator ecosystem by means of Lotka-Volterra dynamical models [1,2,3]
In order to reflect the dynamical behavior of mathematical models depending on the past history, time delay is usually incorporated into model, which can be utilized to mathematically describe hunting delay, maturation delay, and gestation delay for population within prey predator ecosystem
Individuals in each stage are identical in biological characteristics, and some vital rates of individuals in a population almost always depend on stage structure
Summary
Much research efforts have been extensively made on interaction and coexistence mechanism of population in prey predator ecosystem by means of Lotka-Volterra dynamical models [1,2,3]. When the delay kernel function G(s) admits the weak generic kernel, Song and Yuan [10] and Ma et al [12] discussed the local asymptotical stability of interior equilibrium and Hopf bifurcations of nonconstant periodic solutions based on linearization method and regarding the discrete hunting delay τ as bifurcation parameter. Gourley and Kuang [22] and Bandyopadhyay and Banerjee [23] formulated a class of general and robust prey predator models with stage structure and constant maturation time delay and performed a systematic mathematical and computational study. They have shown that there is a window in maturation time delay parameter that generates sustainable oscillatory dynamics.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
