Global dynamics of a delayed predator–prey model with stage structure for the predator and the prey

E-ISSN： 1099-1476|38|17|3937-3949

ISSN： 0170-4214

Source： MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Vol.38, Iss.17, 2015-11, pp. : 3937-3949

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Abstract

A delayed predator–prey system with Holling type II functional response and stage structure for both the predator and the prey is investigated. By analyzing the corresponding characteristic equations, the local stability of each feasible equilibrium of the system is discussed, and the existence of a Hopf bifurcation at the coexistence equilibrium is established. By means of the persistence theory for infinite dimensional systems, it is proven that the system is permanent if the coexistence equilibrium exists. By using suitable Lyapunov functions and the LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when both the predator–extinction equilibrium and the coexistence equilibrium do not exist, and that the predator–extinction equilibrium is globally stable when the coexistence equilibrium does not exist. Further, sufficient conditions are obtained for the global stability of the coexistence equilibrium. Numerical simulations are carried out to illustrate the main theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.