A delay digestion process with application in a three-species ecosystem

In a natural ecosystem, specialist predators feed almost exclusively on one specific species of prey which may be possible for a parasitoid. But generalist predators feed on many types of species. It is also well known that the predation rate increases as prey density rises, but eventually levels off due to the predator’s handling time. The response function, thereby, is often assumed to Holling II functional response. In addition, digestion processes of the predation often involve reactions with delays. In view of these facts, a three-species ecosystem with a delay digestion process and Holling functional response is formulated. By analyzing the corresponding characteristic equations, the stability of the equilibria is investigated. Furthermore, Hopf bifurcations occurring at the positive equilibrium under some conditions are demonstrated. The consequence of global stability of the positive equilibrium is that predation will not irreversibly change the system. That is, as long as preys are not made extinct by excessive predation of their predator, the system is able to recover. Numerical simulations are carried out to illustrate our theoretical results. Meanwhile, they indicate that time delay is harmless for permanence of populations even thought it has a tendency to produce oscillations.