Global Stability and Bifurcation Analysis of a Harvested Stage Structure Predator-Prey System with Linear Functional Response

Abstract

The global properties of a harvested stage-structured predator-prey model with linear functional response and constant delay are studied using Lyapunov functions and LaSalle’s invariance principle. It is shown that time delay and the harvesting effort can cause a stable equilibrium to become unstable. A condition which leads to the extinction of the predators is indicated. We show also, that the
predator coexists with prey permanently if and only if the predator’s recruitment rate at the peak of prey abundance is larger than its harvesting rate. By choosing the delay as a bifurcation parameter, we show both analytically and numerically that Hopf bifurcation can occur as the delay crosses some critical value. We, also present results on positivity and boundedness of the solution.

Key words and phrases. Stage structure; Globally stable; Hopf bifurcation; Harvesting; Population model; Lyapunov function.

2000 Mathematics Subject Classification. 92D25, 46A45