Stability, Nonlinear Oscillations and Bifurcation in a Delay-Induced Predator-Prey System with Harvesting

摘要

A harvested predator-prey system that incorporates feedback delay in prey growth rate is studied. Consideringdelay as a parameter, we investigate the effect of delay on thestability of the coexisting equilibrium. It is observed that thereexists a critical length of the delay parameter below whichthe coexistence equilibrium is stable and above which it isunstable. A Hopf bifurcation exists when delay crosses thecritical value. By applying the normal form theory and thecenter manifold theorem, we determine the explicit formulaethat demonstrate the stability and direction of the bifurcatingperiodic solutions. Computer simulations have been carriedout to illustrate different analytical findings. Our simulationresults indicate that the Hopf bifurcation is supercritical and thebifurcating periodic solutions are unstable for the consideredparameter values. The system, however, can be stabilized fromits unstable oscillatory state only by lowering the intrinsicgrowth rate of prey population.