Qualitative Analysis in A Predator-prey Model with Sigmoidal Type Functional Response

In this paper, a cyclic predator-prey system with Sigmoidal type functional response is considered. The stability of the positive equilibrium and existence of Hopf bifurcation is studied by analyzing the distribution of the roots of associated characteristic equation. It is shown that the positive equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the positive equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. An explicit formula for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations is derived, using the normal form theory and center manifold argument. Finally, numerical simulations supporting the theoretical results are carried out. Key–Words: Predator-prey system, Stability, Hopf bifurcation, Sigmoidal type functional response, Time delay