Delay Fishing Items Predator - Prey Model Dynamics

In recent decades, more and more scholars have considered studying biological systems by maths model, and discovering their dynamical behaviors. From the view of human's practical needs, a ratio-dependent model with time delayed harvesting is dis-cussed in Chapter 2. By the Hopf bifurcation theory, we choose the delay as bifurcation parameter and find that when the delay passes through the critical value, the stability of positive equilibrium changes and bifurcates a family of periodic orbits. Using Hassard method and the center manifold theory, further, we can further obtain the direction and stability of the periodic orbits.Recently, the chaos in biological systems is investigated by many authors. In Chapter 3, we study the dynamical properties of a Hasting-Powell model with time delayed harvesting. Similarly, we take the delay as the bifurcation parameter and find that when the delay passes through the critical value, the positive equilibrium changes from stable to unstable. On some parameters periodic solutions appear, even chaos can be observed. Finally a numerical simulation is given to support our results.