Stabilization of a Chaotic Predator-Prey System Considering Fractal Boundaries

Abstract

In this paper, the control scheme, which stabilizes the oscillating behavior including chaos, is discussed associated with a discrete predator-prey system described by the two-dimensional nonlinear difference equation. First, the mathematical model is given based on the stock-recruitment concept. It is shown through the stability analysis that the constant and constant rate harvestings of predators contribute to prevent oscillating behavior of the system. Secondly, recognizing the fact that a set of initial conditions of the system often exhibits the fractal boundary, its mechanism is clarified and the necessary condition for existence of fractal boundaries is analytically obtained. Finally, it is shown that, as the amount of the constant harvesting of predators increases, fractal boundaries disappear, while, under the constant rate harvesting, the complexity of fractal boundaries increases.