Mathematics analysis and chaos in an ecological model with an impulsive control strategy

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, an ecological model with an impulsive control strategy is established. By using the theories of impulsive equation, small amplitude perturbation skills and comparison technique, we get the condition which guarantees the global asymptotical stability of the lowest-level prey and mid-level predator eradication periodic solution. It is proved that the system is permanent. Further, influences of the impulsive perturbation on the inherent oscillation are studied numerically, which shows rich dynamics, such as period-doubling bifurcation, period-halving bifurcation, chaotic band, narrow or wide periodic window, chaotic crises,etc. Moreover, the computation of the largest Lyapunov exponent demonstrates the chaotic dynamic behavior of the model. At the same time, we investigate the qualitative nature of strange attractor by using Fourier spectra. All these results may be useful for study of the dynamic complexity of ecosystems.