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Species permanence and dynamical behavior analysis of an impulsively controlled ecological system with distributed time delay. (English) Zbl 1198.34171
Summary: On the basis of the theories and methods of ecology and ordinary differential equations, an ecological system with impulsive harvest and distributed time delay is established. By using the theories of impulsive equations, small amplitude perturbation skills and comparison techniques, we get a condition which guarantees the global asymptotical stability of the prey-\((x)\) eradication and predator-\((z)\) eradication periodic solution. Further, the influences of the impulsive perturbation on the inherent oscillation are studied numerically, and shows rich dynamics, such as period-doubling bifurcation, chaotic bands, periodic windows, chaotic crises, etc. Moreover, the computation of the largest Lyapunov exponent shows the chaotic dynamic behavior of the model. Meanwhile, we investigate the qualitative nature of the strange attractor by using Fourier spectra. All of these results may be useful in the study of the dynamic complexity of ecosystems.

MSC:
34K20 Stability theory of functional-differential equations
34K45 Functional-differential equations with impulses
92D40 Ecology
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