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Chaos in a Lotka-Volterra predator-prey system with periodically impulsive ratio-harvesting the prey and time delays. (English) Zbl 1130.37042

Summary: We introduce and study a Lotka-Volterra predator-prey system with impulsive ratio-harvesting the prey and time delays. By using Floquet theory and small amplitude perturbation skills, we discuss the boundary periodic solutions for predator-prey system under periodic pulsed conditions. The stability analysis of the boundary periodic solution yields an invasion threshold of the predator. Further, by use of the coincidence degree theorem and its related continuous theorem we prove the existence of the positive periodic solutions of the system when the value of the coefficient is large than the threshold. Finally, by comparing bifurcation diagrams with different bifurcation parameters, we show that the impulsive effect and the time delays bring to the system to be more complex, which experiences a complex process of cycles \(\rightarrow \) quasi-periodic oscillation \(\rightarrow \) periodic doubling cascade \(\rightarrow \) chaos.

MSC:

37N25 Dynamical systems in biology
92D25 Population dynamics (general)
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C28 Complex behavior and chaotic systems of ordinary differential equations
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