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Bifurcation and complexity in a ratio-dependent predator-prey chemostat with pulsed input. (English) Zbl 1150.34009

Summary: A three dimensional ratio-dependent chemostat model with periodically pulsed input is considered. By using the stroboscopic map and Floquet theorem, an exact periodic with positive concentration of substrate and predator in the absence of prey is obtained. When \(\beta\) is less than some critical value the boundary periodic solution \( (x_s (t), 0, z_s (t))\) is locally stable, and when \(\beta\) is larger than the critical value there are periodic oscillations in substrate, prey and predator. Increasing the impulsive period \(\tau\), the system undergoes a series of period-doubling bifurcations leading to chaos, which implies that the dynamical behavior of the periodically pulsed ratio-dependent predator-prey ecosystem are very complex.

MSC:

34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92D25 Population dynamics (general)
34C60 Qualitative investigation and simulation of ordinary differential equation models
34A37 Ordinary differential equations with impulses
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