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Analysis of a Tessiet type food chain chemostat with \(k\)-times’ periodically pulsed input. (English) Zbl 1217.34066
Summary: We introduce and study a Tessiet type food chain chemostat, which contains with predator, prey and \(k\)-times’ periodically pulsed substrate. We investigate the subsystem with substrate and prey and study the stability of the periodic solutions, which are the boundary periodic solutions of the system. The stability analysis of the boundary periodic solution yields an invasion threshold. By use of standard techniques of bifurcation theory, we prove that above this threshold there are periodic oscillations in substrate, prey, and predator. Simple cycles may give way to chaos in a cascade of period-doubling bifurcations. Furthermore, by comparing bifurcation diagrams with different bifurcation parameters, we can see that the impulsive system shows two kinds of bifurcations, whose are period-doubling and period-halfing. When impulsive period is small, there exists quasiperiodic oscillation in the impulsive system.

MSC:
34C23 Bifurcation theory for ordinary differential equations
92D25 Population dynamics (general)
37N25 Dynamical systems in biology
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