Hopf Bfurcation Theory

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Abstract. In this paper, a classical model describing a food web in a chemostat involving three species competing for non-reproducing, growth rate-limiting nutrient in which one of the competitors predates on one of the other competitors is considered. Quantitative analyses of non-negativity and boundedness of solution trajectories, dissipativity, and behavior around equilibria, global stability and persistence of the model equations are analyzed. We present the global stability of equilibria by constructing a Lyapunov function. Hopf bifurcation theory is applied. Keywords: Chemostat; Food web; Global stability; Hopf bifurcation; Dissipative. 1. Introduction In microbiology and population biology, the laboratory device chemostat extensively…show more content…
Since one can measure the control parameter easily, the device has various applications in ecology and population biology. It can be viewed as a simple lake system in ecology while it serves as a laboratory bio-reactor in chemical engineering used for investigations in genetically altered cell. As for example, the prey (bacteria) consumes nutrient (waste) while the predator (ciliates) feeds on the prey in waste water treatment process. It is of mathematical interest to construct models with chemostat. The dynamics of chemostat model with nutrient uptake is of Monod kinetics play an important role in population ecology. After the first introduction of chemostat the researchers have paid their attention to develop mathematical theories of models in it. Qualitative analyses of predator-prey models in chemostat both from the experimental and the modeling aspect describe by set of differential equations were studied by many authors (Aris and…show more content…
In (Butler and Wolkowicz, 1986), they considered the assumptions that the response functions are general monotone functions. But in (Wolkowicz, 2006), the response functions that satisfy the law of mass action were assumed. On the other hand, a food web in chemostat where the predator predates on both of the preys only and the preys consume the nutrient had been studied theoretically by (Nasrin and Rana, 2011), experimentally in lab by (Jost et al., 1973) and numerically by (Vayenas and Pavlou, 1999). In all of their models, they employed either general monotone functions or Michaelis-Menten kinetics to describe nutrient uptake and competitor (prey) growth. In this paper, we introduce a predator population and restrict our attention to the case of only two competitors. We are going to examine the dynamics of a food web where the competitors consume the nutrient and the predator consumes one of the competitors as prey in the chemostat that incorporates general response functions. Our system can be considered as resulting from the predator-prey system (El-Owaidy and Moniem, 2003) by adding another competitor or from the competition system (Li, 1999) by adding a competitor preying upon one of the competitors. In this work we confine our

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