Research Paper On Hopf Bifurcation

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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 uses as a research technique to culture microorganisms continuously under nutrient limitation in a controlled environment in order to study the general…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

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