The objective of this study is to analyze a model of competition for one resource in the chemostat with general inter-specific density-dependent growth rates, taking into account the predator-prey relationship. This relationship is characterized by the fact that the prey species promotes the growth of the predator species which in turn inhibits the growth of the first species. The model is a three-dimensional system of ordinary differential equations. With the same dilution rates, the model can be reduced to a planar system where the two models have the same local and even global behavior. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. Using the nullcline method, we present a geometric characterization of the existence and stability of all equilibria showing the multiplicity of coexistence steady states. The bifurcation diagrams illustrate that the steady states can appear or disappear only through saddle-node or transcritical bifurcations. Moreover, the operating diagrams describe the asymptotic behavior of this system by varying the control parameters and show the effect of the inhibition of predation on the emergence of the bistability region and the reduction until the disappearance of the coexistence region by increasing this inhibition parameter.
<p style='text-indent:20px;'>We study an interspecific, density-dependent model of two species competing for a single nutrient in a chemostat, allowing for a predator-prey relationship between them. We have previously examined the system in the absence of species mortality, showing that multiple positive steady states can appear and disappear through a saddle-node or transcritical bifurcation. In this paper we include mortality. We give a complete analysis for the existence and local stability of all steady states of the three-dimensional system that cannot be reduced to two dimensional ones. Specializing the forms of the rate functions, we show how mortality destabilizes the positive steady state and that stable limit cycles emerge through supercritical Hopf bifurcations. To describe how the process behaves with respect to the choice of dilution rate and input concentration as control parameters, we determine the operating diagram theoretically and also numerically by using the software package MATCONT. The bifurcation diagram based on the input concentration shows various types of bifurcations of steady states, and coexistence either at a positive steady state or via sustained oscillations.</p>
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