The objective of this study is to analyze a model of competition for one resource in the chemostat with general interspecific 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.

中文翻译：

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恒化器中捕食者 - 猎物关系的种间密度依赖性模型

本研究的目的是分析恒化器中一种资源的竞争模型，该模型具有一般的种间密度依赖性生长速率，同时考虑到捕食者 - 猎物的关系。这种关系的特点是猎物物种促进了捕食者物种的生长，而捕食者物种反过来又抑制了第一个物种的生长。该模型是常微分方程的三维系统。在相同的稀释率下，模型可以简化为一个平面系统，其中两个模型具有相同的局部甚至全局行为。根据运行参数确定平面内简化模型的所有稳态的存在和稳定条件。使用 nullcline 方法，我们提出了所有平衡的存在和稳定性的几何表征，显示了共存稳态的多样性。分岔图说明稳态只能通过鞍节点或跨临界分岔出现或消失。此外，操作图通过改变控制参数描述了该系统的渐近行为，并通过增加该抑制参数显示了抑制捕食对双稳态区域出现和减少直到共存区域消失的影响。