种间密度依赖模型的分析与操作图
Analysis and operating diagram of an interspecific density-dependent model
AI总结：
本文分析恒化器中两种微生物竞争单一资源的种间密度依赖模型，证明正稳态唯一且不稳定、至多一物种存活，并解析构造操作图，数值验证还发现Bogdanov-Takens和Zero-Hopf等新分岔。
摘要：
本文研究恒化器中两种微生物物种竞争单一资源的模型，该模型包含一般的种间密度依赖增长率，且每个物种具有不同的移除率。我们给出了所有稳态存在性、唯一性和局部稳定性的充分必要条件。我们证明，若正稳态存在，则它是唯一的且不稳定。在这种情况下，系统表现出双稳态，过程的行为依赖于初始条件。我们的数学分析证明，至多只有一个物种能够存活，这证实了竞争排斥原理。我们得出结论：即使在两个物种的动力学中考虑死亡率，仅在经典恒化器模型中添加种间竞争也不足以显示两个物种的共存。此外，我们从理论和数值两方面重点研究了操作图，该操作图根据过程的两个操作参数——稀释率和底物输入浓度——描绘了每个稳态的存在性和稳定性。利用我们的数学分析，我们通过绘制分隔各个区域的曲线来解析地构造操作图。我们使用MATCONT软件进行的数值方法验证了这些理论结果，但它揭示了通过改变两个参数而出现的新分岔，如Bogdanov-Takens分岔和Zero-Hopf分岔。分岔分析表明，所有稳态只能通过跨临界分岔出现或消失。
英文摘要：
This paper studies a two microbial species model in competition for a single resource in the chemostat including general interspecific density-dependent growth rates with distinct removal rates for each species. We give the necessary and sufficient conditions of existence, uniqueness, and local stability of all steady states. We show that a positive steady state, if it exists, then it is unique and unstable. In this case, the system exhibits a bi-stability where the behavior of the process depends on the initial condition. Our mathematical analysis proves that at most one species can survive which confirms the competitive exclusion principle. We conclude that adding only interspecific competition in the classical chemostat model is not sufficient to show the coexistence of two species even considering mortality in the dynamics of two species. Otherwise, we focus on the study, theoretically and numerically, of the operating diagram which depicts the existence and the stability of each steady state according to the two operating parameters of the process which are the dilution rate and the input concentration of the substrate. Using our mathematical analysis, we construct analytically the operating diagram by plotting the curves that separate their various regions. Our numerical method using MATCONT software validates these theoretical results but it reveals new bifurcations that occur by varying two parameters as Bogdanov-Takens and Zero-Hopf bifurcations. The bifurcation analysis shows that all steady states can appear or disappear only through transcritical bifurcations.
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