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<p>In this paper, we study a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat, where the plasmid-bearing population produces an allelopathic toxin that is lethal to its plasmid-free competitor. The model incorporates general monotonic growth rate functions and distinct removal rates for both species. We provide a complete analysis of the existence and local stability of all steady states in the four-dimensional system. With identical removal rates and Monod-type growth functions, the model was previously investigated by Hsu and Waltman. They showed that the system could exhibit two positive equilibria and conjectured that one of them is locally asymptotically stable whenever it exists, while the other is unstable. We confirm this conjecture in the present work. By including the different removal rates, it is shown that one of the positive equilibria destabilizes with the emergence of a stable limit cycle through supercritical Hopf bifurcations. Moreover, the operating diagram, which describes some asymptotic behavior of the model by varying the operating parameters, is presented. The bifurcation diagram as a function of the input concentration illustrates the various types of bifurcations of equilibria and the coexistence either around a positive equilibrium or sustained oscillations.</p>

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Mortality can induce limit cycles in a chemostat competition model between plasmid-bearing and plasmid-free organisms with allelopathy

  • 期刊论文

  • 发表日期：

    2026年1月24日

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  • Mohamed Dellal

摘要

In this paper, we study a model of competition between plasmid-bearing and plasmid-free organisms in a chemostat, where the plasmid-bearing population produces an allelopathic toxin that is lethal to its plasmid-free competitor. The model incorporates general monotonic growth rate functions and distinct removal rates for both species. We provide a complete analysis of the existence and local stability of all steady states in the four-dimensional system. With identical removal rates and Monod-type growth functions, the model was previously investigated by Hsu and Waltman. They showed that the system could exhibit two positive equilibria and conjectured that one of them is locally asymptotically stable whenever it exists, while the other is unstable. We confirm this conjecture in the present work. By including the different removal rates, it is shown that one of the positive equilibria destabilizes with the emergence of a stable limit cycle through supercritical Hopf bifurcations. Moreover, the operating diagram, which describes some asymptotic behavior of the model by varying the operating parameters, is presented. The bifurcation diagram as a function of the input concentration illustrates the various types of bifurcations of equilibria and the coexistence either around a positive equilibrium or sustained oscillations.