<p>This paper centers on a chemostat model with distributed delay, where the maximal growth rate of microorganisms adheres to the log-normal Ornstein–Uhlenbeck process. Our analysis relies on the linear chain technique. We first show the existence and uniqueness of the global positive solution. Subsequently, via constructing appropriate Lyapunov functions, we determine sufficient condition for the existence of stationary distribution. From a biological point of view, the existence of stationary distribution suggests that the microorganisms have the ability to survive over an extended period. Furthermore, for the case of weak kernel and strong kernel, we derive the exact expression of local probability density function through matrix transformation respectively. In addition, the condition for extinction of the microorganisms is given. Theoretical analysis underscores that the dynamics of the model under consideration are governed by two pivotal parameters <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1237_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{0}^{s}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1237_Article_IEq2.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{0}^{e}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mi>e</mi> </msubsup> </math></EquationSource> </InlineEquation>, which are interconnected with <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1237_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> of corresponding deterministic system. Finally, several numerical examples are provided to corroborate the theoretical analysis results and illustrate the impact of main parameters on the dynamics.</p>

Dynamical Properties of a Chemostat Model with Log-Normal Ornstein–Uhlenbeck Process and Distributed Delay

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发表日期：

2025年2月18日

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Miaomiao Gao,
Daqing Jiang

摘要

This paper centers on a chemostat model with distributed delay, where the maximal growth rate of microorganisms adheres to the log-normal Ornstein–Uhlenbeck process. Our analysis relies on the linear chain technique. We first show the existence and uniqueness of the global positive solution. Subsequently, via constructing appropriate Lyapunov functions, we determine sufficient condition for the existence of stationary distribution. From a biological point of view, the existence of stationary distribution suggests that the microorganisms have the ability to survive over an extended period. Furthermore, for the case of weak kernel and strong kernel, we derive the exact expression of local probability density function through matrix transformation respectively. In addition, the condition for extinction of the microorganisms is given. Theoretical analysis underscores that the dynamics of the model under consideration are governed by two pivotal parameters $R_{0}^{s}$ $R_{0}^{s}$ and $R_{0}^{e}$ $R_{0}^{e}$ , which are interconnected with $R_{0}$ $R_{0}$ of corresponding deterministic system. Finally, several numerical examples are provided to corroborate the theoretical analysis results and illustrate the impact of main parameters on the dynamics.