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Analysis and operating diagram of an interspecific density-dependent model

Tahani Mtar, Radhouane Fekih-Salem

TL;DR

This work analyzes a two-species chemostat model with interspecific density-dependent growth and distinct removal rates, addressing coexistence vs. exclusion under the classical ecological framework. By deriving necessary and sufficient conditions for all steady states, it proves the competitive exclusion principle holds: at most one species can persist, with a unique unstable positive equilibrium ${\mathcal{E}}^*$ when it exists, yielding bi-stability between the boundary equilibria ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ in certain regions. The authors construct an analytic operating diagram in the $(\,S_{in},D\,)$ plane delineated by curves corresponding to break-even conditions and intersection of growth curves, and validate it with MATCONT numerical continuation, which additionally uncovers Bogdanov-Takens and Zero-Hopf bifurcations. The results show that mortality alone cannot guarantee coexistence under interspecific interference, highlighting the need for intra- or alternative interaction mechanisms to sustain two species in a chemostat.

Abstract

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.

Analysis and operating diagram of an interspecific density-dependent model

TL;DR

This work analyzes a two-species chemostat model with interspecific density-dependent growth and distinct removal rates, addressing coexistence vs. exclusion under the classical ecological framework. By deriving necessary and sufficient conditions for all steady states, it proves the competitive exclusion principle holds: at most one species can persist, with a unique unstable positive equilibrium when it exists, yielding bi-stability between the boundary equilibria and in certain regions. The authors construct an analytic operating diagram in the plane delineated by curves corresponding to break-even conditions and intersection of growth curves, and validate it with MATCONT numerical continuation, which additionally uncovers Bogdanov-Takens and Zero-Hopf bifurcations. The results show that mortality alone cannot guarantee coexistence under interspecific interference, highlighting the need for intra- or alternative interaction mechanisms to sustain two species in a chemostat.

Abstract

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.
Paper Structure (8 sections, 10 theorems, 18 equations, 5 figures, 11 tables)

This paper contains 8 sections, 10 theorems, 18 equations, 5 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Mathematical model
  3. Existence and stability of steady states
  4. Operating diagram
  5. Bifurcation analysis and numerical simulations
  6. Conclusion
  7. Proofs
  8. Parameters used in numerical simulations

Key Result

Proposition 2.3

Assume that hyp1hyp2 hold. For any nonnegative initial condition, the solution of system DDInterSpecMod exists for all $t \geq 0$, remains nonnegative and is bounded. Let $D_{\min}=\min (D,D_1,D_2)$. The set is positively invariant and is a global attractor for the dynamics DDInterSpecMod.

Figures (5)

  • Figure 1: Relative positions of $\tilde{x}_i$ and $\bar{x}_i$, $i=1,2$: (a) Case 1 of \ref{['Cases123']} no intersection, (b) Case 2 of \ref{['Cases123']} a unique intersection, (c) Case 3 of \ref{['Cases123']} no intersection. The red [resp. blue] color is for the LES [resp. unstable] steady state.
  • Figure 1: MAPLE: (a) operating diagram of \ref{['DDInterSpecMod']} in the $(S_{in},D)$-plane. (b) Magnification on the regions $\mathcal{J}_2$ and $\mathcal{J}_4$ when $(S_{in},D) \in [0,1]\times[0,2]$.
  • Figure 1: (a) One-parameter bifurcation diagram of \ref{['DDInterSpecMod']} in the variable $S$ with $D$ as the bifurcation parameter and $S_{in}=1$. (b) Magnification when $D \in [\sigma_3,\sigma_4]$. Blue dashed curves correspond to unstable steady states and red solid curves to LES steady states. The green solid diamonds represent the transcritical bifurcations.
  • Figure 2: MATCONT: (a) Operating diagram of \ref{['DDInterSpecMod']}. (b) Magnification of the two-parameter bifurcations at points ZH and BT when $(S_{in},D) \in [0,1]\times[0,2]$.
  • Figure 2: SCILAB: the three-dimensional space $(S, x_1, x_2)$ of \ref{['DDInterSpecMod']} according to regions in the operating diagram of \ref{['FigDOMaple', 'Fig-DOMatc']} when $S_{in}=1$. (a) Convergence to $\mathcal{E}_1$ for $(S_{in},D)\in \mathcal{J}_5$. (b) Convergence to $\mathcal{E}_2$ for $(S_{in},D)\in \mathcal{J}_4$. (c) Bistability of $\mathcal{E}_1$ and $\mathcal{E}_2$ when $(S_{in},D)\in \mathcal{J}_3$.

Theorems & Definitions (14)

  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Remark 3.5
  • Proposition 3.6
  • Corollary 3.7
  • Proposition 4.1
  • Proposition 5.1
  • ...and 4 more