Global dynamics and bifurcation analysis of a chemostat model with obligate mutualism and mortality

TL;DR

Overall, this study highlights the critical role of mortality in shaping complex dynamics in obligate mutualism, producing multistability and oscillatory coexistence patterns that may better represent natural microbial or ecological systems.

Abstract

We propose a system of differential equations modeling the competition between two obligate mutualistic species for a single nutrient in a chemostat. Each species promotes the growth of the other, and growth occurs only in the presence of its partner. The three-dimensional model incorporates interspecific density-dependent growth functions and distinct removal rates. We perform a mathematical analysis by characterizing the multiplicity of equilibria and deriving conditions for their existence and stability. Using MatCont, we construct numerical operating diagrams in the parameter space of dilution rate and input substrate concentration, providing a global view of the qualitative dynamics of the system. One-parameter bifurcation diagrams with respect to the input substrate then reveal a variety of dynamical transitions, including saddle-node, Hopf, limit point of cycles LPC, period-doubling PD, and homoclinic bifurcations. When mortality is included, the system exhibits a richer dynamical repertoire than in the mortality-free case, with stable and unstable periodic orbits, tri-stability between equilibria and limit cycles, and several codimension-two bifurcations, including Bogdanov-Takens (BT), cusp of cycles (CPC), resonance points (R1 and R2), and generalized Hopf GH points. These features allow coexistence not only around positive equilibria but also along stable limit cycles, reflecting more realistic ecological dynamics. In contrast, neglecting mortality restricts coexistence to equilibria only. Overall, this study highlights the critical role of mortality in shaping complex dynamics in obligate mutualism, producing multistability and oscillatory coexistence patterns that may better represent natural microbial or ecological systems.

Figures (10)

• Figure 1: Graphical illustration of the existence of solutions of equation \ref{['EquExis_xji']}. Depending on the parameter values, the horizontal line $y=D_i$ may intersect the curve $x_j\mapsto f_i(S_{in}-\tfrac{D_j}{D}x_j,x_j)$ at zero, one, or two points.
• Figure 1: (a) Existence and instability of two positive equilibria for $\sigma_1 < S_{in} < \sigma_6$ with $(S_{in},D)=(3,0.2)$; (b) sign change of $c_4(S_{in})$ for $S_{in} \geq \sigma_6$ with $D=0.2$; (c) evolution of a pair of complex-conjugate eigenvalues as $S_{in}$ decreases. Unstable equilibria are indicated in blue.
• Figure 1: Phase-plane representation of the reduced system \ref{['ModelDiEqualModRed']} illustrating the global stability of the washout equilibrium $E_0$, in the case where the nullclines $\gamma_1$ and $\gamma_2$ do not intersect.
• Figure 1: (a) Operating diagram of \ref{['ModelDiEgaux']} obtained with MatCont. (b) Corresponding one-parameter bifurcation diagram of the substrate concentration $S$ as a function of $S_{in}$ for $D=0.2$.
• Figure 1: MatCont results for $D=0.195$. (a--c) One-parameter bifurcation diagrams of system \ref{['ModelMutualism']} with respect to $S_{in}$, displaying the equilibrium branches of the substrate $S$ and the biomasses $x_1$ and $x_2$. Stable solutions are shown in red and unstable ones in blue. (d--f) Enlarged views near critical parameter values, highlighting the Hopf (H), homoclinic (Hom), and period-doubling (PD) bifurcations, together with the associated branches of periodic solutions.

Theorems & Definitions (31)

• Proposition 2.4
• Lemma 2.5
• Proof 1
• Lemma 2.7
• Proposition 2.8
• Proof 2
• Proposition 2.9
• Proof 3
• Lemma 2.10
• Proof 4