Multiple Local and Global Bifurcations and Their Role in Quorum Sensing Dynamics

TL;DR

The study addresses the complexity of quorum-sensing dynamics by formulating a three-component model coupling autoinducer concentration to two bacterial subtypes and by performing detailed bifurcation analyses. It develops a constant-autoinducer reduction to a 2D system, identifies parameter regimes with multiple steady states, and characterizes stability and local bifurcations (Hopf, Bogdanov--Takens). Extending to the full 3D system reveals Shilnikov-type homoclinic chaos and a rich network of codimension-one and -two bifurcations organized around BT points, with two-parameter continuations mapping regions of steady, periodic, and chaotic dynamics. The findings demonstrate how QS can exhibit robust synchronisation as well as chaotic fluctuations, offering insights for predicting and potentially controlling bacterial collective behaviour; future work could incorporate time delays and migration to broaden the model's realism.

Abstract

Quorum sensing governs bacterial communication, playing a crucial role in regulating population behaviour. We propose a mathematical model that uncovers chaotic dynamics within quorum sensing networks, highlighting challenges to predictability. The model explores interactions between autoinducers and two bacterial subtypes, revealing oscillatory dynamics in both a constant autoinducer sub-model and the full three-component model. In the latter case, we find that the complicated dynamics can be explained by the presence of homoclinic Shilnikov bifurcations. We employed a combination of normal form analysis and numerical continuation methods to analyse the system.

Figures (10)

• Figure 1: Nullclines sketches for key parameter values of $e$, where $b>K>a$. Panels (a) and (c) show three steady-states, where two are positive (within first quadrant) and the third one is the origin; in panel (b) three positive steady-states are depicted; and panel (d) shows only one positive steady-state. Panels (a) and (c) correspond to scenarios approximately at two saddle-node bifurcations. Here $\tilde{f}(u,v)=0$ and $\tilde{g}(u,v)=0$ stand for $\dot{u}=0$ and $\dot{v}=0$ in \ref{['eq:4.11']}, respectively.
• Figure 2: Bifurcation diagrams for bifurcation parameters $e$ and $b$. Dashed and solid curves correspond to unstable and stable steady-states, respectively, where heavily dashed branches gather repeller steady-states while saddle steady-states are collected by slightly dashed branches, and solid grey branch amass stable limit-cycles. (a) With $b=1.4$ and $a=0.042$ fixed, we identify two saddle-node (SN) points located at $e_1\approx1.948551$ and $e_2\approx3.497398$. (b) Upon fixing $e=3$ and $a=0.042$, we find two SN points at $b_1\approx1.128239$ and $b_2\approx1.512641$ as well as a Hopf (H) point at $b_3=1.168852$ which terminates at $b_4\approx1.239672$. Parameter value for $K=0.043$.
• Figure 3: Two parameter continuation. The solid curve corresponds to the SN locus. A CP point is located at $(e,a)\approx(1.723644,0.123118)$ and a BT point at $(e,a)\approx(2.490826,0.066303)$. Other parameters values: $b=1.4$ and $K=0.043$.
• Figure 4: Two-parameter continuation for parameters $b$ and $e$. Four possible scenarios are categorised as follows. I: Stable steady-state, the population converges to a positive population density; IIa: Oscillation dynamics, the bacteria approach a periodic solution; IIb: Oscillation dynamics and stable steady-state (bistability), this region shows oscillation dynamics and a stable node. The bold dashed curve captures homoclinic nonlinear transitions (hom); III: Stable steady-state, no periodic solutions arise as orbits asymptotically converge to a steady state; IV: Two stable steady-states (bistability), there are two stable steady-states, a node and a focus, which are separated by a saddle point. The remaining parameters are set to: $a=0.042$ and $K=0.043$.
• Figure 5: A solution sample for time-dependent parameter $b$: (a) a solution with $b$ linearly ranging from $b=1$ to $b=3$ with $e=1$ fixed; (b) a solution with parameter $b$ going backwards from $b=3$ to $b=1$ with $e=1$ fixed; (c) parameter $b$ linearly increases from $b=1.0$ to $b=1.5$ with $e=1.5$ fixed. Initial conditions are set to $(u,v)=(0.5,0.5)$. Other parameters values: $a=0.042$ and $K=0.043$.

Theorems & Definitions (11)

• Proposition 1
• Proposition 2
• Proof 1
• Theorem 3
• Lemma 4
• Proof 2
• Lemma 5
• Proof 3
• Lemma 6
• Proof 4