Transitions to Intermittent Chaos in Quorum Sensing Dynamics

TL;DR

This paper analyzes a QS-inspired system with two heterogeneous delays $\tau_1$ and $\tau_2$ representing distinct response times of motile and static subpopulations. A nonlinear time-delayed model with an activator–inhibitor structure is analyzed via a pseudo-characteristic polynomial and Hopf-bifurcation criteria, revealing how delay differences qualitatively reshape stability and enable self-sustained oscillations. Numerically, the authors map Hopf points, torus and fold bifurcations, and uncover intermittent chaos—characterized by laminar–chaotic bursts—driven by the interaction of dual delays. The findings shed light on how temporal heterogeneity can induce complex dynamical transitions in QS networks, with implications for modulating intercellular communication and designing temporally controlled synthetic systems.

Abstract

This study analyses the dynamical consequences of heterogeneous temporal delays within a quorum sensing-inspired (QS-inspired) system, specifically addressing the differential response kinetics of two sub-populations to signalling molecules. A nonlinear delay differential equation (DDE) model, predicated upon an activator-inhibitor framework, is formulated to represent the interspecies interactions. Key analytical techniques, including the derivation of the pseudo-characteristic polynomial and the determination of Hopf bifurcation criteria, are employed to investigate the stability properties of steady-state solutions. The analysis reveals the critical role of multiple, dissimilar delays in modulating system dynamics and inducing bifurcations. Numerical simulations, conducted in conjunction with analytical results, reveal the emergence of periodic self-sustained oscillations and intermittent chaotic behaviour. These observations emphasise the intricate relationship between temporal heterogeneity and the stability landscape of systems exhibiting QS-inspired dynamics. This interplay highlights the capacity for temporal variations to induce complex dynamical transitions within such systems. These findings assist to the comprehension of temporal dynamics within these and related systems, and may contribute to the development of strategies aimed at modulating intercellular communication and engineering synthetic biological systems with temporal control.

Figures (8)

• Figure 1: Existence of purely imaginary eigenvalues for the pseudo-polynomial \ref{['quasy']}. Such solutions are guaranteed for $\omega>0$ values that satisfy \ref{['quasyDef']}; that is, where functions $\xi(\omega)$ (black curve) and $\cos(\omega h-\alpha(\omega))$ with fixed $h$ (blue curve) intersect. Cases for equilibria $E_1$ and $E_2$ are considered separately. (a) Analysis of intersections at $E_1$ for $h$ values of $124, 262, 400$. (b) Similar solution visualisations for $E_2$ with $h=2\times10^{4}, 3\times10^{4}. 4\times10^{4}$. Parameter values as in Table \ref{['table:Ap1']}.
• Figure 2: The pseudo-characteristic polynomial \ref{['quasy']} possesses at least one purely imaginary root when $\omega$ simultaneously satisfies the geometric conditions \ref{['triang']}. This occurs within two distinct frequency intervals: a lower interval, $\Omega_1=[0,0.0466554]$, and a higher interval, $\Omega_2=[0.136335,0.187117]$, indicated by the grey bands in panel (a). Panels (b) and (c) illustrate all positive time delays for which the pseudo-polynomial roots lie on the imaginary axis, parameterised by $\omega$. Specifically, panel (b) shows trajectories for lower frequencies ($\omega\in\Omega_1$), while panel (c) displays those for higher frequencies ($\omega\in\Omega_2$). Asterisks mark the $(\tau_1,\tau_2)$ pairs where $\tau_1=52$. The direction in which roots cross the imaginary axis, as a function of time delay variation, is characterised by the transversality parameter \ref{['transv']}: blue curves indicate a crossing from left to right on the imaginary axis, while red curves denote a crossing from right to left.
• Figure 3: Extended stability and periodic orbit continuation. The system's stability and periodic orbit extensions with respect to the varying delay parameter $\tau_2$, with $\tau_1=52$ held fixed. The primary Hopf bifurcation $HB_0$ and ten secondary Hopf bifurcations ($HB_i$, for $i = 1, \dots, 10$) are identified, and their locations and criticality satisfactorily correspond with those predicted in Figure \ref{['fig:paramUnified']}(c). The point labelled $HB$, situated between $HB_6$ and $HB_7$, corresponds to the first $HB$ point (not extended) shown in Figure \ref{['fig:paramUnified']}(b). Periodic orbit branches emanate from each Hopf point. Stable branches, representing either periodic or steady-state orbits, are indicated in blue, while unstable branches are shown in red. Asterisks mark detected torus $TR$ and fold $LP$ bifurcation points along these periodic orbit branches.
• Figure 4: Top Panel: this panel presents a two-parameter continuation of Hopf bifurcations within the range $\tau_2 \in [0,200]$, with $\tau_1=52$ fixed. The emergence of intersecting islands as $\tau_2$ increases suggests the onset of complex oscillations. Panels (1)-(4): to further investigate the complexity in the time evolution scenario, the temporal dynamics of the system variables $u(t)$, $v(t)$, and $w(t)$ (red, black, and blue lines, respectively) are computed for selected parameter values: (1) $\tau_2 = 80$, $b = 1.5$; (2) $\tau_2 = 105$, $b = 1.36$; (3) $\tau_2 = 115$, $b = 1.36$; and (4) $\tau_2 = 160$, $b = 1.2$. Other parameter values are as in Table \ref{['table:Ap1']}. While panels (1) through (3) display qualitatively similar temporal dynamics, panel (4) indicates the onset of complex, non-periodic oscillations.
• Figure 5: Upper row: Poincaré map computed on the cross-section $u=2$ (black dots) after transient time, as $\tau_2$ is slowly varied; Pearson correlation coefficient (blue bars) measuring the correlation between two initially nearby $w$-component orbits. Lower row: frequency-amplitude plots and projections of representative orbits onto the ($u$, $v$) plane for three key distinct values of $\tau_2$; the central panel depicts a quasi-periodic orbit, while the left-hand and right-hand panels show stable periodic orbits. Other parameter values as in Table \ref{['table:Ap1']}.

Theorems & Definitions (3)

• Proposition 3.1
• Proposition 3.2
• Proposition 3.3