Nonlinear competition avoidance favors coexistence in microbial populations

TL;DR

This work addresses how density-dependent motility can enable coexistence among competitively interacting microbes. The authors introduce a minimal metapopulation model in which each species exhibits a sigmoidal, competitor-density–driven escape in motility, parameterized by a threshold $\Delta$ and steepness $\xi$, and couple this to standard Lotka–Volterra growth and interspecific competition. Through analysis of a two-patch system and simulations on two-dimensional lattices, they show that nonlinear motility can sustain long-term coexistence via spatial segregation, even when the well-mixed system would predict competitive exclusion, with the coexistence region expanding with system size and nonlinearity and disappearing if the sigmoid becomes linear. The findings reveal a general mechanism—the competition–colonization trade-off arising from nonlinear, quorum-sensing–like motility—that can generate diverse spatial patterns (rings, checkerboards) and promote biodiversity in microbial communities, linking ecological theory to active-matter pattern formation. These results offer a framework to understand how spatial structure and density-dependent motility contribute to stable coexistence in complex microbial ecosystems.

Abstract

Bacteria regulate their motility through a variety of mechanisms, including quorum sensing (QS) and other density-dependent responses mediated by diffusible signals. While nonlinear density-dependent motility is well known in active-matter theory to generate nonequilibrium spatial patterns, its consequences for the coexistence of growing, interacting species remain less explored. Here we develop a minimal spatially structured model for two strongly competing species in which local demographic interactions are coupled to an escape response: each species increases its motility nonlinearly (sigmoidal) with the local abundance of its competitor. We show that this sigmoidal motility regulation promotes optimal spatial self-organization and can sustain long term coexistence via segregation, even in parameter regimes that yield competitive exclusion in well-mixed Lotka-Volterra dynamics. On two-dimensional lattices, the interplay between demographic competition and density-dependent motility generates a range of emergent patterns, including regimes in which the weaker competitor counterintuitively has higher total abundance. Overall, our results identify nonlinear, competitor-induced motility as a fundamental mechanism capable of sustaining coexistence in competing microbial populations.

Figures (9)

• Figure 1: Schematic illustration of the model. (a) Demographic dynamics within each patch, including intrinsic growth at rate $r$, carrying capacity $K$, and intra- and inter-specific competition terms $A_{ii}$ and $A_{ij}$, respectively. (b) Sigmoidal escape response functions [eq. \ref{['eq:mu']}] for the species 1 with $A_{12}=1.1$ (blue line) and for species 2 for $A_{21}=2.0$ (red line), considering $\Delta=0.5$ and $\xi=10$. (c) Initial configuration of the system in the case of the two-patches setting, in which both species are initially present in patch $\alpha$ and may subsequently migrate to patch $\beta$ with species-specific migration rates $\mu_1^{\alpha\beta}$ and $\mu_2^{\alpha\beta}$.
• Figure 2: (a) Phase diagram distinguishing segregated coexistence from competitive exclusion as a function of the motility threshold $\Delta$ and the interaction ratio $A_{21}/A_{12}$ for two neighboring patches, in the theoretical limits $\xi \to \infty$ and $\mu_0 \to 0$. The background heatmap is obtained from numerical integration of Eqs. \ref{['eq:model']}, while dashed lines indicate analytical predictions (see Appendix A). The three markers correspond to the parameter values $A_{21}/A_{12}=1.4$ and $\Delta = 0.12,\ 0.8,\ 1.8$, for which representative temporal dynamics are shown in panels (b–d). (b–d) Representative time series illustrating the different dynamical outcomes. Symbol ▼ denotes competitive exclusion resulting from the migration of both species to patch $\beta$; ★ indicates coexistence driven by the migration of species 2; and ◆ corresponds to competitive exclusion when neither species migrates. All simulations use $A_{11}=A_{22}=K=r=1$, $\mu_S=0.1$, and $A_{12}=1.1$.
• Figure 3: (a) Range of motility thresholds, $\Delta_{\text{max}} - \Delta_{\text{min}}$, that permit segregated coexistence as a function of the linear diffusion rate $\mu_0$, shown for different values of the interspecific competition ratio $A_{21}/A_{12}$. (b) Range of motility thresholds, $\Delta_{\text{max}} - \Delta_{\text{min}}$, as a function of the steepness parameter $\xi$ of the sigmoidal escape response, again for fixed values of $A_{21}/A_{12}$. The dashed lines indicate the corresponding asymptotic values obtained in the limit $\xi \to \infty$. All curves in both panels are obtained from numerical integration of Eqs. \ref{['eq:model']} with parameters $A_{11} = A_{22} = K = r = 1$, $\mu_S = 0.1$, and $A_{12} = 1.1$.
• Figure 4: Heatmaps for lattices with different side lengths ($L = 3, 5, 7, 11, 21, 51$) showing the equilibrium difference in total abundances between species 1, $x_1=\sum_\alpha x_1^\alpha$, and species 2, $x_2=\sum_\alpha x_2^\alpha$. This difference is normalized by the total available area $L \times L$ and we call the normalized difference $\chi\equiv(x_1-x_2)/L^2$. Blueish regions indicate dominance of species 1, whereas reddish regions indicate dominance of species 2. As in figure \ref{['fig2']}, the activation threshold $\Delta$ is reported on the $x$-axis and the ratio of competition coefficients $A_{21}/A_{12}$ on the $y$-axis. The gold dashed line represents the analytical prediction obtained for the two-patch case without linear diffusion (see figure \ref{['fig2']}). Each cell in the heatmap corresponds to a numerical integration of eq. \ref{['eq:model']} with parameter values $A_{11} = A_{22} = K = r = 1$, $A_{12}=1.1$, $\xi=100$, $\mu_0 = 0.001$, and $\mu_S = 0.1$. Initial conditions are $x_1^\alpha(0) = x_2^\alpha(0) = 0.001$ in the central cell and zero elsewhere. For all lattices, the mobility matrices ${\bf M}$ are constructed considering a Moore neighborhood.
• Figure 5: (a-c) Final total abundances of species 1 ($x_1$) and species 2 ($x_2$), normalized by the lattice size $L^2$, as a function of $A_{21}/A_{12}$, for three values of the activation thresholds $\Delta = 0.60,\ 0.80,\ 1.00$. (d-f) Stationary spatial patterns for species 1 (blue) and species 2 (red) on a two-dimensional lattice with side length $L = 51$. Rows correspond to activation thresholds $\Delta = 0.60,\ 0.80,\ 1.00$, and columns to increasing values of the competition ratio $A_{21}/A_{12}$. All results are obtained from numerical integration of the model with parameters $A_{11} = A_{22} = K = r = 1$, $A_{12}=1.1$, $\xi=100$, $\mu_0=0.001$, and $\mu_S=0.1$. Initial conditions place both species at density $0.001$ in the central cell and zero elsewhere. The mobility matrix ${\bf M}$ is constructed considering a Moore neighborhood.