Travelling waves of invasion in microbial communities with phenotypic switching

Problem Statement
Microbial communities often exhibit complex spatial structures where clonal bacterial populations divide into different phenotypes. While the ecological importance of this subpopulation structure is recognized, particularly regarding species coexistence in well-mixed environments, its effect on spatial competition dynamics remains less understood. Specifically, this paper investigates how phenotypic switching—both stochastic and responsive to competitors—affects the travelling waves of invasion that occur when one bacterial species invades another in a spatially extended habitat. The study focuses on a minimal model involving two competing species: Species A (a single phenotype) and Species B, which can switch between a growing phenotype (B) and a persister phenotype (P) that is resilient to competition.

Methodology
The authors develop a minimal reaction-diffusion model to describe the spatial competition between the two species. The model consists of three coupled partial differential equations tracking the abundances of the growing phenotype ($B$), the persister phenotype ($P$), and the competitor ($A$). The dynamics include:

• Logistic growth and Lotka-Volterra competition terms.
• Stochastic switching rates between $B$ and $P$.
• Responsive switching from $B$ to $P$ induced by the presence of competitor $A$.
• Diffusive terms representing bacterial motility.

The analysis proceeds via two complementary approaches:

1. Analytical Derivation: The authors transform the system into a set of ordinary differential equations using a travelling-wave coordinate $z = x - ct$. They analyze the linearized dynamics near the homogeneous steady states (where only one species exists) to derive necessary algebraic conditions for the existence of heteroclinic connections (travelling waves). This involves calculating the eigenvalues of the Jacobian matrices at the wave's leading and trailing edges to determine the minimum wave speed ($c_{min}$) required for non-negative solutions.
2. Numerical Simulation: The full reaction-diffusion system and a corresponding "averaged" model (which collapses the phenotypic subpopulations into a single effective population) are integrated numerically in MATLAB. The authors randomly sample parameter spaces to compare the actual wave speeds ($c^*$) selected by the dynamics against the analytically derived minimum speeds and against the speeds predicted by the averaged model.

Key Contributions and Results
The paper yields two primary, somewhat counterintuitive findings regarding the effect of phenotypic switching on invasion speed:

1. Invasion by Competitors ($c > 0$): When the competitor species (A) invades the switching species (B), the presence of phenotypic switching has no effect on the speed of the travelling wave. The minimum wave speed for the full model with switching is identical to that of the averaged model without switching. This result holds regardless of the rate of responsive switching ($\beta$).

2. Invasion of Competitors ($c < 0$): When the switching species (B) invades the competitor (A), phenotypic switching can increase the speed of the invasion wave. Specifically, the wave speed in the full model can exceed the speed predicted by the averaged model, provided the rate of responsive switching is not too high. This suggests that the ability to switch to a resilient phenotype can act as an offensive mechanism, accelerating the takeover of a competitor's territory.

The analytical results establish that the minimum wave speed is determined by the linearized dynamics at the leading edge of the wave. For $c > 0$, the switching terms do not appear in the leading-order expression for the minimum speed. For $c < 0$, the switching terms modify the effective growth and diffusion parameters, allowing for faster propagation under specific conditions.

Significance and Claims
The authors conclude that bacterial persistence, often viewed as a defensive strategy to survive stress or antibiotics, can function as an offensive ecological strategy in spatial competition. By switching to a persister phenotype, a population can paradoxically speed up its invasion of a competitor. This finding challenges the intuitive expectation that a "hump" of a resilient phenotype would slow down an invading wave by acting as a buffer.

The paper situates this result within the broader context of phenotypic variation stabilizing coexistence in well-mixed populations, suggesting that the same mechanism (switching to a rare phenotype) can also drive spatial expansion. The authors note that while their model is one-dimensional, it is relevant to bacterial movement in porous media like soil. They identify the mechanism of wave speed selection in systems with phenotypic switching as an open mathematical question for future research, particularly regarding how immigration from switching affects the selection of wave speeds in classical population dynamics models where $\dot{X}=0$ if $X=0$.