Fixation and extinction in time-fluctuating spatially structured metapopulations

TL;DR

This work analyzes fixation and extinction dynamics of competing wild-type and mutant strains in time-fluctuating, spatially structured metapopulations arranged on regular circulation graphs. By combining coarse-grained Moran-type descriptions with piecewise-deterministic Markov process (PDMP) approximations and Monte Carlo simulations, it reveals that environmental switching can render mutant fixation probabilities dependent on migration rate and produce non-monotonic responses with switching rate $ u$. The study distinguishes weak and strong bottleneck regimes, showing that weak bottlenecks yield migration-dependent fixation times while strong bottlenecks couple extinction and competition, enabling near-optimal regimes to eradicate mutants without risking metapopulation collapse. The results offer a general, graph-agnostic framework for eco-evolutionary dynamics in fluctuating environments and suggest strategies for targeted mutant removal in microbial or cellular systems, with potential extensions to more complex spatial topologies.

Abstract

Bacteria evolve in volatile environments and complex spatial structures. Migration, fluctuations and environmental variability therefore have a significant impact on the evolution of microbial populations. Here, we consider a class of spatially explicit metapopulation models arranged as regular (circulation) graphs where wild-type and mutant cells compete in a time-fluctuating environment in which demes (subpopulations) are connected by slow cell migration. The carrying capacity is the same at each deme and endlessly switches between two values associated to harsh and mild environmental conditions. Environmental variability can lead to population bottlenecks, following which the population is prone to fluctuation-induced extinction. Here, we analyse how slow migration, spatial structure, and fluctuations affect the phenomena of fixation and extinction on clique, cycle, and square lattice metapopulations. When the carrying capacity remains large, bottlenecks are weak and deme extinction can be ignored. The dynamics is thus captured by a coarse-grained description within which the probability and mean time of fixation are obtained analytically. This allows us to show that, in contrast to static environments, the mutant fixation probability depends on the rate of migration. We also show that the fixation probability and mean fixation time can exhibit a non-monotonic dependence on the switching rate. When the carrying capacity is small under harsh conditions, bottlenecks are strong, and the metapopulation evolution is shaped by the coupling of deme extinction and strain competition. This yields rich dynamical scenarios, among which we identify the best conditions to eradicate mutants without dooming the metapopulation to extinction. We offer an interpretation of these findings in the context of an idealised treatment strategy and discuss possible generalisations of our models.

Figures (13)

• Figure 1: Metapopulation dynamics in a static environment. (a) Examples of metapopulation graphs: a clique, cycle, and grid (from left to right). Neighbouring demes are connected by migration (double arrows). Initially, there is one mutant deme (red/light) and $\Omega -1$ wild-type demes (blue / dark), and all demes have the same constant carrying capacity $K$. (b) Dynamics in a single deme. Left: Wild-type $W$ cells (blue / dark) compete with mutants of type $M$ (red / light). When $K$ is small, the deme is prone to extinction. When $K$ is large, both types coexist prior to $W$ or $M$ fixation. Top right: Realisations of the rescaled deme size $n/K$ vs. time $t$ for $K=5$ (orange/light) and $K=100$ (green/dark) illustrating how $n$ fluctuates about $K$. Bottom right: Fraction of $M$ cells vs. $t$ in a deme with $K=100$. Deme extinction is not observed. Transient coexistence of $W$ and $M$ is followed by the fixation of $W$ (blue traces) or $M$ (red traces). Here $s=0.01$. (c) Invasion of $W$ deme by an $M$ cell: Any $M$ cell of a deme migrates to a neighbouring $W$ site with migration rate $m$ after a mean time $\Delta t = 1/(mK)$, and then type $M$ either quickly fixates, producing a new $M$ deme (right), or does not fix leaving the pair of $M$ and $W$ demes unchanged (left). The same picture holds for the invasion of an $M$ deme by a $W$ cell; see text. (d) Deme recolonisation (here for the clique): Deme extinction occurs after a mean time $\tau_E$, and empty demes are then recolonised by an invader from a neighbouring surviving deme after $\Delta t \sim 1/(mK)$. A recolonised deme is rapidly taken over (in $\Delta t \sim \mathcal{O}(\ln(K))$). (e) Coarse-grained description of the metapopulation dynamics: Each deme is always either fully $W$ (blue / dark) or $M$ (red / light) or empty (white). In this description, different scenarios arise, shown for the clique. Competition-dominated regime: all demes are occupied and there is always fixation of $W$ or $M$. Extinction-dominated regime: there are frequent deme extinctions and the metapopulation quickly goes extinct.
• Figure 2: (a): Competition-dominated dynamics, $\psi\gg 1$. (Top left) $M$ fixation probability $\phi$ vs. constant carrying capacity $K$; (Bottom left) unconditional mean fixation time $\theta$ vs. $K$; (Top right) $\phi$ vs. per capita migration rate $m$; (Bottom right) $\theta$ vs. $m$. Markers are simulation results and lines are predictions of Eq. \ref{['eq:static-fixation']} for $s=0.1$ (blue) and $s=0.01$ (red) on a clique (solid lines / crosses), cycle (dashed lines / circles), and grid (dotted lines / triangles). In (left), $m=10^{-4}, \Omega=16$, and in (right), $K=50, \Omega=16$. In (top), markers for the same $s$ are almost indistinguishable indicating independence of the spatial structure. (b): Extinction-dominated dynamics, $\psi<1$. (Top) Mean extinction time of a single deme $\tau_E$ vs. $K$ ($m=0$). Circles are simulation data, line shows the predictions of Eq. \ref{['eq:tauE']}. (Bottom) Metapopulation mean extinction time $\theta_E$ vs. $K$ for $\Omega=16$ and $m=10^{-2}$ (blue) and $10^{-4}$ (red). Markers are simulation results and thick lines are predictions of Eq. \ref{['eq:S9']} for cliques (solid lines / crosses), cycles (dashed lines / circles), and grids (dotted lines / triangles). Thin dashed vertical lines are guides to the eye showing $\psi=1$ for $m=10^{-2}$ (blue) and $10^{-4}$ (red). Selection plays no role in this regime, so simulation data for (b) has been obtained with $s=0$. In all panels, there is initially one $M$ deme and $\Omega -1$ demes occupied by $W$. In panels (a,top) and (b,bottom), dashed lines overlap with solid lines and so are not visible. Error bars are plotted in each case but are typically too small to see.
• Figure 3: (a) Left: single deme in time-switching environment. The carrying capacity $K(t)$ encodes environmental variability by switching between $K=K_+$ (mild environment, green / light) and $K(t)=K_-<K_+$ (harsh environment, orange / dark) at symmetric rate $\nu$ (see also Appendix \ref{['appendix:bias']}). Communities are larger in the mild environment. When $K(t)$ switches at an intermediate rate $\nu\lesssim 1$, each deme experiences bottlenecks prior to deme fixation at an average frequency $\nu/2$; see text. Right: $n$ and $K$ vs. time in the intermediate switching regime where the size $n$ of a deme undergoes bottlenecks. Parameters are: $K_+=200$, $\nu=0.05$ and $K_-=100$ (top) and $K_-=5$ (bottom). The bottlenecks are weak when $\psi(m,K_-)\gg1$ (top, right) where deme extinction is unlikely. When $\psi(m,K_-)<1$, there are strong bottlenecks and each deme can go extinct in the harsh environment (bottom, right). (b) Clique metapopulation with $\Omega=6$ connected demes (double arrows). All demes have the same time-switching carrying capacity $K(t)$ encoding environmental variability across the metapopulation, with each deme in the same environmental state. (c) Example evolution across two nearest-neighbour demes in a switching environment subject to strong bottlenecks in the intermediate switching regime; see text. Starting in the mild environment where $K=K_+$, the carrying capacity switches to $K_-$ (harsh environment) after $t\sim 1/\nu$. Following the $K_+\to K_-$ switch, each deme size decreases and each subpopulation is subject to strong demographic fluctuations and hence prone to extinction. In the absence of recolonisation of empty demes, effective only in the mild state, all demes go extinct. If there is a switch back to the mild environment $K_-\to K_+$ prior to total extinction, empty demes can be rescued by migration and recolonised by incoming $W$ or $M$ cells from neighbouring demes. In the sketch, an empty deme is recolonised by a mutant in the mild environment and becomes an $M$ deme. The cycle continues until the entire metapopulation consists of only $W$ or $M$ demes, or metapopulation extinction.
• Figure 4: Fixation probability $\Phi^{\rm G}$ and mean fixation time $\Theta^{\rm G}$ against switching rate $\nu$ for various parameters. Each panel shows $\Phi^{\rm G}$ vs. $\nu$ (top) and $\Theta^{\rm G}$ vs. $\nu$ (bottom). Markers show simulation results and lines are predictions of Eq. \ref{['eq:weak-bottleneck-solution']}. (a,b) $\Phi^{\rm clique}(\nu)$ and $\Theta^{\rm clique}(\nu)$ for a clique metapopulation and different values of $m$ in (a) and $s$ in (b). (a) $m=10^{-5}$ (red), $m=10^{-4}$ (blue), $m=10^{-3}$ (yellow), and $s=0.01$. (b) $s=10^{-3}$ (red), $s=10^{-2}$ (blue), $s=10^{-1}$ (yellow), and $m=10^{-4}$. Dashed black lines are guides to the eye showing $\Phi_{0,\infty}$ in (a,top) and $\Theta_{0,\infty}$ in (a,bottom); see text. Other parameters are $\Omega=16$, $K_+=200$, and $K_-=20$. (c) $\Phi^{\rm G}(\nu)$ and $\Theta^{\rm G}(\nu)$ for clique (red, crosses), cycle (blue, circles), and grid (yellow, triangles) metapopulations. Other parameters are $\Omega=16$, $K_+=200$, $K_-=20$, $s=0.01$, $m=10^{-4}$. (d) $\Phi^{\rm clique}(\nu)$ and $\Theta^{\rm clique}(\nu)$ for a clique metapopulation with $K_+=200$ (red), $K_+=500$ (blue), and $K_+=1000$ (yellow). Deviations occur for $\Theta$ with $K_+=1000$ since the slow-migration condition is not satisfied in the mild environment. Other parameters are $\Omega=16$, $K_-=20$, and $s=0.01$, $m=10^{-4}$. In all examples, there is initially a single $M$ deme and $\Omega -1$ others of type $W$; see text.
• Figure 5: Typical single realisations of $N/\Omega$ (black), $N_{M}/\Omega$ (red), $N_{W}/\Omega$ (blue), and $K(t)$ (grey) against time for different values of $K_-$ and $\nu$. (a,b): Here, $\nu=10^{-4}$ and $K_-=8$. In (a), $K=K_-$ at $t=0$ and $M$ and then $W$ quickly go extinct. In (b), $K=K_+$ at $t=0$ and $M$ fixes the population while $W$ goes extinct. (c,d): Here, $\nu=10^{-2}$ and $K_-=8$. In (c), mutants survive the first few bottlenecks but their abundance is low leading to the fixation of $W$ and removal of $M$ after four bottlenecks ($t\gtrsim 1000$). In (d), mutants survive the first bottlenecks and spread in the mild state where they recolonise and invade demes. They are eventually able to fix the population. (e,f): Here, $\nu=10$, and $K_-=4$ in (e) and $K_-=10$ in (f). $K(t)$ switches very frequently and is not shown for clarity. In (e), the deme size is $n\approx 2K_-=8$ and the dynamics is dominated by deme extinction leading to the rapid extinction of the metapopulation. In (e), the deme size is $n\approx 2K_-=20$ and there is $M/W$ competition that leads to fixation of $M$ and extinction of $W$ after a typical time $t\sim \theta^{\rm clique}(2K_-)\gtrsim 10^4$ (not shown). Similar results are obtained on other regular graphs ${\rm G}$; see text. Other parameters are $\Omega=10$, $s=0.1$, $m=10^{-4}$, and $K_+=200$. In all panels, initially there is a single $M$ deme and $\Omega -1$ demes occupied by $W$.