Invading activity fronts stabilize excitable systems against stochastic extinction

Abstract

Stochastic chemical reaction or population dynamics in finite systems often terminates in an absorbing state. Yet in large spatially extended systems, the time to reach species extinction (or fixation) becomes exceedingly long. Tuning control parameters may diminish the survival probability, rendering species coexistence susceptible to stochastic extinction events. In inhomogeneous settings, where a vulnerable subsystem is diffusively coupled to an adjacent stable patch, the former is reanimated through continuous influx from the interfaces, provided the absorbing region sustains spreading activity fronts. We demonstrate this generic elimination of finite-size extinction instabilities via immigration flux in predator-prey, epidemic spreading, and cyclic competition models.

Figures (4)

• Figure 1: (a) Coexistence (white), prey fixation (dark blue) and total extinction (with increasing probability indicated in darker brown shades) regimes in the stochastic LV model on a $L \times L$ square lattice as function of $L$ and the predation probability $\lambda$ (with fixed $\mu = 0.1$, $\sigma = 0.2$, and $K = 3$). The (largely) size-independent genuine (DP) predator extinction transition of type (i) occurs at $\lambda_c \approx 0.3$. For large $\lambda > 0.5$, one observes stochastic type (ii) total population extinction events with strongly $L$-dependent probability Swailem23Distefano25. (b) Schematic dependence on system size $N$ of invasion front traversal $t_\mathrm{inv} \sim N^{1/d}$ (red dashed) and mean extinction times $t_\mathrm{ext}$ for robust (black, $c_1 > c_2$) and vulnerable (blue) (sub-)systems prone to be stochastically driven into an absorbing state. For $t_\mathrm{ext,1}> t > t_\mathrm{inv}$ (shaded red), an excitable patch subject to stochastic extinction for $t > t_\mathrm{ext,2}$ (shaded blue) may be reanimated and stabilized via diffusive coupling to an active region.
• Figure 2: Snapshots of a single Monte Carlo simulation run for two diffusively coupled LV systems on a square lattice (height $L_\parallel = 200$, periodic boundary conditions), (a) initialized ($t = 0$) with a random distribution of single predator (red) and prey (blue) individuals. The smaller left patch (width $L_\perp = 100$, carrying capacity $K = 1$) resides in a stable active predator-prey coexistence state characterized by persistent radially propagating activity waves. The larger right subsystem ($L_\perp = 200$, $K = 3$) is prone to fluctuation-driven total population extinction driven by initially abundant local predation events; as has happened by (b) $t = 260$ MCS (empty sites: black). Yet roughly planar invasion fronts emanating from both interfaces with the active region enter the empty, excitable subsystem, collide at (c) $t = 800$ MCS, generate wider population waves, and subsequently stabilize the vulnerable area in perpetuity at the higher species densities pertinent to $K = 3$, as shown at (d) $t = 3000$ MCS) Distefano25Movies. Time evolution of the population densities averaged over $21$ independent runs in the (e) stable and (f) vulnerable regions.
• Figure 3: Snapshots of a single Monte Carlo simulation run for two diffusively coupled SIS regions of equal size on a square lattice ($L_\parallel = L_\perp = 100$, periodic boundary conditions, carrying capacity $K = 1$), initialized ($t = 0$) with a single infectious (red) seed immersed in a susceptible (blue) population, subject to the infection probabilities $\lambda = 0.115$ (left region), $\lambda = 0.109$ (right subsystem) and recovery probability $\gamma = 0.1$, which sets the right patch very close to the epidemic threshold. (a) By $t = 1300$ MCS, the epidemic is spreading in the left subsystem, but has gone extinct on the right. The snapshots at (b) $t = 2900$ and (c) $4000$ MCS show the infection fronts reaching the active region's boundaries and spreading into the inactive, susceptible domain (right); even though the diffusive coupling was turned off at $t = 3000$ MCS. (d) In the (quasi-)stationary regime ($t = 6000$ MCS), both isolated regions sustain the epidemic outbreak, with a lower infectious population density $n_I$ on the right. The inset in (a) shows the temporal increase of $n_I(t)$ in both subsystems (averaged over the surviving runs in the more robust left ecosystem) Shabani25Movies.
• Figure 4: Snapshots of a single Monte Carlo simulation run for two diffusively coupled SIR (left) and SIS (right) systems of equal size on a square lattice ($L_\parallel=L_\perp=100$, periodic boundary conditions, carrying capacity $K = 1$), initialized ($t = 0$) with a single infectious (red) seed immersed in a susceptible (blue) population, subject to the infection probabilities $\lambda = 0.210$ (left region), $\lambda = 0.109$ (right) and recovery probability $\gamma = 0.1$ (on the left, recovered individuals are indicated black). (a) At $t = 400$ MCS, the epidemic is spreading in the left SIR patch, but has gone extinct on the SIS side. (b) The snapshot at $t = 1000$ MCS depicts incipient infection fronts emanating from the interfaces and invading into the inactive susceptible SIS region (right). (c) By $t = 2000$ MCS steps, the outbreak has run its course in the SIR region and reached its final state with a majority of recovered sites (black) interspersed by untouched susceptible (blue) patches, while the epidemic continues to spread in the SIS domain. (d) At longer run times (here $t = 6000$ MCS), both effectively decoupled systems persist in their respective (quasi-)stationary states, with infection events occuring continuously and persistently in the right SIS patch. The inset in (a) shows the population densities $n_I + n_R$ (left region) and $n_I$ (right) Shabani25Movies.