Minimizing the population extinction risk by migration

Front propagation in a spatial system of weakly interacting networks

We consider the spread of epidemic in a spatial metapopulation system consisting of weakly interacting patches. Each local patch is represented by a network with a certain node degree distribution and individuals can migrate between neighboring patches. Stochastic particle simulations of the SIR model show that after a short transient, the spatial spread of epidemic has a form of a propagating front. A theoretical analysis shows that the speed of front propagation depends on the effective diffusion coefficient and on the local proliferation rate similarly to fronts described by the Fisher-Kolmogorov equation. To determine the speed of front propagation, first, the early-time dynamics in a local patch is computed analytically by employing degree based approximation for the case of a constant disease duration. The resulting delay differential equation is solved for early times to obtain the local growth exponent. Next, the reaction diffusion equation is derived from the effective master equation and the effective diffusion coefficient and the overall proliferation rate are determined. Finally, the fourth order derivative in the reaction diffusion equation is taken into account to obtain the discrete correction to the front propagation speed. The analytical results are in a good agreement with the results of stochastic particle simulations.

Characterizing outbreak vulnerability in a stochastic SIS model with an external disease reservoir

In this article, we take a mathematical approach to the study of population-level disease spread, performing a quantitative and qualitative investigation of an SISκ model which is a susceptible-infectious-susceptible ( SIS ) model with exposure to an external disease reservoir. The external reservoir is non-dynamic, and exposure from the external reservoir is assumed to be proportional to the size of the susceptible population. The full stochastic system is modelled using a master equation formalism. A constant population size assumption allows us to solve for the stationary probability distribution, which is then used to investigate the predicted disease prevalence under a variety of conditions. By using this approach, we quantify outbreak vulnerability by performing the sensitivity analysis of disease prevalence to changing population characteristics. In addition, the shape of the probability density function is used to understand where, in parameter space, there is a transition from disease free, to disease present, and to a disease endemic system state. Finally, we use Kullback–Leibler divergence to compare our semi-analytical results for the SISκ model with more complex susceptible-infectious-recovered ( SIR ) and susceptible-exposed-infectious-recovered ( SEIR ) models.

Extinctions of coupled populations, and rare event dynamics under non-Gaussian noise

The survival of natural populations may be greatly affected by environmental conditions that vary in space and time. We look at a population residing in two locations (patches) coupled by migration, in which the local conditions fluctuate in time. We report on two findings. First, we find that, unlike rare events in many other systems, here the histories leading to a rare extinction event are not dominated by a single path. We develop the appropriate framework, which turns out to be a hybrid of the standard saddle-point method, and the Donsker-Varadhan formalism which treats rare events of atypical averages over a long time. It provides a detailed description of the statistics of histories leading to the rare event and the mean time to extinction. The framework applies to rare events in a broad class of systems driven by non-Gaussian noise. Second, applying this framework to the population-dynamics model, we find a phase transition in its extinction behavior. Strikingly, a patch which is a sink (where individuals die more than are born) can nonetheless reduce the probability of extinction, even if it lowers the average population's size and growth rate.

Two-level modeling of quarantine

Continuum models of epidemics do not take into account the underlying microscopic network structure of social connections. This drawback becomes extreme during quarantine when most people dramatically decrease their number of social interactions, while others (like cashiers in grocery stores) continue maintaining hundreds of contacts per day. We formulate a two-level model of quarantine. On a microscopic level, we model a single neighborhood assuming a star-network structure. On a mesoscopic level, the neighborhoods are placed on a two-dimensional lattice with nearest-neighbors interactions. The modeling results are compared with the COVID-19 data for several counties in Michigan (USA) and the phase diagram of parameters is identified.

Extinction risk of a metapopulation under bistable local dynamics

We study the extinction risk of a fragmented population residing on a network of patches coupled by migration, where the local patch dynamics includes deterministic bistability. Mixing between patches is shown to dramatically influence the population's viability. We demonstrate that slow migration always increases the population's global extinction risk compared to the isolated case, while at fast migration synchrony between patches minimizes the population's extinction risk. Moreover, we discover a critical migration rate that maximizes the extinction risk of the population, and identify an early-warning signal when approaching this state. Our theoretical results are confirmed via the highly efficient weighted ensemble method. Notably, our theoretical formalism can also be applied to studying switching in gene regulatory networks with multiple transcriptional states.

Optimal dispersal in ecological dynamics with Allee effect in metapopulations

We introduce a minimal agent-based model to understand the effects of the interplay between dispersal and geometric constraints in metapopulation dynamics under the Allee Effect. The model, which does not impose nonlinear birth and death rates, is studied both analytically and numerically. Our results indicate the existence of a survival-extinction boundary with monotonic behavior for weak spatial constraints and a nonmonotonic behavior for strong spatial constraints so that there is an optimal dispersal that maximizes the survival probability. Such optimal dispersal has empirical support from recent experiments with engineered bacteria.

Migration alters oscillatory dynamics and promotes survival in connected bacterial populations

Migration influences population dynamics on networks, thereby playing a vital role in scenarios ranging from species extinction to epidemic propagation. While low migration rates prevent local populations from becoming extinct, high migration rates enhance the risk of global extinction by synchronizing the dynamics of connected populations. Here, we investigate this trade-off using two mutualistic strains of E. coli that exhibit population oscillations when co-cultured. In experiments, as well as in simulations using a mechanistic model, we observe that high migration rates lead to synchronization whereas intermediate migration rates perturb the oscillations and change their period. Further, our simulations predict, and experiments show, that connected populations subjected to more challenging antibiotic concentrations have the highest probability of survival at intermediate migration rates. Finally, we identify altered population dynamics, rather than recolonization, as the primary cause of extended survival.

Tuning Spatial Profiles of Selection Pressure to Modulate the Evolution of Drug Resistance

Spatial heterogeneity plays an important role in the evolution of drug resistance. While recent studies have indicated that spatial gradients of selection pressure can accelerate resistance evolution, much less is known about evolution in more complex spatial profiles. Here we use a stochastic toy model of drug resistance to investigate how different spatial profiles of selection pressure impact the time to fixation of a resistant allele. Using mean first passage time calculations, we show that spatial heterogeneity accelerates resistance evolution when the rate of spatial migration is sufficiently large relative to mutation but slows fixation for small migration rates. Interestingly, there exists an intermediate regime-characterized by comparable rates of migration and mutation-in which the rate of fixation can be either accelerated or decelerated depending on the spatial profile, even when spatially averaged selection pressure remains constant. Finally, we demonstrate that optimal tuning of the spatial profile can dramatically slow the spread and fixation of resistant subpopulations, even in the absence of a fitness cost for resistance. Our results may lay the groundwork for optimized, spatially resolved drug dosing strategies for mitigating the effects of drug resistance.

Resonant activation of population extinctions

Understanding the mechanisms governing population extinctions is of key importance to many problems in ecology and evolution. Stochastic factors are known to play a central role in extinction, but the interactions between a population's demographic stochasticity and environmental noise remain poorly understood. Here we model environmental forcing as a stochastic fluctuation between two states, one with a higher death rate than the other. We find that, in general, there exists a rate of fluctuations that minimizes the mean time to extinction, a phenomenon previously dubbed "resonant activation." We develop a heuristic description of the phenomenon, together with a criterion for the existence of resonant activation. Specifically, the minimum extinction time arises as a result of the system approaching a scenario wherein the severity of rare events is balanced by the time interval between them. We discuss our findings within the context of more general forms of environmental noise and suggest potential applications to evolutionary models.

Noise-induced switching and extinction in systems with delay

We consider the rates of noise-induced switching between the stable states of dissipative dynamical systems with delay and also the rates of noise-induced extinction, where such systems model population dynamics. We study a class of systems where the evolution depends on the dynamical variables at a preceding time with a fixed time delay, which we call hard delay. For weak noise, the rates of interattractor switching and extinction are exponentially small. Finding these rates to logarithmic accuracy is reduced to variational problems. The solutions of the variational problems give the most probable paths followed in switching or extinction. We show that the equations for the most probable paths are acausal and formulate the appropriate boundary conditions. Explicit results are obtained for small delay compared to the relaxation rate. We also develop a direct variational method to find the rates. We find that the analytical results agree well with the numerical simulations for both switching and extinction rates.

Spontaneous formation of large clusters in a lattice gas above the critical point

We consider clustering of particles in the lattice gas model above the critical point. We find the probability for large density fluctuations over scales much larger than the correlation length. This fundamental problem is of interest in various biological contexts such as quorum sensing and clustering of motile, adhesive, cancer cells. In the latter case, it may give a clue to the problem of growth of recurrent tumors. We develop a formalism for the analysis of this rare event employing a phenomenological master equation and measuring the transition rates in numerical simulations. The spontaneous clustering is treated in the framework of the eikonal approximation to the master equation.

Non-Markovian effects in the first-passage dynamics of obstructed tracer particle diffusion in one-dimensional systems

The standard setup for single-file diffusion is diffusing particles in one dimension which cannot overtake each other, where the dynamics of a tracer (tagged) particle is of main interest. In this article, we generalize this system and investigate first-passage properties of a tracer particle when flanked by identical crowder particles which may, besides diffuse, unbind (rebind) from (to) the one-dimensional lattice with rates k(off) (k(on)). The tracer particle is restricted to diffuse with rate k(D) on the lattice and the density of crowders is constant (on average). The unbinding rate k(off) is our key parameter and it allows us to systematically study the non-trivial transition between the completely Markovian case (k(off) ≫ k(D)) to the non-Markovian case (k(off) ≪ k(D)) governed by strong memory effects. This has relevance for several quasi one-dimensional systems. One example is gene regulation where regulatory proteins are searching for specific binding sites on a crowded DNA. We quantify the first-passage time distribution, f(t) (t is time), numerically using the Gillespie algorithm, and estimate f(t) analytically. In terms of k(off) (keeping k(D) fixed), we study the transition between the two known regimes: (i) when k(off) ≫ k(D) the particles may effectively pass each other and we recover the single particle result f(t) ∼ t(-3/2), with a reduced diffusion constant; (ii) when k(off) ≪ k(D) unbinding is rare and we obtain the single-file result f(t) ∼ t(-7/4). The intermediate region displays rich dynamics where both the characteristic f(t) - peak and the long-time power-law slope are sensitive to k(off).

Nonmonotonic effects of migration in subdivided populations

The influence of migration on the stochastic dynamics of subdivided populations is still an open issue in various evolutionary models. Here, we develop a self-consistent mean-field-like method in order to determine the effects of migration on relevant nonequilibrium properties, such as the mean fixation time. If evolution strongly favors coexistence of species (e.g., balancing selection), the mean fixation time develops an unexpected minimum as a function of the migration rate. Our analysis hinges only on the presence of a separation of time scales between local and global dynamics, and therefore, it carries over to other nonequilibrium processes in physics, biology, ecology, and social sciences.

Demographic stochasticity and evolution of dispersion I. Spatially homogeneous environments

The selection of dispersion is a classical problem in ecology and evolutionary biology. Deterministic dynamical models of two competing species differing only in their passive dispersal rates suggest that the lower mobility species has a competitive advantage in inhomogeneous environments, and that dispersion is a neutral characteristic in homogeneous environments. Here we consider models including local population fluctuations due to both individual movements and random birth and death events to investigate the effect of demographic stochasticity on the competition between species with different dispersal rates. In this paper, the first of two, we focus on homogeneous environments where deterministic models predict degenerate dynamics in the sense that there are many (marginally) stable equilibria with the species' coexistence ratio depending only on initial data. When demographic stochasticity is included the situation changes. A novel large carrying capacity ([Formula: see text]) asymptotic analysis, confirmed by direct numerical simulations, shows that a preference for faster dispersers emerges on a precisely defined [Formula: see text] time scale. We conclude that while there is no evolutionarily stable rate for competitors to choose in these models, the selection mechanism quantified here is the essential counterbalance in spatially inhomogeneous models including demographic fluctuations which do display an evolutionarily stable dispersal rate.

Synchronization-induced persistence versus selection for habitats in spatially coupled ecosystems

Critical population phase transitions, in which a persistent population becomes extinction-prone owing to environmental changes, are fundamentally important in ecology, and their determination is a key factor in successful ecosystem management. To persist, a species requires a suitable environment in a sufficiently large spatial region. However, even if this condition is met, the species does not necessarily persist, owing to stochastic fluctuations. Here, we develop a model that allows simultaneous investigation of extinction due to either stochastic or deterministic reasons. We find that even classic birth-death processes in spatially extended ecosystems exhibit phase transitions between extinction-prone and persistent populations. Sometimes these are first-order transitions, which means that environmental changes may result in irreversible population collapse. Moreover, we find that higher migration rates not only lead to higher robustness to stochastic fluctuations, but also result in lower sustainability in heterogeneous environments by preventing efficient selection for suitable habitats. This demonstrates that intermediate migration rates are optimal for survival. At low migration rates, the dynamics are reduced to metapopulation dynamics, whereas at high migration rates, the dynamics are reduced to a multi-type branching process. We focus on species persistence, but our results suggest a unique method for finding phase transitions in spatially extended stochastic systems in general.

Survival of the scarcer

We investigate extinction dynamics in the paradigmatic model of two competing species A and B that reproduce (A→2A, B→2B), self-regulate by annihilation (2A→0, 2B→0), and compete (A+B→A, A+B→B). For a finite system that is in the well-mixed limit, a quasistationary state arises which describes coexistence of the two species. Because of discrete noise, both species eventually become extinct in time that is exponentially long in the quasistationary population size. For a sizable range of asymmetries in the growth and competition rates, the paradoxical situation arises in which the numerically disadvantaged species according to the deterministic rate equations survives much longer.