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Demographic (shot) noise in population dynamics scales with the square root of the population size. This process is very important, as it yields an absorbing state at zero field, but simulating it, especially on spatial domains, is a non-trivial task. Here we compare the results of two operator-splitting techniques suggested for simulating the corresponding Langevin equation, one by Pechenik and Levine (PL) and the other by Dornic, Chate and Mu~noz (DCM). We identify an anomalously strong bias toward the active phase in the numerical scheme of DCM, a bias which is not present in the alternative scheme of PL. This bias strongly distorts the phase diagram determined via the DCM procedure for the range of time-steps used in such simulations. We pinpoint the underlying cause in the inclusion of the diffusion, treated as an on-site decay with a constant external source, in the stochastic part of the algorithm. Treating the diffusion deterministically is shown to remove this unwanted bias while keeping the simulation algorithm stable, thus a hybrid numerical technique, in which the DCM approach to diffusion is applied but the diffusion is simulated deterministically, appears to be optimal.
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We study simple stochastic scenarios, based on birth-and-death Markovian processes, that describe populations with Allee effect, to account for the role of demographic stochasticity. In the mean-field deterministic limit we recover well-known deterministic evolution equations widely employed in population ecology. The mean-time to extinction is in general obtained by the Wentzel-Kramers-Brillouin (WKB) approximation for populations with strong and weak Allee effects. An exact solution for the mean time to extinction can be found via a recursive equation for special cases of the stochastic dynamics. We study the conditions for the validity of the WKB solution and analyze the boundary between the weak and strong Allee effect by comparing exact solutions with numerical simulations.
We study the dynamics of predator-prey systems where prey are confined to a single region of space and where predators move randomly according to a power-law (Levy) dispersal kernel. Site fidelity, an important feature of animal behaviour, is incorporated in the model through a stochastic resetting dynamics of the predators to the prey patch. We solve in the long time limit the rate equations of Lotka-Volterra type that describe the evolution of the two species densities. Fixing the demographic parameters and the Levy exponent, the total population of predators can be maximized for a certain value of the resetting rate. This optimal value achieves a compromise between over-exploitation and under-utilization of the habitat. Similarly, at fixed resetting rate, there exists a Levy exponent which is optimal regarding predator abundance. These findings are supported by 2D stochastic simulations and show that the combined effects of diffusion and resetting can broadly extend the region of species coexistence in ecosystems facing resources scarcity.
We discuss similarities and differences between systems of interacting players maximizing their individual payoffs and particles minimizing their interaction energy. Long-run behavior of stochastic dynamics of spatial games with multiple Nash equilibria is analyzed. In particular, we construct an example of a spatial game with three strategies, where stochastic stability of Nash equilibria depends on the number of players and the kind of dynamics.
Inferring the processes underlying the emergence of observed patterns is a key challenge in theoretical ecology. Much effort has been made in the past decades to collect extensive and detailed information about the spatial distribution of tropical rainforests, as demonstrated, e.g., in the 50 ha tropical forest plot on Barro Colorado Island, Panama. These kind of plots have been crucial to shed light on diverse qualitative features, emerging both at the single-species or the community level, like the spatial aggregation or clustering at short scales. Here, we build on the progress made in the study of the density correlation functions applied to biological systems, focusing on the importance of accurately defining the borders of the set of trees, and removing the induced biases. We also pinpoint the importance of combining the study of correlations with the scale dependence of fluctuations in density, which are linked to the well known empirical Taylors power law. Density correlations and fluctuations, in conjunction, provide an unique opportunity to interpret the behaviors and possibly to allow comparisons between data and models. We also study such quantities in models of spatial patterns and, in particular, we find that a spatially explicit neutral model generates patterns with many qualitative features in common with the empirical ones.
In genetic circuits, when the mRNA lifetime is short compared to the cell cycle, proteins are produced in geometrically-distributed bursts, which greatly affects the cellular switching dynamics between different metastable phenotypic states. Motivated by this scenario, we study a general problem of switching or escape in stochastic populations, where influx of particles occurs in groups or bursts, sampled from an arbitrary distribution. The fact that the step size of the influx reaction is a-priori unknown, and in general, may fluctuate in time with a given correlation time and statistics, introduces an additional non-demographic step-size noise into the system. Employing the probability generating function technique in conjunction with Hamiltonian formulation, we are able to map the problem in the leading order onto solving a stationary Hamilton-Jacobi equation. We show that bursty influx exponentially decreases the mean escape time compared to the usual case of single-step influx. In particular, close to bifurcation we find a simple analytical expression for the mean escape time, which solely depends on the mean and variance of the burst-size distribution. Our results are demonstrated on several realistic distributions and compare well with numerical Monte-Carlo simulations.
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