Stochastic effects on extinction and pattern formation in the three-species cyclic May–Leonard model
Serrao, Shannon Reuben
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We study the fluctuation effects in the seminal cyclic predator-prey model in population dynamics due to Robert May and Warren Leonard both in the zero-dimensional and two-dimensional spatial version. We compute the mean time to extinction of a stable set of coexisting populations driven by large fluctuations. We see that the contribution of large fluctuations to extinction can be captured by a quasi-stationary approximation and the Wentzel–Kramers–Brillouin (WKB) eikonal ansatz. We see that near the Hopf bifurcation, extinctions are fast owing to the flat non-Gaussian distribution whereas away from the bifurcation, extinctions are dominated by large fluctuations of the fat tails of the distribution. We compare our results to Gillespie simulations and a single-species theoretical calculation. In addition, we study the spatio-temporal pattern formation of the stochastic May--Leonard model through the Doi-Peliti coherent state path integral formalism to obtain a coarse-grained Langevin description, i.e. the Complex Ginzburg Landau equation with stochastic noise in one complex field. We see that when one restricts the internal reaction noise to small amplitudes, one can obtain a simple form for the stochastic noise correlations that modify the Complex Ginzburg Landau equation. Finally, we study the effect of coupling a spatially extended May--Leonard model in two dimensions with symmetric predation rates to one with asymmetric rates that is prone to reach extinction. We show that the symmetric region induces otherwise unstable coexistence spiral patterns in the asymmetric May--Leonard lattice. We obtain the stability criterion for this pattern induction as we vary the strength of the extinction inducing asymmetry. This research was sponsored by the Army Research Office and was accomplished under Grant Number W911NF-17-1-0156.
General Audience Abstract
In the field of ecology, the cyclic predator-prey patterns in a food web are relevant yet independent to the hierarchical archetype. We study the paradigmatic cyclic May--Leonard model of three species, both analytically and numerically. First, we employ well--established techniques in large-deviation theory to study the extinction of populations induced by large but rare fluctuations. In the zero--dimensional version of the model, we compare the mean time to extinction computed from the theory to numerical simulations. Secondly, we study the stochastic spatial version of the May--Leonard model and show that for values close to the Hopf bifurcation, in the limit of small fluctuations, we can map the coarse-grained description of the model to the Complex Ginsburg Landau Equation, with stochastic noise corrections. Finally, we explore the induction of ecodiversity through spatio-temporal spirals in the asymmetric version of the May--Leonard model, which is otherwise inclined to reach an extinction state. This is accomplished by coupling to a symmetric May-Leonard counterpart on a two-dimensional lattice. The coupled system creates conditions for spiral formation in the asymmetric subsystem, thus precluding extinction.
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