Rare extinction events in cyclic predator–prey games

Abstract

In the May–Leonard model of three cyclically competing species, we analyze the statistics of rare events in which all three species go extinct due to strong but rare fluctuations. These fluctuations are from the tails of the probability distribution of species concentrations. They render a coexistence of three populations unstable even if the coexistence is stable in the deterministic limit. We determine the mean time to extinction (MTE) by using a WKB-ansatz in the master equation that represents the stochastic description of this model. This way, the calculation is reduced to a problem of classical mechanics and amounts to solving a Hamilton–Jacobi equation with zero-energy Hamiltonian. We solve the corresponding Hamilton’s equations of motion in six-dimensional phase space numerically by using the Iterative Action Minimization Method. This allows to project on the optimal path to extinction, starting from a parameter choice where the three-species coexistence-fixed point undergoes a Hopf bifurcation and becomes stable. Specifically for our system of three species, extinction events can be triggered along various paths to extinction, differing in their intermediate steps. We compare our analytical predictions with results from Gillespie simulations for two-species extinctions, complemented by an analytical calculation of the MTE in which the remaining third species goes extinct. From Gillespie simulations we also analyze how the distributions of times to extinction (TE) change upon varying the bifurcation parameter. Our results shed some light on the sensitivity of the TE to system parameters. Even within the same model and the same dynamical regime, which allows a stable coexistence of species in the deterministic limit, the MTE depends on the distance from the bifurcation point in a way that contains the system size dependence in the exponent. It is challenging and worthwhile to quantify how rare the rare events of extinction are.

Rare desynchronization events in power grids: On data implementation and dimensional reductions

“…The WKB-approach applies to fluctuations which are rare. For example, when WKB is applied to large populations in the context of evolutionary game theory [31,32,33], large fluctuations are suppressed if the system size N is large. Typical fluctuations in population sizes are of O( √ N ), rare ones are of O(N ).…”

Rare desynchronization events in power grids: on data implementation and dimensional reductions

“…The WKB-approach applies to fluctuations which are rare. For example, when WKB is applied to large populations in the context of evolutionary game theory [9,34,35], large fluctuations are suppressed if the system size N is large. Typical fluctuations in population sizes are of O( √ N), rare ones are of O(N).…”

“…ϕ(t 1 ) = ϕ * and ϕ(t n ) = ϕ s for the phase angles. Here ϕ * and ϕ s are only determined up to a global phase shift, which we set according to equations (34) and (35). Together, this discretizes Hamilton's equations into a set of 2 × 6 × n equations, which can be solved with an off-the-shelf solver.…”