How Colored Environmental Noise Affects Population Extinction

Environmental noise can cause an exponential reduction in the mean time to extinction (MTE) of an isolated population. We study this effect on an example of a stochastic birth-death process with rates modulated by a colored (that is, correlated) Gaussian noise. A path integral formulation yields a transparent way of evaluating the MTE and finding the optimal realization of the environmental noise that determines the most probable path to extinction. The population-size dependence of the MTE changes from exponential in the absence of the environmental noise to a power law for a short-correlated noise and to no dependence for long-correlated noise. We also establish the validity domains of the white-noise limit and adiabatic limit.

Figure 1

Various regimes of extinction on the plane of rescaled parameters $V$ and $T$. The dashed lines are schematic borders of the adiabatic, Eq. (16), and white-noise, Eq. (14) limits. The dotted line is the border of the weak-noise, Eq. (17), regime. The shaded area is a crossover region.

Figure 2

Zero-energy trajectories of the Hamiltonian $H$ (the dashed line), of $H_v$ (the dotted line), and of both $H$ and $H_v$ (the solid lines). The shadowed area is the extinction action for the short-correlated environmental noise, leading to Eq. (14).

Figure 3

Optimal realizations of the environmental noise in the limit of short (the dashed line) and long (the solid line) correlations of the noise. The duration of the catastrophe for the short-correlated noise is $\tilde{t}\approx t_r\ln (Kv{t}_c/\mu )$.