Population extinction risk in the aftermath of a catastrophic event

We investigate how a catastrophic event (modeled as a temporary fall of the reproduction rate) increases the extinction probability of an isolated self-regulated stochastic population. Using a variant of the Verhulst logistic model as an example, we combine the probability generating function technique with an eikonal approximation to evaluate the exponentially large increase in the extinction probability caused by the catastrophe. This quantity is given by the eikonal action computed over "the optimal path" (instanton) of an effective classical Hamiltonian system with a time-dependent Hamiltonian. For a general catastrophe the eikonal equations can be solved numerically. For simple models of catastrophic events analytic solutions can be obtained. One such solution becomes quite simple close to the bifurcation point of the Verhulst model. The eikonal results for the increase in the extinction probability caused by a catastrophe agree well with numerical solutions of the master equation.