Asymptotic analysis of first passage time problems inspired by ecology.

Abstract

A hybrid asymptotic-numerical method is formulated and implemented to accurately calculate the mean first passage time (MFPT) for the expected time needed for a predator to locate small patches of prey in a 2-D landscape. In our analysis, the movement of the predator can have both a random and a directed component, where the diffusivity of the predator is isotropic but possibly spatially heterogeneous. Our singular perturbation methodology, which is based on the assumption that the ratio [Formula: see text] of the radius of a typical prey patch to that of the overall landscape is asymptotically small, leads to the derivation of an algebraic system that determines the MFPT in terms of parameters characterizing the shapes of the small prey patches together with a certain Green's function, which in general must be computed numerically. The expected error in approximating the MFPT by our semi-analytical procedure is smaller than any power of [Formula: see text], so that our approximation of the MFPT is still rather accurate at only moderately small prey patch radii. Overall, our hybrid approach has the advantage of eliminating the difficulty with resolving small spatial scales in a full numerical treatment of the partial differential equation (PDE). Similar semi-analytical methods are also developed and implemented to accurately calculate related quantities such as the variance of the mean first passage time (VMFPT) and the splitting probability. Results for the MFPT, the VMFPT, and splitting probability obtained from our hybrid methodology are validated with corresponding results computed from full numerical simulations of the underlying PDEs.