Capture of a diffusing lamb by a diffusing lion when both return home

TL;DR

This paper models the pursuit of a diffusing lamb by a diffusing lion using random walk theory, analyzing how resetting protocols affect the distribution of their meeting points, with implications for stochastic process understanding.

Contribution

It introduces a novel analysis of diffusing predator-prey dynamics with resetting, revealing how different resetting protocols shape the distribution of their intersection points.

Findings

Without resets, intersection location follows a Cauchy distribution.

Resetting creates a mixed distribution with exponential tails and a core dependent on reset type.

Poisson resets lead to Cauchy core; sharp resets lead to Gaussian core.

Abstract

A diffusing lion pursues a diffusing lamb when both of them are allowed to get back to their homes intermittently. Identifying the system with a pair of vicious random walkers, we study their dynamics under Poissonian and sharp resetting. In absence of any resets, the location of intersection of the two walkers follows a Cauchy distribution. In presence of resetting, the distribution of the location of annihilation is composed of two parts: one in which the trajectories cross without being reset (center) and the other where trajectories are reset at least once before they cross each other (tails). We find that the tail part decays exponentially for both the resetting protocols. The central part of the distribution, on the other hand, depends on the nature of the restart protocol, with Cauchy for Poisson resetting and Gaussian for sharp resetting. We find good agreement of the analytical results with numerical calculations.