Drowsy Cheetah Hunting Antelopes: A Diffusing Predator Seeking Fleeing Prey

TL;DR

This paper models a predator-prey system with a cheetah and two antelopes as diffusing particles, analyzing the probability of the predator never catching the prey based on their initial positions.

Contribution

It introduces a mathematical framework using the backward Fokker-Planck equation to analyze a three-particle diffusion system with drift.

Findings

Derived the probability of the cheetah never catching an antelope.

Quantified how initial separations influence capture likelihood.

Provided analytical expressions for long-term outcomes.

Abstract

We consider a system of three random walkers (a `cheetah' surrounded by two `antelopes') diffusing in one dimension. The cheetah and the antelopes diffuse, but the antelopes experience in addition a deterministic relative drift velocity, away from the cheetah, proportional to their distance from the cheetah, such that they tend to move away from the cheetah with increasing time. Using the backward Fokker-Planck equation we calculate, as a function of their initial separations, the probability that the cheetah has caught neither antelope after infinite time.