Optimizing a jump-diffusion model of a starving forager

TL;DR

This paper investigates a hybrid jump-diffusion model of a starving forager on a one-dimensional lattice, revealing that a mixed movement strategy optimizes forager lifetime through a combination of jumping and diffusing behaviors.

Contribution

It introduces a hybrid model combining diffusion and jumping for a forager, analyzing how mixed strategies can maximize survival time.

Findings

Forager lifetime approximates cover time plus starvation time when starvation threshold exceeds lattice size.

The lifetime varies non-monotonically with jump probability, indicating an optimal mix of movement modes.

A combined jump and diffusion strategy maximizes forager survival, as shown through analysis and simulations.

Abstract

We analyze the movement of a starving forager on a one-dimensional periodic lattice, where each location contains one unit of food. As the forager lands on sites with food, it consumes the food, leaving the sites empty. If the forager lands consecutively on $s$ empty sites, then it will starve. The forager has two modes of movement: it can either diffuse, by moving with equal probability to adjacent sites on the lattice, or it can jump to a uniformly randomly chosen site on the lattice. We show that the lifetime $T$ of the forager in either paradigm can be approximated by the sum of the cover time $\tau_{\rm cover}$ and the starvation time $s$, when $s$ far exceeds the number $n$ of lattice sites. Our main findings focus on the hybrid model, where the forager has a probability of either jumping or diffusing. The lifetime of the forager varies non-monotonically according to $p_j$, the probability of jumping. By examining a small system, analyzing a heuristic model, and using direct numerical simulation, we explore the tradeoff between jumps and diffusion, and show that the strategy that maximizes the forager lifetime is a mixture of both modes of movement.