Optimizing a jump-diffusion model of a starving forager

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.