An exactly solvable predator prey model with resetting

Abstract

We study a simple model of a diffusing particle (the prey) that on encounter with one of a swarm of diffusing predators can either perish or be reset to its original position at the origin. We show that the survival probability of the prey up to time $t$ decays algebraically as $\sim t^{-\theta (p,\gamma )}$ where the exponent $\theta$ depends continuously on two parameters of the model, with $p$ denoting the probability that a prey survives upon encounter with a predator and $\gamma =D_A/(D_A+D_B)$ where $D_A$ and $D_B$ are the diffusion constants of the prey and the predator respectively. We also compute exactly the probability distribution $P(N|t_c)$ of the total number of encounters till the capture time $t_c$ and show that it exhibits an anomalous large deviation form $P(N|t_c)\sim t_c^{-\Phi \left(\frac{N}{\ln t_c}=z\right)}$ for large $t_c$. The rate function $\Phi (z)$ is computed explicitly. Numerical simulations are in excellent agreement with our analytical results.

arXiv comment18 pages, 3 figures, accepted for Journal of Physics A Special Issue "Stochastic Resetting: Theory and Applications" 2022

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