An exactly solvable predator prey model with resetting

TL;DR

This paper introduces an exactly solvable predator-prey model with resetting, revealing how survival probability decays and encounter distributions depend on model parameters, supported by analytical and numerical results.

Contribution

It provides an exact analytical solution for a predator-prey model with resetting, including survival probability decay and encounter distribution, advancing understanding of stochastic resetting in ecological models.

Findings

Survival probability decays algebraically with a parameter-dependent exponent.

Exact distribution of total encounters exhibits anomalous large deviation behavior.

Analytical results are validated by numerical simulations.

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.