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.

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

References in corpus (14)

• First Passage Under Restart
• First order transition for the optimal search time of Lévy flights with resetting
• Optimal mean first-passage time for a Brownian searcher subjected to resetting: experimental and theoretical results
• Diffusion with resetting in arbitrary spatial dimension
• Dynamical transition in the temporal relaxation of stochastic processes under resetting
• Random walks with preferential relocations to places visited in the past and their application to biology
• First passages for a search by a swarm of independent random searchers
• Monotonous continuous-time random walks with drift and stochastic reset events
• Interacting Brownian Motion with Resetting
• Phase transitions in large deviations of reset processes
• Record statistics for random walks and Lévy flights with resetting
• Real Roots of Random Polynomials and Zero Crossing Properties of Diffusion Equation
• Statistics of the Number of Zero Crossings : from Random Polynomials to Diffusion Equation
• Searching for clusters of targets under stochastic resetting