[Paper Review] Predator-prey dynamics: Chasing by stochastic resetting

J. Quetzalcóatl Toledo-Marín, Denis Boyer+1 more|arXiv (Cornell University)|Dec 4, 2019

Diffusion and Search Dynamics11 citations

TL;DR

This paper studies a one-dimensional predator-prey system where a Brownian predator uses stochastic resetting to jump to previously visited prey positions, significantly reducing the mean first-encounter time with the prey. Unlike pure diffusion, which leads to divergent capture times, resetting ensures finite mean capture times that are minimized at an optimal resetting rate, demonstrating how memory-based search strategies enhance search efficiency.

ABSTRACT

We analyze predator-prey dynamics in one dimension in which a Brownian predator adopts a chasing strategy that consists in stochastically resetting its current position to locations previously visited by a diffusive prey. We study three different chasing strategies, namely, active, uniform and passive which lead to different diffusive behaviors of the predator in the absence of capture. When capture is considered, regardless of the chasing strategy, the mean first-encounter time is finite and decreases with the resetting rate. This model illustrates how the use of cues significantly improves the efficiency of random searches. We compare numerical simulations with analytical calculations and find excellent agreement.

Motivation & Objective

• To investigate how stochastic resetting to past prey positions improves search efficiency in a one-dimensional predator-prey system.
• To analyze the first-encounter time statistics when the predator uses memory of past prey locations for relocation.
• To compare different resetting strategies—active, uniform, and passive—on predator diffusion and capture time.
• To establish that resetting leads to finite mean capture times, contrasting with the divergent capture times in pure diffusion.
• To validate analytical results with numerical simulations using a kinetic Monte Carlo approach.

Proposed method

• Model the prey’s motion as an overdamped Brownian motion with diffusion coefficient $ D_y $, governed by $ dy/dt = \xi_y(t) $.
• Model the predator’s motion as a combination of free diffusion and stochastic resets to past prey positions, governed by a dichotomic process $ \sigma(t) $ at rate $ Q $.
• Use a memory kernel $ \phi(s;t) $ to describe the probability of selecting a past prey position $ y(s) $ at time $ s \leq t $, with uniform or biased sampling.
• Derive the Fokker-Planck equation for the relative position $ z(t) = x(t) - y(t) $, incorporating resetting effects and solve for the first-passage time distribution.
• Apply the saddle point method to approximate the cumulative distribution of first-encounter times, truncating the infinite sum over resets.
• Perform kinetic Monte Carlo simulations with Poisson-distributed reset times and Lévy-Smirnov distributed first-passage times to validate analytical results.

Experimental results

Research questions

• RQ1 How does stochastic resetting to past prey positions affect the mean first-encounter time between predator and prey?
• RQ2 What is the optimal resetting rate that minimizes the mean capture time in this memory-based search strategy?
• RQ3 How do different resetting strategies—active, uniform, passive—affect the predator’s diffusive behavior and capture efficiency?
• RQ4 Why does the mean number of resets for first capture saturate after only two resets, even for low resetting rates?
• RQ5 To what extent do analytical approximations based on saddle point methods agree with numerical simulations?

Key findings

• The mean first-encounter time between predator and prey is finite when stochastic resetting is used, in contrast to the divergent mean capture time in pure diffusion.
• The mean capture time decreases with increasing resetting rate and reaches a minimum at an optimal rate, demonstrating improved search efficiency.
• For $ Q \geq 0.1 $, the cumulative first-encounter probability saturates after only two resets, indicating that capture is highly likely within two jumps.
• The saddle point approximation for the number of resets yields excellent agreement with numerical simulations across a wide range of $ Q $ values.
• The approximation $ z_i = \sqrt{2D_y i / Q} $ for the relative distance after $ i $ resets is validated by simulations, supporting the truncation of the infinite sum in the first-encounter time calculation.
• Numerical simulations using kinetic Monte Carlo methods confirm the analytical predictions, showing excellent agreement across different resetting rates and initial conditions.