From the distributions of times of interactions to preys and predators dynamical systems

TL;DR

This paper develops a stochastic predator-prey model with non-exponential interaction times, using an age-structured approach to derive a Markovian framework and analyze the macroscopic dynamics, including classical and new functional responses.

Contribution

It introduces an age-structured stochastic model with non-exponential times and proves convergence to a two-dimensional dynamical system, extending classical predator-prey models.

Findings

Convergence of the stochastic model to a deterministic two-dimensional system.

Recovery of classical functional responses within the new framework.

Identification of novel functional responses influenced by predator food availability.

Abstract

We consider a stochastic individual based model where each predator searches during a random time and then manipulates its prey or rests. The time distributions may be non-exponential. An age structure allows to describe these interactions and get a Markovian setting. The process is characterized by a measure-valued stochastic differential equation. We prove averaging results in this infinite dimensional setting and get the convergence of the slow-fast macroscopic prey predator process to a two dimensional dynamical system. We recover classical functional responses. We also get new forms arising in particular when births and deaths of predators are affected by the lack of food.