Branching processes with competition and generalized Ray Knight Theorem

TL;DR

This paper models population dynamics with nonlinear birth and death rates, deriving a diffusion limit and a Ray-Knight representation involving local times of reflected Brownian motion, extending classical results to interacting populations.

Contribution

It introduces a new diffusion approximation for interacting populations with nonlinear rates and provides a Ray-Knight type representation involving local times of a reflected Brownian motion.

Findings

Population size converges to a diffusion process in the large population limit.

Provides a Ray-Knight representation for the diffusion in terms of local times.

Extends classical Ray-Knight theorems to models with interaction and nonlinearity.

Abstract

We consider a discrete model of population with interaction where the birth and death rates are non linear functions of the population size. After proceeding to renormalization of the model parameters, we obtain in the limit of large population that the population size evolves as a diffusion solution of the SDE Z^x_t =x+\int_0^t f(Z^x_s)ds+2\int_0^t\int_0^{Z^x_s}W(ds,du), where W(ds,du) is a time space white noise on ([0,\infty))^2. We give a Ray-Knight representation of this diffusion in terms of the local times of a reflected Brownian motion H with a drift that depends upon the local time accumulated by H at its current level, through the function f'/2.