Branching Processes in a L\xe9vy Random Environment

S Palau,J C Pardo

doi:10.1007/s10440-017-0120-7

Abstract

In this paper, we introduce branching processes in a Lévy random environment. In order to define this class of processes, we study a particular class of non-negative stochastic differential equations driven by a white noise and Poisson random measures which are mutually independent. Following similar techniques as in Dawson and Li (Ann. Probab. 40:813–857, 2012) and Li and Pu (Electron. Commun. Probab. 17(33):1–13, 2012), we obtain existence and uniqueness of strong local solutions of such stochastic equations. We use the latter result to construct continuous state branching processes with immigration and competition in a Lévy random environment as a strong solution of a stochastic differential equation. We also study the long term behaviour of two interesting examples: the case with no immigration and no competition and the case with linear growth and logistic competition.

Highlights

In many biological systems, when the population size is large enough, many birth and death events occur
Bansaye et al [3] studied more general CB-processes in random environment which are driven by Lévy processes whose paths are of bounded variation and the branching mechanism possesses a first moment condition
They were called CB-processes with catastrophes motivated by the fact that the presence of a negative jump in the random environment represents that a proportion of the population, following the dynamics of the CB-process, is killed

Summary

Introduction

In many biological systems, when the population size is large enough, many birth and death events occur. Where M is a Poisson random measure with intensity dsF (dθ ) Inspired in this model, Bansaye et al [3] studied more general CB-processes in random environment which are driven by Lévy processes whose paths are of bounded variation and the branching mechanism possesses a first moment condition. Bansaye et al [3] studied more general CB-processes in random environment which are driven by Lévy processes whose paths are of bounded variation and the branching mechanism possesses a first moment condition They were called CB-processes with catastrophes motivated by the fact that the presence of a negative jump in the random environment represents that a proportion of the population, following the dynamics of the CB-process, is killed. We study its long time behaviour and the Laplace transform of its first passage time below a level under the assumption that the environment has no negative jumps

Stochastic Differential Equations

CBI-Processes with Competition in a Lévy Random Environment

Long Term Behaviour of CB-Processes in a Lévy Random Environment

Population Model with Competition in a Lévy Random Environment