Trait-dependent branching particle systems with competition and multiple offspring

TL;DR

This paper develops a mathematical model for populations with trait-dependent reproduction and competition, allowing multiple offspring and trait-influenced reproduction laws, and derives large population limits despite complex dependencies.

Contribution

It introduces a new trait-dependent branching particle system with multiple offspring and competition, and establishes a superprocess limit using a generalized Dawson's Girsanov Theorem.

Findings

Established a superprocess limit for the population model.

Developed a generalized Girsanov Theorem for non-branching dependent processes.

Provided a tractable framework for trait-structured population dynamics.

Abstract

In this work we model the dynamics of a population that evolves as a continuous time branching process with a trait structure and ecological interactions in form of mutations and competition between individuals. We generalize existing microscopic models by allowing individuals to have multiple offspring at a reproduction event. Furthermore, we allow the reproduction law to be influenced both by the trait type of the parent as well as by the mutant trait type. We look for tractable large population approximations. More precisely, under some natural assumption on the branching and mutation mechanisms, we establish a superprocess limit as solution of a well-posed martingale problem. Standard approaches do not apply in our case due to the lack of the branching property, which is a consequence of the dependency created by the competition between individuals. For showing uniqueness we therefore had to develop a generalization of Dawson's Girsanov Theorem that may be of independent interest.