A compact containment result for nonlinear historical superprocess approximations for population models with trait-dependence

TL;DR

This paper extends previous models of population dynamics with traits by establishing a compact containment condition for nonlinear historical superprocess approximations, accommodating more general trait spaces and mutation dependencies.

Contribution

It generalizes the compact containment result to Polish trait-spaces and mutation kernels dependent on parent traits, enhancing the theoretical framework for population models.

Findings

Established a compact containment condition in a more general trait space setting

Derived path properties of individuals in the population models

Provided bounds on trait-history modulus of continuity

Abstract

We consider an approximating sequence of interacting population models with branching, mutation and competition. Each individual is characterized by its trait and the traits of its ancestors. Birth- and death-events happen at exponential times. Traits are hereditarily transmitted unless mutation occurs. The present model is an extension of the model used in [M\'el\'eard and Tran, EJP, 2012], where for large populations with small individual biomasses and under additional assumptions, the diffusive limit is shown to converge to a nonlinear historical superprocess. The main goal of the present article is to verify a compact containment condition in the more general setup of Polish trait-spaces and general mutation kernels that allow for a dependence on the parent's trait. As a by-product, a result on the paths of individuals is obtained. An application to evolving genealogies on marked metric measure spaces is mentioned where genealogical distance, counted in terms of the number of births without mutation, can be regarded as a trait. Because of the use of exponential times in the modeling of birth- and death-events the analysis of the modulus of continuity of the trait-history of a particle plays a major role in obtaining appropriate bounds.