Graphical Representation of some Duality Relations in Stochastic Population Models

TL;DR

This paper develops a unified stochastic framework using graphical representations to analyze duality relations in population models, linking resampling-selection, branching-coalescing, and logistic growth processes.

Contribution

It introduces a novel unified stochastic approach with graphical methods to understand dualities in various population models.

Findings

Identifies duality relations between particle process components.

Provides a graphical representation for duality in stochastic population models.

Unifies different duality concepts within a single stochastic framework.

Abstract

We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. http://www.ams.org/mathscinet-getitem?mr=MR2123250) and for the self-duality of Feller's branching diffusion with logistic growth (cf. math/0509612). The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation. We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above.