On a population model with memory

TL;DR

This paper introduces a population model with memory, where individuals retain ancestral information, and demonstrates that such memory generally leads to equal or faster growth compared to memoryless models, using spectral analysis and Markov chain techniques.

Contribution

It develops a novel population model incorporating full ancestral memory and compares its growth rate to traditional models, revealing always at least equal growth.

Findings

Memory model grows at least as fast as memoryless model

Spectral radii comparison shows growth advantage with memory

Analysis uses biased Markov chains and ergodic law

Abstract

Consider first a memoryless population model described by the usual branching process with a given mean reproduction matrix on a finite space of types. Motivated by the consequences of atavism in Evolutionary Biology, we are interested in a modification of the dynamics where individuals keep full memory of their forebears and procreation involves the reactivation of a gene picked at random on the ancestral lineage. By comparing the spectral radii of the two mean reproduction matrices (with and without memory), we observe that, on average, the model with memory always grows at least as fast as the model without memory. The proof relies on analyzing a biased Markov chain on the space of memories, and the existence of a unique ergodic law is demonstrated through asymptotic coupling.