On a population model with memory
Jean Bertoin
TL;DR
The paper studies a memory-augmented, multitype branching process on a finite type space, where reproduction activates a randomly chosen ancestral type according to $\tau$, yielding a memory-driven mean operator $\boldsymbol m$ and spectral radius $\boldsymbol r$. Using a many-to-one formula tied to a biased Markov chain and an asymptotic coupling argument, it proves $\boldsymbol r \ge r$, i.e., incorporating memory cannot reduce the long-term average growth rate. It constructs a size-biased, ergodic framework through a biased chain with transition kernel $\overline Q$ and establishes a unique invariant law $\overline{\boldsymbol{\sigma}}$, with marginals linked to the Perron–Frobenius eigenvectors via $\sigma(t)=h(t)\varrho(t)$. The analysis handles both finite and unbounded memory support, leveraging truncations, a many-to-one representation, and an asymptotic coupling to overcomeChallenges arising from the memory of the entire ancestry. Overall, memory activation is shown to be intrinsically beneficial for population growth in this model, underscoring atavism as a robust driver of evolutionary dynamics in the absence of selection or environmental changes.
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.