Mutation--selection balance on an infinite trait space: confinement, drift and equilibrium
Abstract
We study a trait-structured population model incorporating mutation, selection and density-dependent regulation on a countably infinite trait space. The underlying stochastic process is a continuous-time Markov chain in which individuals reproduce at a trait-independent rate, offspring traits are determined by a mutation kernel on ℤ, mortality depends on trait, and births are progressively suppressed as the population approaches a fixed population ceiling. Using results for density-dependent Markov population processes with countably many types, we derive a deterministic approximation in the form of an infinite system of nonlinear differential equations. We establish existence and positive invariance of solutions, and investigate the equilibrium structure of the deterministic system. A fundamental distinction emerges between bounded and confining mortality profiles. When mortality remains bounded, mutation may continually transport mass through the trait space and a stationary trait distribution need not exist. In contrast, when mortality increases without bound as the absolute value of the trait index becomes large, the operator L=D−1P, where P and D govern mutation and mortality, is compact. By combining compactness with Kre\uın-Rutman theory for compact positive operators on Banach lattices, we show that L has an algebraically simple principal eigenvalue with a strictly positive eigenvector, and we derive a threshold condition for the existence of a non-zero equilibrium. In this regime the equilibrium is unique, and its trait distribution is determined by the principal eigenvector of L. Numerical experiments support the theoretical results and illustrate the contrasting behaviours associated with bounded and confining mortality profiles.
Related
- Human mutation field reveals an equilibrium-like structure with irreversible circulation
- Genetic weight of a mutant in a biparental Moran model with infinite selection
- Mutation breeding for quantitative traits in the era of genomic selection
- Mutation Order and Selection Shape Intratumor Heterogeneity in Tumor Evolution
- Causal inference clarifies the roles of background selection and mutation rate variation in shaping human genetic diversity
BibTeX
@misc{pollett2026mutationselection,
title = {{Mutation--selection balance on an infinite trait space: confinement, drift and equilibrium}},
author = {Phil. Pollett},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02086},
eprint = {2609.02086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02086},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mutation--selection balance on an infinite trait space: confinement, drift and equilibrium
AU - Phil. Pollett
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02086
UR - https://arxiv.org/abs/2609.02086
AB - We study a trait-structured population model incorporating mutation, selection and density-dependent regulation on a countably infinite trait space. The underlying stochastic process is a continuous-time Markov chain in which individuals reproduce at a trait-independent rate, offspring traits are determined by a mutation kernel on ℤ, mortality depends on trait, and births are progressively suppressed as the population approaches a fixed population ceiling. Using results for density-dependent Markov population processes with countably many types, we derive a deterministic approximation in the form of an infinite system of nonlinear differential equations. We establish existence and positive invariance of solutions, and investigate the equilibrium structure of the deterministic system. A fundamental distinction emerges between bounded and confining mortality profiles. When mortality remains bounded, mutation may continually transport mass through the trait space and a stationary trait distribution need not exist. In contrast, when mortality increases without bound as the absolute value of the trait index becomes large, the operator L=D^−1P, where P and D govern mutation and mortality, is compact. By combining compactness with Kre\uın-Rutman theory for compact positive operators on Banach lattices, we show that L has an algebraically simple principal eigenvalue with a strictly positive eigenvector, and we derive a threshold condition for the existence of a non-zero equilibrium. In this regime the equilibrium is unique, and its trait distribution is determined by the principal eigenvector of L. Numerical experiments support the theoretical results and illustrate the contrasting behaviours associated with bounded and confining mortality profiles.
ER -