On mathematical theory of selection: Continuous time population dynamics

TL;DR

This paper develops a mathematical framework for modeling continuous-time selection in inhomogeneous populations, enabling explicit solutions and effective computation of statistical traits, with applications to classical models.

Contribution

It introduces a new method reducing complex selection models to a system of ODEs, facilitating explicit solutions and analysis of inhomogeneous population dynamics.

Findings

Explicit solutions for inhomogeneous Malthusian, Ricker, and logistic models.

A reduction of complex models to an escort system of ODEs.

Effective computation of statistical characteristics like mean fitness.

Abstract

Mathematical theory of selection is developed within the frameworks of general models of inhomogeneous populations with continuous time. Methods that allow us to study the distribution dynamics under natural selection and to construct explicit solutions of the models are developed. All statistical characteristics of interest, such as the mean values of the fitness or any trait can be computed effectively, and the results depend in a crucial way on the initial distribution. The developed theory provides an effective method for solving selection systems; it reduces the initial complex model to a special system of ordinary differential equations (the escort system). Applications of the method to the Price equations are given; the solutions of some particular inhomogeneous Malthusian, Ricker and logistic-like models used but not solved in the literature are derived in explicit form.