Nonlinear deterministic equations in biological evolution

TL;DR

This paper reviews deterministic nonlinear models of biological evolution, highlighting their complex behaviors like multiple equilibria and phase transitions, especially in sexual populations, and discusses cases with analytical solutions.

Contribution

It provides a comprehensive overview of nonlinear deterministic evolution models, emphasizing analytical solutions and complex dynamics in biological populations.

Findings

Nonlinear equations arise in complex biological evolution models.

Solutions exhibit multiple equilibria and phase transitions.

Analytical understanding is available for certain models.

Abstract

We review models of biological evolution in which the population frequency changes deterministically with time. If the population is self-replicating, although the equations for simple prototypes can be linearised, nonlinear equations arise in many complex situations. For sexual populations, even in the simplest setting, the equations are necessarily nonlinear due to the mixing of the parental genetic material. The solutions of such nonlinear equations display interesting features such as multiple equilibria and phase transitions. We mainly discuss those models for which an analytical understanding of such nonlinear equations is available.