# Adaptive Fitness Landscape for Replicator Systems: To Maximize or not to Maximize

## TL;DR

This paper investigates when the adaptive landscape metaphor accurately describes evolutionary dynamics in replicator systems and explores the structure of mean fitness even when it is not maximized.

## Contribution

It identifies conditions under which the adaptive landscape metaphor holds exactly and analyzes the implications of mean fitness structure on system dynamics.

## Key findings

- Adaptive landscape accurately models some replicator dynamics.
- Conditions for exact correspondence between dynamics and fitness maximization.
- Insights into mean fitness landscape structure and its evolutionary implications.

## Abstract

Sewall Wright's adaptive landscape metaphor penetrates a significant part of evolutionary thinking. Supplemented with Fisher's fundamental theorem of natural selection and Kimura's maximum principle, it provides a unifying and intuitive representation of the evolutionary process under the influence of natural selection as the hill climbing on the surface of mean population fitness. On the other hand, it is also well known that for many more or less realistic mathematical models this picture is a sever misrepresentation of what actually occurs. Therefore, we are faced with two questions. First, it is important to identify the cases in which adaptive landscape metaphor actually holds exactly in the models, that is, to identify the conditions under which system's dynamics coincides with the process of searching for a (local) fitness maximum. Second, even if the mean fitness is not maximized in the process of evolution, it is still important to understand the structure of the mean fitness manifold and see the implications of this structure on the system's dynamics. Using as a basic model the classical replicator equation, in this note we attempt to answer these two questions and illustrate our results with simple well studied systems.