Dynamical Systems and Fitness Maximization in Evolutionary Biology

Abstract

Five years after Charles Darwin put forward his theory of Natural Selection, Herbert Spencer coined the phrase “survival of the fittest.” Survival of the fittest implies that individuals with highest fitness will survive and will pass on their traits – as required for evolution. Since then, scholars have struggled to mathematically model genetic change, inheritance, and selection, and further to define “fitness” to understand how fitness changes in a population over time. For about 90 years, the goal has been to prove that fitness continually increases – fitness maximization – in line with traditional expectation.

This chapter presents the history of this issue to date, including proving a new simple and elegant formula for the fundamental theorem of natural selection with mutations, and a new application of the fundamental theorem of dynamical systems to evolutionary models which constrains the possible concept of fitness maximization.

Taking a mathematical modeling perspective, we present both experimental genetics and mathematical models. Field biology researchers have observed that generally populations are either in stasis or are in fitness decline, and many mathematicians modeling genetics have rejected the very idea of general fitness maximization. We consider a variety of mutation-selection models, from Fisher’s early work, mutation-limited models that consider one mutation at a time, models that consider a distribution of simultaneous mutations, to the most comprehensive numerical simulations. We conclude that fitness is best understood in terms of net functionality (not just reproduction rate), and that fitness maximization is not a robust biological principle.