What is Darwin's mathematical formula?

Answer 1

Abstract. A mathematical formulation of Darwin's theory of evolutionary

optimization through variation and selection is derived in terms

of conventional ODEs that can be interpreted as chemical kinetics of

evolution. Variation in form of mutation and recombination operates on

genotypes being DNA or RNA sequences, whereas phenotypes, which are

represented by organisms or molecular structures, are the target of selection.

The impact of recombination on optimization is briefly sketched.

Differential equations modeling selection in populations with correct

replication and mutation are derived from the molecular mechanisms

of polynucleotide replication. The analysis of these ODEs reveals restrictions

of the optimization principle caused by mutation. Error propagation

over generations sets a limit to mutation rates in evolution,

which manifests itself in the form of a phase transition-like phenomenon

characterized as error threshold. Conditions on fitness landscapes for

the occurrence of error thresholds derived from numerical investigations

are presented: Smooth fitness landscapes show no error thresholds but

gradual transitions, sufficiently steep landscapes and rugged landscapes

sustain error thresholds. Sharp transitions are also found with realistic

landscapes combining ruggedness and neutrality. Lethal mutants may

lead to extinction of populations and set another upper limit to mutation

rates in form of an extinction threshold through lethal mutagenesis.