Journal of Mathematical Biology

, Volume 30, Issue 1, pp 73–87

# Cross entropy minimization in uninvadable states of complex populations

• Immanuel M. Bomze
Article

DOI: 10.1007/BF00168008

Bomze, I.M. J. Math. Biol. (1991) 30: 73. doi:10.1007/BF00168008

## Abstract

Selection is often. viewed as a process that maximizes the average fitness of a population. However, there are often constraints even on the phenotypic level which may prevent fitness optimization. Consequently, in evolutionary game theory, models of frequency dependent selection are investigated, which focus on equilibrium states that are characterized by stability (or uninvadability) rather than by optimality. The aim of this article is to show that nevertheless there is a biologically meaningful quantity, namely cross (fitness) entropy, which is optimized during the course of evolution: a dynamical model adapted to evolutionary games is presented which has the property that relative entropy decreases monotonically, if the state of a (complex) population is close to an uninvadable state. This result may be interpreted as if evolution has an “order stabilizing” effect.

### Key words

Frequency dependent selectionEvolutionary game theoryReplicator dynamics
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