Structural Instability in Evolutionary Population Biology

Abstract

In recent applications of population-genetic and game-theoretic concepts to behavioural ecology, which have become known as “sociobiology”, theoretical problems arise from the fact that neo-Darwinian evolution involves processes such as mutation and natural selection which proceed at quite different rates. These problems can be circumscribed as follows. On the one hand, gene selection can in principle be described by dynamic differential equations of the form

$$ \frac{{d{{X}_{i}}}}{{dt}} = {{G}_{i}}\left( {C;{{X}_{1}},...,{{X}_{m}}} \right),1\; \leqq i \leqq m, $$

(V.1)

covering the classical selection equations of Fisher, Haldane and Wright. This approach assumes m alleles A₁,...,A_m at certain loci, with respective numbers X₁,..., X _m of copies in the population, whereby G _; is in general a highly non-linear function of X ₁,..., X _m , of the fitness matrix C,and of epistatic and other effects (Akin 1979). Theoretical interest concentrates on the possibility of asymptotically stable stationary states, or equilibria, of (V.1) with

$$ {{G}_{i}} = 0,\quad 1\; \leqq i \leqq m, $$

(V.2)

because under certain conditions these equilibria correspond to local maxima of the population mean fitness towards which the gene pool will evolve. Dynamic equations of the type (V.1) have also been used to analyse frequency-dependent selection at the phenotypic level, especially selection of behavioural strategies of intraspecific cooperation and competition. In this approach, which is known as “evolutionary game dynamics” (Taylor and Jonker 1978; Schuster et al. 1981; Zeeman 1981; Hofbauer and Sigmund 1984), X₁,..., X _m are the respective numbers of competitors in an m-strategy game, and C is the pay-off matrix of the game evaluated in terms of Darwinian fitness.