A G-function approach to fitness minima, fitness maxima, evolutionarily stable strategies and adaptive landscapes

We use a fitness-generating function (G-function) approach to evolutionary games. The G-function allows for simultaneous consideration of strategy dyn amics and population dynamics. In contrast to approaches using a separate f itness function for each strategy, the G-function automatically expands and contracts the dimensionality of the evolutionary game as the number of ext ant strategies increases or decreases. In this way, the number of strategie s is not fixed but emerges as part of the evolutionary process. We use the G-function to derive conditions for a strategy's (or a set of strategies) r esistance to invasion and convergence stability. In hopes of relating the p roliferation of ESS-related terminology, we define an ESS as a set of strat egies that is both resistant to invasion and convergent-stable, With our de finition of ESS, we show the following: (1) Evolutionarily unstable maxima and minima are not achievable from adaptive dynamics. (2) Evolutionarily st able minima are achievable from adaptive dynamics and allow for adaptive sp eciation and divergence by additional strategies - in this sense, these min ima provide transition points during an adaptive radiation and are therefor e unstable when subject to small mutations. (3) Evolutionarily stable maxim a are both invasion-resistant and convergent-stable. When global maxima on the adaptive landscape are at zero fitness, these combinations of strategie s make up the ESS. We demonstrate how the number of co-existing strategies (coalition) emerges when seeking an ESS solution. The Lotka-Volterra compet ition model and Monod model of competition are used to illustrate combinati ons of invasion resistance and convergence stability, adaptive speciation a nd evolutionarily 'stable' minima, and the diversity of co-existing strateg ies that can emerge as the ESS.