A G-function approach to fitness minima, fitness maxima, evolutionary 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 dynamics and population dynamics. In contrast to approaches using a separate fitness function for each strategy, the G-function automatically expands and contracts the dimensionality of the evolutionary game as the number of extant strategies increases or decreases. In this way, the number of strategies 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) resistance to invasion and convergence stability. In hopes of relating the proliferation of ESS-related terminology, we define an ESS as a set of strategies that is both resistant to invasion and convergent-stable. With our definition of ESS, we show the following: (1) Evolutionarily unstable maxima and minima are not achievable from adaptive dynamics. (2) Evolutionarily stable minima are achievable from adaptive dynamics and allow for adaptive speciation and divergence by additional strategies - in this sense, these minima provide transition points during an adaptive radiation and are therefore unstable when subject to small mutations. (3) Evolutionarily stable maxima are both invasion-resistant and convergent-stable. When global maxima on the adaptive landscape are at zero fitness, these combinations of strategies make up the ESS. We demonstrate how the number of co-existing strategies (coalition) emerges when seeking an ESS solution. The Lotka-Volterra competition model and Monod model of competition are used to illustrate combinations of invasion resistance and convergence stability, adaptive speciation and evolutionarily 'stable' minima, and the diversity of co-existing strategies that can emerge as the ESS.