In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, a companion paper explored relevant features of the probability simplex of allelic frequencies for a population which is genetically homogeneous except possibly at a single locus.

The Shahshahani metric is modified in this paper to produce a measure of distance near an arbitrary frequency F in the allelic simplex which can be used when some alleles are given zero weight by F. The equation of evolution for the modified metric can then be used to show that certain sets of frequencies (corresponding to equilibrium mean strategies) act as local attractors, as long as the mean strategies corresponding to those sets are non-singular or even, in most cases, singular. We identify conditions under which the measure of distance from an initial frequency to a nearby set of equilibrium frequencies corresponding to exceptional mean strategies might increase, either temporarily or for a protracted length of time.

In order to determine the robustness of the mean-covariance approach to exploring behavioural models of sexual diploid biological populations which are based on the evolutionarily stable strategy (ESS) concept, relevant features of the probability simplex of allelic frequencies for a population with genetic variability at a single locus are explored. Singularities and related properties of mappings from the space of allele frequencies to the space of strategy frequencies are examined, and related to a certain covariance measure of variability present in the population.

A companion paper builds on this characterization to establish that previous claims of stability in fact hold under slightly weaker conditions than initially indicated. The pair of papers also determines conditions under which unstable equilibria can occur, and establishes that these conditions are exceptional in practice.

Extensions of ESS theory to situations well outside the classical formulation often assume, as a convenience, that the population being modelled is, in some sense, monomorphic. While this assumption is in keeping with the original approach used in developing the theory, it is rendered less plausible by the observation that the original models do not preclude the possibility of polymorphism, a potentially serious omission. We consider a generalisation of the classical bilinear fitness function, and examine the circumstances that will tend to favour monomorphism.

The evolutionarily stable strategy for a given payoff matrix contest, although originally determined in terms of a haploid population, has been shown elsewhere to correspond to an equilibrium of the mean strategy of a diploid population. In this note, the equilibrium is shown to be locally stable for diploid populations. This local stability is demonstrated primarily by relating the behaviour of the perturbed diploid population to one, or in some cases two, associated haploid populations.

The assumption of arbitrary and biologically implausible inheritance patterns in sexual diploid populations can yield population models in which convergence of a population's mean strategy to an evolutionarily stable strategy will not occur, even though this strategy is attainable with the correct choice of gametic frequencies. The present paper investigates the effect of imposing a simplifying and biologically reasonable restriction on the assumed inheritance patterns; namely, that dominance or underdominance occurs.

In evolutionarily stable strategy models, transmission of one's strategy to one's offspring has routinely been assumed to be exact, whether this transmission is genetic or by instruction. If inexact transmission of strategy is possible, however, the nature of the mode of transmission becomes important. This paper demonstrates analytically that the possibility of learning one's strategy can increase rather than decrease eventual strategy diversity and that the resulting population mean strategy can differ appreciably from an evolutionarily stable strategy. These results are found to be consistent with simulation results reported in the literature.

A simple argument demonstrates that the mean strategy of a diploid sexual population at evolutionary equilibrium can be expected to be an evolutionarily stable strategy (ESS) in the formal sense. This result follows under a wide set of models of genetic inheritance of strategy (including sexual selection) provided that the ESS is both attainable and maintainable.

A class of Lyapunov functions is used to demonstrate that strategy stability occurs in complex randomly mating diploid populations. Strategies close to the evolutionarily stable strategy tend to fare better than more remote strategies. If convergence in mean strategy to an evolutionarily stable strategy is not possible, evolution will continue until all strategies in use lie on a unique face of the convex hull of available strategies.

The results obtained are also relevant to the haploid parthenogenetic case.

The changes in diversity of competitive strategies in a Maynard Smith population model with mixed strategies are related to the changes in population mean strategy. The effects of slight mutations in strategy frequencies, and of slight perturbations of the contest payoff rules are then investigated, and found to increase and decrease diversity respectively (to a third-order approximation). A relation among mutational effects, payoff perturbation effects and stable population diversity is suggested.

The possibility that interspecific contest tactics might be able to stabilize two species predator-prey populations is investigated using a matrix model of the population. It is found that if the interspecific tactics have no influence on intraspecific encounters, the non-trivial equilibrium population composition is necessarily unstable. The result provides some insight into a reported instability of a predator-prey population with ‘stable' strategies.