Strategically developed phenotypes and the evolution of signals
In the first part of this thesis, a general one dimensional theory of strategically determined phenotypes is developed and applied to biological signalling games. Abstract modular modelling techniques are utilised to solve hitherto analytically intractable problems including error-prone signalling, and how to incorporate genetic features into optimization models. Links are drawn between previous biological models, such as the War of Attrition and Strategic Handicap mod- els. Mistakes in previous biological models are recognised and, where possible, rectified. A number of novel insights into biological phenomena arising from the models are presented, including analyses of: when free signals are possible; honest signalling of future paternal investment; dimorphic signals; the effects of the mechanisms of female discrimination in sexual selection on male signalling strategies; and the effects of relatedness on the magnitude and stability of equi- librium signalling strategies. It is argued that Zahavi's proposed demarcation between signal selection and natural selection is unjustifiable from a theoretical perspective. The second part of the thesis concerns the epistatic handicap process of sex- ual selection. Unlike the conditional and revealing handicap mechanisms, the epistatic or 'Zahavian' handicap mechanism of sexual selection has hitherto found scant support in the theoretical literature, as it appeared to function only under the most extreme conditions. A continuous game theory model, a quantitative genetics model, and a three locus major gene model are presented which show that the epistatic handicap mechanism can function, independent of the Fisher process of sexual selection, under reasonable assumptions. More- over, the game theory model illuminates the connection between the strategic and epistatic handicap mechanisms. The quantitative genetics and major gene models, together with a fourth model, are also used to show that a general argu- ment concerning indirect genetic correlations, which has appeared in a number of papers on sexual selection, is specious. Finally, a general theorem on games that satisfy the single-crossing condition (also known as the sorting, Spence-Mirrless, or constant sign condition) which underlies many of the results presented in the first part of the thesis is proven in appendix C. Applying a limit result to this general theorem provides a new proof of, and extensions to, Nash's existence result for equilibria to strategic- form games without having to resort to Kakutani's fixed point theorem.