Geometric progression

In species which the reproductive success of one sex depends heavily on winning the concession of the other, as is evident with many polygamous birds, sexual selection will act by increasing the degree of preference to which it is due, with the consequence that both the trait preferred and the intensity of preference will be increased together with ever-increasing velocity. This process causing a fervent and rapid evolution of both the conspicuous ornamentation and the preference for such, until so arrested directly or indirectly by bionomic Natural Selection reasons. Thus, in many cases a positive feedback loop of sexual selection is created, resulting with exorbitant physical structures in the non-limited sex, the most notorious example being the peacock (shown above). It is important to note that while a peacock may have exorbitant plumage, the peahen has even more exorbitant taste for such.

Initially to start the process, there would be a correlation between the trait and higher fitness. Two previously isolated species, A and B, could come to inhabit the same area resulting in some hybridization. In this situation reproductive isolation will be favored. If the mean value of a trait e.g. tails, in species A, is larger than those of species B, selection would favor females of species A with preference for large tails. Once started the process could continue past the need for species isolation.

The peahen will desire to copulate with the most attractive Peacock so that her progeny, if male, will be attractive to females in the next generation. Additionally the Peacock will desire to copulate with a Peahen that finds him attractive so that if the progeny is female, preference for his degree of ornamentation remains present in the next generation. Since the rate of change in preference is proportioned according to the highest average degree of taste amongst females, and that females desire to best other members of the sex, it creates an additive effect in the cyclical process that will yield exponential increases, in both sexes, if unchecked.

R.A.Fisher in The Genetical Theory of Natural Selection was the first to articulate this process in a game theoretic style treatment.

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plumage development in the male, and sexual preference for such developments in the female, must thus advance together, and so long as the process is unchecked by severe counterselection, will advance with ever-increasing speed. In the total absence of such checks, it is easy to see that the speed of development will be proportional to the development already attained, which will therefore increase with time exponentially, or in geometric progression. —Ronald Fisher, 1930

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The exponential element, which is the kernel of the thing, arises from the rate of change in hen taste being proportional to the absolute average degree of taste. —Ronald Fisher, 1932 [5]

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It is important to notice that the conditions of relative stability brought about by these or other means, will be far longer duration than the process in which the ornaments are evolved. In most existing species the runaway process must have been already checked, and we should expect that the more extraordinary developments of sexual plumage are not due like most characters to a long and even course of evolutionary progress, but to sudden spurts of change. —Ronald Fisher, 1930

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Since R.A.Fishers initial conceptual model of the 'run-away' process, various others have continued the work on modeling an accurate mathematical proof. Notably R.Lande^[6] & P.O'Donald.