Uyeda, Josef C.; Arnold, Stevan J.; Hohenlohe, Paul A.; and Louise S Mead. In prep. Speciation by drift in female mating preferences.
Overview:
There are two main goals of this paper, both of which have been neglected despite a multitude of papers examining Fisherian sexual selection in the past 30 years. They are as follows:
One of the most famous verbal models of sexual selection is R.A. Fisher's Runaway selection model. This model predicted that female preference and male ornaments would evolve larger phenotypic values at ever increasing speed as a result of assortative mating and trait covariance. This model was formalized with a mathematical model developed by R. Lande (1981).
CLICK HERE FOR AN OVERVIEW OF LANDE (1981)
Using quantitative genetic parameters such as the variance-covariance matrix (the G-matrix), Lande showed that the runaway process was a special case of a more general model. The alternative to the runaway is a stable case in which populations "walk-to" a line of equilibrium, in which viability selection exactly counters the strength of sexual selection. Populations off the line of equilibrium will be pushed to the line. Once on the line, Lande predicted that populations could drift along the line of equilibrium in a process similar to a diffusion process, thereby providing a mechanism for speciation. While Lande's results have a clear connection to speciation, it is difficult to ascertain just how much reproductive isolation is obtained. Certainly, if reproductive isolation evolves quickly for a wide range of parameter values, we are more likely to believe that it can have evolutionary consequences than if it is prohibitively slow under most realistic parameter values.
Connecting to Arnold et al. (1996)
In Arnold et al. (1996), the authors develop a model that corresponds well to empirical measures of reproductive isolation. They connect the distribution of male and female trait values to an empirical measure of reproductive isolation, called the Joint Isolation (JI). The statistic effectively ranges from 0 to 2 (although negative values are possible, indicating disassortative mating), with a value of 0 representing no reproductive isolation, and a value of 2 representing complete reproductive isolation. The model allows for isolation asymmetry and assumes a gaussian preference function. The connection between the two models can be seen in the following animation:
FIGURE 1. Connecting Lande (1981) to measures of sexual isolation
Once we have connected these two models, we can simulate the evolution of replicate populations using Lande's dynamic equations, enabling us to observe the changing distribution of joint isolation (JI) with time.
FIGURE 2. Drifting replicate populations and the resulting reproductive isolation