The Selection Gradient of an Infinite-Dimensional Trait

Abstract

The selection gradient is of central importance in evolutionary biology because it quantifies the forces of directional selection acting on a trait. Lande has shown that the selection gradient can be computed as the vector gradient of the log mean fitness when the trait is normally distributed. Using the framework of Gaussian processes and reproducing kernel Hilbert spaces, a rigorous definition is developed for the selection gradient of an infinitedimensional trait. Lande's result is then extended to this case.

The adaptive dynamics of function-valued traits

“…The function x g is the selection gradient and is obtained as the functional derivative of f at trait value x (Kirkpatrick and Lofsvold 1992;Gomulkiewicz and Beder 1996)…”

“…Two types of function-valued traits that have received particular attention in the context of this approach are growth trajectories, where the argument of the function-valued trait is age and the trait itself measures expected body size (Kirkpatrick 1988(Kirkpatrick , 1993Lofsvold 1989, 1992;Kirkpatrick et al 1990), and reaction norms, where the argument measures an environmental condition and the function-valued trait characterizes the phenotypes expressed in response to these conditions (Gomulkiewicz and Kirkpatrick 1992). The mathematical structures underlying the evolution of function-valued traits in quantitative genetics have been elucidated by Gomulkiewicz and Beder (1996) and by Beder and Gomulkiewicz (1998); these authors also derived results that facilitate analysis of several interesting classes of fitness functions used in quantitative genetics. Furthermore, the practical relevance of function-valued adaptation for livestock breeding has been pointed out (Kirkpatrick et al 1994;Kirkpatrick 1997).…”

Up hill, down dale: quantitative genetics of curvaceous traits

“…Given a fitness function and the distribution of the trait, we can calculate mean fitness, W , in a population. When relative fitnesses for all individuals are constant over time, the selection gradient is equal to the gradient in mean fitness as a function of the trait mean (Gomulkiewicz & Beder 1996) jðtÞ Z d lnð W Þ=d MðtÞ:…”

Function-valued adaptive dynamics and optimal control theory

“…The theory of optimal control has been used in the context of function-valued traits in game theory (Hamelin and Lewis, 2010), in quantitative genetics (Gomulkiewicz and Kirkpatrick, 1992;Gomulkiewicz and Beder, 1996;Beder and Gomulkiewicz, 1998;Jaffrézic and Pletcher, 2000;Kingsolver et al, 2001), and in life-history theory (Perrin and Sibly, 1993;Gilchrist et al, 2006). The novel feature considered in this article is the extension of the methods of optimal control theory to problems with environmental feedback, which is essential for tackling biologically realistic models.…”