A continuum formulation of the ideal free distribution and its implications for population dynamics

The ideal free distribution is a description of how organisms would distribute themselves in space if they were free to move so as to maximize fitness. The standard formulation of the ideal free distribution envisions the environment as consisting of finitely many discrete habitats. In this paper, a version of the ideal free distribution is derived for the case where the environment is a continuum. The continuum formulation allows computation of average fitness at the population level by taking account of both local fitness and the spatial distribution of the population. An example shows that the average fitness may have a different form than the local fitness; in particular, if local fitness is described by a logistic equation at each location, the average fitness may obey the theta-logistic equation of F. J. Ayala et al. (1973, Theor. Popul. Biol. 4, 331-356). This gives a mechanistic derivation of the theta-logistic equation.