Journal of Applied Mathematics
Volume 2005 (2005), Issue 4, Pages 403-424

Evolutionary distributions in adaptive space

Yosef Cohen1,2

1Department of Fisheries and Wildlife, University of Minnesota, St. Paul 55108, MN, USA
2Department of Desert Ecology, Institute for Dryland Environmental Research, Ben-Gurion University of the Negev, Sede Boqer 84490, Israel

Received 18 January 2005

Copyright © 2005 Yosef Cohen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An evolutionary distribution (ED), denoted by z(x,t), is a distribution of density of phenotypes over a set of adaptive traits x. Here x is an n-dimensional vector that represents the adaptive space. Evolutionary interactions among phenotypes occur within an ED and between EDs. A generic approach to modeling systems of ED is developed. With it, two cases are analyzed. (1) A predator prey inter-ED interactions either with no intra-ED interactions or with cannibalism and competition (both intra-ED interactions). A predator prey system with no intra-ED interactions is stable. Cannibalism destabilizes it and competition strengthens its stability. (2) Mixed interactions (where phenotypes of one ED both benefit and are harmed by phenotypes of another ED) produce complete separation of phenotypes on one ED from the other along the adaptive trait. Foundational definitions of ED, adaptive space, and so on are also given. We argue that in evolutionary context, predator prey models with predator saturation make less sense than in ecological models. Also, with ED, the dynamics of population genetics may be reduced to an algebraic problem. Finally, extensions to the theory are proposed.