Journal of Applied Mathematics
Volume 2005 (2005), Issue 4, Pages 403-424
Evolutionary distributions in adaptive space
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 , is a distribution of density of phenotypes over a set of adaptive
traits . Here is an -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.