The Principle of Partial Control in Reaction-Diffusion Models for the Evolution of Dispersal

Lee Altenberg

Submitted September 16, 2012.


  • Principal 1: The principle of partial control (Altenberg, 1984)

    When variation has only partial control over the transformations occurring on types under selection, then it may be possible for the part which it controls to evolve an increase in rates.

  • Principal 2: Partial Control and Induced Directed Variation (Altenberg, 2012)

    Undirected variation of a transformation process, i.e. equal scaling of all transition probabilities by a rate m, may act effectively like directed variation toward fitter types due to dynamics induced by other transformation processes and selection, so that increases in m increase the population growth rate r.

Abstract

Studies using reaction-diffusion models for the evolution of dispersal have classified their behavior in terms of the categories of conditional vs. unconditional dispersal, and the operators of diffusion, nonlocal diffusion, advection, and spatial heterogeneity of growth rates. Recent results on resolvent positive operators reveal a different basis to classify their behavior: it focuses on the form of variation in the operators rather than the form of the operators themselves. When the variation consists of equal scaling of all dispersal and advection operators, selection favors reduced dispersal. But other forms of variation in the operators may select for increased dispersal. When variation has only partial control over the operators it may act effectively as though it were directed toward fitter habitats. The \emph{principle of partial control} provides a heuristic for variation that favors increased dispersal. While some results for the classification of variation that produces departure from reduction have been obtained for finite matrices, the general classification problem remains open for both finite and infinite dimensional spaces.
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