Rate distributions and survival of the fittest: a formulation on thespace of measures

Azmy S. Ackleh; Ben G. Fitzpatrick; Horst R. Thieme

doi:10.3934/dcdsb.2005.5.917

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2005, Volume 5, Issue 4: 917-928. Doi: 10.3934/dcdsb.2005.5.917

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Rate distributions and survival of the fittest: a formulation on the space of measures

1.
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504-1010
2.
Department of Mathematics, Loyola Marymount University, Los Angeles, California 90045
3.
Department of Mathematics and Statistics, Arizona State University, Tempe, Arizona 85287-1804

Received: May 31, 2004

Revised: April 30, 2005

Published: October 2005

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Abstract

In this paper we address the basic mathematical properties of a general population model having distributed growth and mortality rates. The problem considered generalizes previous efforts [3] in three ways. First, our model involves nonlinear growth and mortality terms. Second, the parameter space is assumed to be any compact subset of (0,∞) x (0, ∞), and third, the solutions of the rate distribution model are constructed in spaces of measures. The latter point is particularly appropriate for the asymptotic behavior, in which survival of the fittest manifests itself as a Dirac delta measure being the attractor of the dynamical system. As opposed to previous approaches to these problems, the measure space formulation allows the (weakly) stable equilibrium to be a point in the state space.

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Mathematics Subject Classification: Primary: 92D25.

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