Steady states in hierarchicalstructured populations with distributed states at birth

József Z. Farkas; Peter Hinow

doi:10.3934/dcdsb.2012.17.2671

Article Contents

2012, Volume 17, Issue 8: 2671-2689. Doi: 10.3934/dcdsb.2012.17.2671

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Steady states in hierarchical structured populations with distributed states at birth

József Z. Farkas^1, and
Peter Hinow^2,

1.
Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA
2.
Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413

Received: February 28, 2011

Revised: July 31, 2011

Published: October 2012

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Abstract

We investigate steady states of a quasilinear first order hyperbolic partial integro-differential equation. The model describes the evolution of a hierarchical structured population with distributed states at birth. Hierarchical size-structured models describe the dynamics of populations when individuals experience size-specific environment. This is the case for example in a population where individuals exhibit cannibalistic behavior and the chance to become prey (or to attack) depends on the individual's size. The other distinctive feature of the model is that individuals are recruited into the population at arbitrary size. This amounts to an infinite rank integral operator describing the recruitment process. First we establish conditions for the existence of a positive steady state of the model. Our method uses a fixed point result of nonlinear maps in conical shells of Banach spaces. Then we study stability properties of steady states for the special case of a separable growth rate using results from the theory of positive operators on Banach lattices.

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Mathematics Subject Classification: 92D25, 47N60, 47D06, 35B35.

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