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Semigroup Analysis of Structured ParasitePopulations

Published online by Cambridge University Press:  28 April 2010

J. Z. Farkas ,
D. M. Green  and
P. Hinow
J. Z. Farkas*
Affiliation:
Department of Computing Science and Mathematics University of Stirling, FK9 4LA, Scotland UK
D. M. Green
Affiliation:
Institute of Aquaculture, University of Stirling, FK9 4LA Scotland, UK
P. Hinow
Affiliation:
Department of Mathematical Sciences, University of Wisconsin – Milwaukee P.O. Box 413, Milwaukee, WI, 53201, USA
*
*Corresponding author. E-mailjzf@maths.stir.ac.uk
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Abstract

Motivated by structured parasite populations in aquaculture we consider a class ofsize-structured population models, where individuals may be recruited into the populationwith distributed states at birth. The mathematical model which describes the evolution ofsuch a population is a first-order nonlinear partial integro-differential equation ofhyperbolic type. First, we use positive perturbation arguments and utilise results fromthe spectral theory of semigroups to establish conditions for the existence of a positiveequilibrium solution of our model. Then, we formulate conditions that guarantee that thelinearised system is governed by a positive quasicontraction semigroup on the biologicallyrelevant state space. We also show that the governing linear semigroup is eventuallycompact, hence growth properties of the semigroup are determined by the spectrum of itsgenerator. In the case of a separable fertility function, we deduce a characteristicequation, and investigate the stability of equilibrium solutions in the general case usingpositive perturbation arguments.

Keywords

aquaculturequasicontraction semigroupspositivityspectral methodsstability

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