Semigroup Analysis of Structured Parasite Populations

¹ Department of Computing Science and Mathematics University of Stirling, FK9 4LA, Scotland UK
² Institute of Aquaculture, University of Stirling, FK9 4LA Scotland, UK
³ Department of Mathematical Sciences, University of Wisconsin – Milwaukee P.O. Box 413, Milwaukee, WI, 53201, USA

^* Corresponding author. E-mail jzf@maths.stir.ac.uk

Abstract

Motivated by structured parasite populations in aquaculture we consider a class of size-structured population models, where individuals may be recruited into the population with distributed states at birth. The mathematical model which describes the evolution of such a population is a first-order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then, we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In the case of a separable fertility function, we deduce a characteristic equation, and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.