Semigroup analysis of structured parasite populations

TL;DR

This paper analyzes a size-structured parasite population model using semigroup theory, establishing conditions for equilibrium existence, stability, and spectral properties of the linearized system.

Contribution

It introduces a novel semigroup framework for analyzing structured parasite populations with distributed states at birth, including stability and spectral analysis.

Findings

Existence of positive equilibrium solutions under certain conditions

Characterization of the linearized system via positive quasicontraction semigroups

Stability analysis based on spectral properties of the generator

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.