Semigroup Analysis of Structured Parasite Populations

J. Z. Farkas; D. M. Green; P. Hinow

Mathematical Modelling of Natural Phenomena (2010)

  • Volume: 5, Issue: 3, page 94-114
  • ISSN: 0973-5348

Abstract

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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.

How to cite

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Farkas, J. Z., Green, D. M., and Hinow, P.. "Semigroup Analysis of Structured Parasite Populations." Mathematical Modelling of Natural Phenomena 5.3 (2010): 94-114. <http://eudml.org/doc/197676>.

@article{Farkas2010,
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.},
author = {Farkas, J. Z., Green, D. M., Hinow, P.},
journal = {Mathematical Modelling of Natural Phenomena},
keywords = {aquaculture; quasicontraction semigroups; positivity; spectral methods; stability},
language = {eng},
month = {4},
number = {3},
pages = {94-114},
publisher = {EDP Sciences},
title = {Semigroup Analysis of Structured Parasite Populations},
url = {http://eudml.org/doc/197676},
volume = {5},
year = {2010},
}

TY - JOUR
AU - Farkas, J. Z.
AU - Green, D. M.
AU - Hinow, P.
TI - Semigroup Analysis of Structured Parasite Populations
JO - Mathematical Modelling of Natural Phenomena
DA - 2010/4//
PB - EDP Sciences
VL - 5
IS - 3
SP - 94
EP - 114
AB - 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.
LA - eng
KW - aquaculture; quasicontraction semigroups; positivity; spectral methods; stability
UR - http://eudml.org/doc/197676
ER -

References

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