Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-23T06:11:03.347Z Has data issue: false hasContentIssue false
  • English
  • Français

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-mail[email protected]
Get access

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
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 3: Mathematical modeling in the medical sciences , 2010 , pp. 94 - 114
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander, U. Schlotterbeck. One-Parameter Semigroups of Positive Operators. Springer-Verlag, Berlin, 1986.
Borges, R., Calsina, À., Cuadrado, S.. Equilibria of a cyclin structured cell population model . Discrete Contin. Dyn. Syst., Ser. B, 11 (2009), No. 3, 613–627. CrossRefGoogle Scholar
Calsina, À. Saldaña, J.. Basic theory for a class of models of hierarchically structured population dynamics with distributed states in the recruitment . Math. Models Methods Appl. Sci., 16 (2006), No. 10, 1695–1722.CrossRefGoogle Scholar
Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, B. de Pagter. One-Parameter Semigroups. North–Holland, Amsterdam, 1987.
Costello, M. J.. Ecology of sea lice parasitic on farmed and wild fish . Trends in Parasitol., 22 (2006), No. 10, 475–483.CrossRefGoogle ScholarPubMed
J. M. Cushing. An Introduction to Structured Population dynamics. SIAM, Philadelphia, 1998.
O. Diekmann, M. Gyllenberg. Abstract delay equations inspired by population dynamics. in “Functional Analysis and Evolution Equations" (Eds. H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, J. von Below). Birkhäuser, 2007, 187–200.
Diekmann, O., Getto, Ph. Gyllenberg, M.. Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars . SIAM J. Math. Anal., 39 (2007), No. 4, 1023–1069.CrossRefGoogle Scholar
K.-J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000.
Farkas, J. Z. Hagen, T.. Stability and regularity results for a size-structured population model . J. Math. Anal. Appl., 328 (2007), No. 1, 119–136.CrossRefGoogle Scholar
Farkas, J. Z. Hagen, T.. Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback . Commun. Pure Appl. Anal., 8 (2009), No. 6, 1825–1839.CrossRefGoogle Scholar
J. Z. Farkas, T. Hagen. Hierarchical size-structured populations: The linearized semigroup approach. to appear in Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal.
Heuch, P. A. Mo, T. A.. A model of salmon louse production in Norway: effects of increasing salmon production and public management measures . Dis. Aquat. Org., 45 (2001), No. 2, 145–152.CrossRefGoogle ScholarPubMed
M. Iannelli. Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori, Pisa, 1994.
T. Kato. Perturbation Theory for Linear Operators. Springer, New York, 1966.
Kubokawa, Y., Ergodic theorems for contraction semi-groups, J. Math. Soc. Japan 27 (1975), 184-193. CrossRefGoogle Scholar
J. A. J. Metz, O. Diekmann. The Dynamics of Physiologically Structured Populations. Springer, Berlin, 1986.
Murray, A. G. Gillibrand, P. A.. Modelling salmon lice dispersal in Loch Torridon, Scotland . Marine Pollution Bulletin, 53 (2006), No. 1-4, 128–135.CrossRefGoogle ScholarPubMed
Prüß, J.. Stability analysis for equilibria in age-specific population dynamics . Nonlin. Anal. TMA 7 (1983), No. 12, 1291–1313.Google Scholar
Revie, C. W., Robbins, C., Gettinby, G., Kelly, L. Treasurer, J. W.. A mathematical model of the growth of sea lice, Lepeophtheirus salmonis, populations on farmed Atlantic salmon, Salmo salar L., in Scotland and its use in the assessment of treatment strategies . J. Fish Dis. 28 (2005), 603–613.CrossRefGoogle ScholarPubMed
Tucker, C. S., Norman, R., Shinn, A., Bron, J., Sommerville, C. Wootten, R.. A single cohort time delay model of the life-cycle of the salmon louse Lepeophtheirus salmonis on Atlantic salmon Salmo salar . Fish Path. 37 (2002), No. 3-4, 107–118.CrossRefGoogle Scholar
Tully, O. Nolan, D. T.. A review of the population biology and host-parasite interactions of the sea louse Lepeophtheirus salmonis (Copepoda: Caligidae) . Parasitology, 124 (2002), 165–182.CrossRefGoogle Scholar
G. F. Webb. Theory of nonlinear age-dependent population dynamics. Marcel Dekker, New York, 1985.
K. Yosida. Functional analysis. Springer, Berlin, 1995.