Semigroup Analysis of Structured Parasite Populations

J. Z. Farkas¹ ; D. M. Green² ; P. Hinow³

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

Mathematical modelling of natural phenomena, Tome 5 (2010) no. 3, pp. 94-114.

Résumé

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.

Détail

DOI : 10.1051/mmnp/20105307

Bibliographie

[1] 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.

[2] R. Borges, À. Calsina, S. Cuadrado Equilibria of a cyclin structured cell population model Discrete Contin. Dyn. Syst., Ser. B 2009 613 627

[3] À. Calsina, J. Saldaña Basic theory for a class of models of hierarchically structured population dynamics with distributed states in the recruitment Math. Models Methods Appl. Sci. 2006 1695 1722

[4] Ph. Clément, H. J. A. M Heijmans, S. Angenent, C. J. van Duijn, B. de Pagter. One-Parameter Semigroups. North–Holland, Amsterdam, 1987.

[5] M. J. Costello Ecology of sea lice parasitic on farmed and wild fish Trends in Parasitol. 2006 475 483

[6] J. M. Cushing. An Introduction to Structured Population dynamics. SIAM, Philadelphia, 1998.

[7] 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.

[8] O. Diekmann, Ph. Getto, M. Gyllenberg Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars SIAM J. Math. Anal. 2007 1023 1069

[9] K.-J. Engel, R. Nagel. One-Parameter Semigroups for Linear Evolution Equations. Springer, New York, 2000.

[10] J. Z. Farkas, T. Hagen Stability and regularity results for a size-structured population model J. Math. Anal. Appl. 2007 119 136

[11] J. Z. Farkas, T. Hagen Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback Commun. Pure Appl. Anal. 2009 1825 1839

[12] 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.

[13] P. A. Heuch, T. A. Mo A model of salmon louse production in Norway: effects of increasing salmon production and public management measures Dis. Aquat. Org. 2001 145 152

[14] M. Iannelli. Mathematical Theory of Age-Structured Population Dynamics. Giardini Editori, Pisa, 1994.

[15] T. Kato. Perturbation Theory for Linear Operators. Springer, New York, 1966.

[16] Y. Kubokawa Ergodic theorems for contraction semi-groups J. Math. Soc. Japan 1975 184 193

[17] J. A. J. Metz, O. Diekmann. The Dynamics of Physiologically Structured Populations. Springer, Berlin, 1986.

[18] A. G. Murray, P. A. Gillibrand Modelling salmon lice dispersal in Loch Torridon, Scotland Marine Pollution Bulletin 2006 128 135

[19] J. Prüß Stability analysis for equilibria in age-specific population dynamics Nonlin. Anal. TMA 1983 1291 1313

[20] C. W. Revie, C. Robbins, G. Gettinby, L. Kelly, J. W. Treasurer 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. 2005 603 613

[21] C. S. Tucker, R. Norman, A. Shinn, J. Bron, C. Sommerville, R. Wootten A single cohort time delay model of the life-cycle of the salmon louse Lepeophtheirus salmonis on Atlantic salmon Salmo salar Fish Path. 2002 107 118

[22] O. Tully, D. T. Nolan A review of the population biology and host-parasite interactions of the sea louse Lepeophtheirus salmonis (Copepoda: Caligidae) Parasitology 2002 165 182

[23] G. F. Webb. Theory of nonlinear age-dependent population dynamics. Marcel Dekker, New York, 1985.

[24] K. Yosida. Functional analysis. Springer, Berlin, 1995.