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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  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, 11 (2009), No. 3, 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., 16 (2006), No. 10, 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., 22 (2006), No. 10, 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., 39 (2007), No. 4, 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., 328 (2007), No. 1, 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., 8 (2009), No. 6, 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., 45 (2001), No. 2, 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. Japan27 (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, 53 (2006), No. 1-4, 128–135. 
  19. J. Prüß. Stability analysis for equilibria in age-specific population dynamics. Nonlin. Anal. TMA7 (1983), No. 12, 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.28 (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.37 (2002), No. 3-4, 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, 124 (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. 

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