Physiologically structured populations with diffusion and dynamicboundary conditions

József Z. Farkas; Peter Hinow

doi:10.3934/mbe.2011.8.503

Article Contents

2011, Volume 8, Issue 2: 503-513. Doi: 10.3934/mbe.2011.8.503

This issue Previous Article Bacteria--phagocyte dynamics, axiomatic modelling and mass-action kinetics Next Article Adaptive response and enlargement of dynamic range

Physiologically structured populations with diffusion and dynamic boundary conditions

József Z. Farkas^1, and
Peter Hinow^2,

1.
Department of Computing Science and Mathematics, University of Stirling, Stirling, FK9 4LA
2.
Department of Mathematical Sciences, University of Wisconsin – Milwaukee, P.O. Box 413, Milwaukee, WI 53201-0413

Received: February 28, 2010

Revised: November 30, 2010

Published: March 2011

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Abstract

We consider a linear size-structured population model with diffusion in the size-space. Individuals are recruited into the population at arbitrary sizes. We equip the model with generalized Wentzell-Robin (or dynamic) boundary conditions. This approach allows the modelling of populations in which individuals may have distinguished physiological states. We establish existence and positivity of solutions by showing that solutions are governed by a positive quasicontractive semigroup of linear operators on the biologically relevant state space. These results are obtained by establishing dissipativity of a suitably perturbed semigroup generator. We also show that solutions of the model exhibit balanced exponential growth, that is, our model admits a finite-dimensional global attractor. In case of strictly positive fertility we are able to establish that solutions in fact exhibit asynchronous exponential growth.

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Mathematics Subject Classification: 92D25, 47N60, 47D06, 35B35.

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References

[1]	R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Elsevier/Academic Press, Amsterdam, Boston, 2003.
[2]	W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, "One-Parameter Semigroups of Positive Operators," Springer-Verlag, Berlin, 1986.
[3]	À. Calsina and 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), 1695-1722.doi: 10.1142/S0218202506001686.
[4]	Ph. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, "One-Parameter Semigroups," North-Holland, Amsterdam 1987.
[5]	J. Dyson, R. Villella-Bressan and G. F. Webb, Asynchronous exponential growth in an age structured population of proliferating and quiescent cells, Math. Biosci., 177&178 (2002), 73-83.
[6]	K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York 2000.
[7]	J. Z. Farkas, D. Green and P. Hinow, Semigroup analysis of structured parasite populations, Math. Model. Nat. Phenom., 5 (2010), 94-114, arXiv:0812.1363.
[8]	J. Z. Farkas and T. Hagen, Asymptotic analysis of a size-structured cannibalism model with infinite dimensional environmental feedback, Commun. Pure Appl. Anal., 8 (2009), 1825-1839, arXiv:0812.1369.
[9]	J. Z. Farkas and T. Hagen, Hierarchical size-structured populations: The linearized semigroup approach, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 17 (2010), 639-657, arXiv:0812.1367.
[10]	J. Z. Farkas and P. Hinow, On a size-structured two-phase population model with infinite states-at-birth, Positivity, 14 (2010), 501-514, arXiv:0903.1649.
[11]	A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, $C_0$-semigroups generated by second order differential operators with general Wentzell boundary conditions, Proc. Amer. Math. Soc., 128 (2000), 1981-1989.doi: 10.1090/S0002-9939-00-05486-1.
[12]	A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.doi: 10.1007/s00028-002-8077-y.
[13]	W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math., 55 (1952), 468-519.doi: 10.2307/1969644.
[14]	W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.doi: 10.1090/S0002-9947-1954-0063607-6.
[15]	D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.
[16]	G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.
[17]	M. Gyllenberg and G. F. Webb, Asynchronous exponential growth of semigroups of nonlinear operators, J. Math. Anal. Appl., 167 (1992), 443-467.doi: 10.1016/0022-247X(92)90218-3.
[18]	K. P. Hadeler, Structured populations with diffusion in state space, Math. Biosci. Eng., 7 (2010), 37-49.doi: 10.3934/mbe.2010.7.37.
[19]	R. Haller-Dintelmann, M. Hieber and J. Rehberg, Irreducibility and mixed boundary conditions, Positivity, 12 (2008), 83-91.doi: 10.1007/s11117-007-2131-5.
[20]	M. Langlais and F. A. Milner, Existence and uniqueness of solutions for a diffusion model of host-parasite dynamics, J. Math. Anal. Appl., 279 (2003), 463-474.doi: 10.1016/S0022-247X(03)00020-9.
[21]	J. A. J. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations," Springer, Berlin, 1986.
[22]	F. A. Milner and C. A. Patton, A diffusion model for host-parasite interaction, J. Comput. Appl. Math., 154 (2003), 273-302.doi: 10.1016/S0377-0427(02)00826-9.
[23]	J. L. Vazquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive type, Comm. Partial Differential Equations, 33 (2008), 561-612.doi: 10.1080/03605300801970960.
[24]	A. D. Ventcel, Semigroups of operators that correspond to a generalized differential operator of second order (Russian), Dokl. Akad. Nauk SSSR (N.S.), 111 (1956), 269-272.
[25]	A. D. Ventcel, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177.doi: 10.1137/1104014.
[26]	R. Waldstätter, K. P. Hadeler and G. Greiner, A Lotka-McKendrick model for a population structured by the level of parasitic infection, SIAM J. Math. Anal., 19 (1988), 1108-1118.doi: 10.1137/0519075.
[27]	G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985.