# American Institute of Mathematical Sciences

2011, 8(2): 503-513. doi: 10.3934/mbe.2011.8.503

## Physiologically structured populations with diffusion and dynamic boundary conditions

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

Received  March 2010 Revised  December 2010 Published  April 2011

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.
Citation: József Z. Farkas, Peter Hinow. Physiologically structured populations with diffusion and dynamic boundary conditions. Mathematical Biosciences & Engineering, 2011, 8 (2) : 503-513. doi: 10.3934/mbe.2011.8.503
##### References:
 [1] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Elsevier/Academic Press, Amsterdam, Boston, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31. doi: 10.1090/S0002-9947-1954-0063607-6.  Google Scholar [15] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar [16] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. A. J. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations," Springer, Berlin, 1986.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] A. D. Ventcel, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177. doi: 10.1137/1104014.  Google Scholar [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.  Google Scholar [27] G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985.  Google Scholar

show all references

##### References:
 [1] R. A. Adams and J. J. F. Fournier, "Sobolev Spaces," 2nd edition, Elsevier/Academic Press, Amsterdam, Boston, 2003.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] K. J. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Springer, New York 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31. doi: 10.1090/S0002-9947-1954-0063607-6.  Google Scholar [15] D. Gilbarg and N. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983.  Google Scholar [16] G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] J. A. J. Metz and O. Diekmann, "The Dynamics of Physiologically Structured Populations," Springer, Berlin, 1986.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [25] A. D. Ventcel, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177. doi: 10.1137/1104014.  Google Scholar [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.  Google Scholar [27] G. F. Webb, "Theory of Nonlinear Age-Dependent Population Dynamics," Marcel Dekker, New York, 1985.  Google Scholar
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