# American Institute of Mathematical Sciences

December  2018, 23(10): 4087-4116. doi: 10.3934/dcdsb.2018127

## Time asymptotics of structured populations with diffusion and dynamic boundary conditions

 UMR 6626 Laboratoire de Mathématiques de Besançon, Université Bourgogne Franche-Comté, Besançon, 25000, France

Received  July 2017 Revised  November 2017 Published  April 2018

This work revisits and extends in various directions a work by J.Z. Farkas and P. Hinow (Math. Biosc and Eng, 8 (2011) 503-513) on structured populations models (with bounded sizes) with diffusion and generalized Wentzell boundary conditions. In particular, we provide first a self-contained $L^{1}$ generation theory making explicit the domain of the generator. By using Hopf maximum principle, we show that the semigroup is always irreducible regardless of the reproduction function. By using weak compactness arguments, we show first a stability result of the essential type and then deduce that the semigroup has a spectral gap and consequently the asynchronous exponential growth property. Finally, we show how to extend this theory to models with arbitrary sizes and point out an open problem pertaining to this extension.

Citation: Mustapha Mokhtar-Kharroubi, Quentin Richard. Time asymptotics of structured populations with diffusion and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4087-4116. doi: 10.3934/dcdsb.2018127
##### References:
 [1] Y. A. Abramovich and C. D. Aliprantis, Problems in Operator Theory, Graduate Studies in Mathematics, 51, American Mathematical Society, Providence, RI, 2002. doi: 10.1090/gsm/051. Google Scholar [2] T. M. Apostol, Mathematical Analysis, Second Edition, Reading, Addison-Wesley Publishing Co, 1974. Google Scholar [3] A. Bartlomiejczyk and H. Leszczyński, Method of lines for physiologically structured models with diffusion, Appl. Numer. Math., 94 (2015), 140-148. doi: 10.1016/j.apnum.2015.03.006. Google Scholar [4] A. Bartlomiejczyk and H. Leszczyński, Structured populations with diffusion and Feller conditions, Math. Biosci. Eng., 13 (2016), 261-279. doi: 10.3934/mbe.2015002. Google Scholar [5] A. Bobrowski, Convergence of One-Parameter Operator Semigroups, New Mathematical Monographs, 30, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316480663. Google Scholar [6] A. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equ., 12 (2012), 495-512. doi: 10.1007/s00028-012-0142-6. Google Scholar [7] A. Calsina and J. Z. Farkas, On a strain-structured epidemic model, Nonlinear Anal. Real World Appl., 31 (2016), 325-342. doi: 10.1016/j.nonrwa.2016.01.014. Google Scholar [8] P. Clément, H. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-Parameter Semigroups, vol. 5, North-Holland Publishing Co., Amsterdam, 1987. Google Scholar [9] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. Google Scholar [10] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 63, Springer-Verlag, 2000. doi: 10.1007/b97696. Google Scholar [11] J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Mathematical Biosciences and Engineering, 8 (2011), 503-513. doi: 10.3934/mbe.2011.8.503. Google Scholar [12] 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 [13] 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 [14] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Berlin, Germany, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar [15] K. P. Hadeler, Structured populations with diffusion in state space, Mathematical Biosciences and Engineering, 7 (2010), 37-49. doi: 10.3934/mbe.2010.7.37. Google Scholar [16] A. Kolmogorov, I. Petrovskii and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. Available from: https://biomath.usu.edu/files/2pd.pdf.Google Scholar [17] P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-78273-5. Google Scholar [18] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM Journal on Applied Mathematics, 19 (1970), 607-628. doi: 10.1137/0119060. Google Scholar [19] J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Springer Berlin Heidelberg, 68 (1986), 136-184. doi: 10.1007/978-3-662-13159-6_4. Google Scholar [20] M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory: New Aspects, vol. 46, World Scientific, 1997. doi: 10.1002/mma.497. Google Scholar [21] M. Mokhtar-Kharroubi, On the convex compactness property for the strong operator topology and related topics, Mathematical Methods in the Applied Sciences, 27 (2004), 687-701. doi: 10.1002/mma.497. Google Scholar [22] M. Mokhtar-Kharroubi, Spectral theory for neutron transport, in Evolutionary Equations with Applications in Natural Sciences (eds. J. Banasiak and M. Mokhtar-Kharroubi), Springer, 2126 (2015), 319-386. doi: 10.1007/978-3-319-11322-7_7. Google Scholar [23] J. D. Murray, Mathematical Biology, Biomathematics, Springer Verlag, Heiderberg, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar [24] R. Nagel, W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184, Springer-Verlag Berlin, 1986. doi: 10.1007/BFb0074922. Google Scholar [25] B. de Pagter, Irreducible compact operators, Mathematische Zeitschrift, 192 (1986), 149-153. doi: 10.1007/BF01162028. Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [27] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, 1984. doi: 10.1007/978-1-4612-5282-5. Google Scholar [28] Q. Richard, Work in progress.Google Scholar [29] G. Schlüchtermann, On weakly compact operators, Mathematische Annalen, 292 (1992), 263-266. doi: 10.1007/BF01444620. Google Scholar [30] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533. Google Scholar [31] J. G. Skellam, The formulation and interpretation of mathematical models of diffusionary processes in population biology, in The Mathematical Theory of the Dynamics of Biological Populations, New York, Academic press, 1973, 63-85.Google Scholar [32] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211. doi: 10.1137/080732870. Google Scholar [33] J. Voigt, On resolvent positive operators and positive $C_0$-semigroups on AL-spaces, Semigroup Forum, 38 (1989), 263-266. doi: 10.1007/BF02573236. Google Scholar [34] 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 [35] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Transactions of the American Mathematical Society, 303 (1987), 751-763. doi: 10.1090/S0002-9947-1987-0902796-7. Google Scholar [36] G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., 1936, Springer, Berlin, 2008, 1-49. doi: 10.1007/978-3-540-78273-5_1. Google Scholar [37] L. W. Weis, A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory, Journal of Mathematical Analysis and Application, 129 (1988), 6-23. doi: 10.1016/0022-247X(88)90230-2. Google Scholar [38] A. D. Wentzell, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177. doi: 10.1137/1104014. Google Scholar

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##### References:
 [1] Y. A. Abramovich and C. D. Aliprantis, Problems in Operator Theory, Graduate Studies in Mathematics, 51, American Mathematical Society, Providence, RI, 2002. doi: 10.1090/gsm/051. Google Scholar [2] T. M. Apostol, Mathematical Analysis, Second Edition, Reading, Addison-Wesley Publishing Co, 1974. Google Scholar [3] A. Bartlomiejczyk and H. Leszczyński, Method of lines for physiologically structured models with diffusion, Appl. Numer. Math., 94 (2015), 140-148. doi: 10.1016/j.apnum.2015.03.006. Google Scholar [4] A. Bartlomiejczyk and H. Leszczyński, Structured populations with diffusion and Feller conditions, Math. Biosci. Eng., 13 (2016), 261-279. doi: 10.3934/mbe.2015002. Google Scholar [5] A. Bobrowski, Convergence of One-Parameter Operator Semigroups, New Mathematical Monographs, 30, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316480663. Google Scholar [6] A. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equ., 12 (2012), 495-512. doi: 10.1007/s00028-012-0142-6. Google Scholar [7] A. Calsina and J. Z. Farkas, On a strain-structured epidemic model, Nonlinear Anal. Real World Appl., 31 (2016), 325-342. doi: 10.1016/j.nonrwa.2016.01.014. Google Scholar [8] P. Clément, H. Heijmans, S. Angenent, C. J. van Duijn and B. de Pagter, One-Parameter Semigroups, vol. 5, North-Holland Publishing Co., Amsterdam, 1987. Google Scholar [9] N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. Google Scholar [10] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 63, Springer-Verlag, 2000. doi: 10.1007/b97696. Google Scholar [11] J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Mathematical Biosciences and Engineering, 8 (2011), 503-513. doi: 10.3934/mbe.2011.8.503. Google Scholar [12] 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 [13] 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 [14] D. Gilbarg and N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Berlin, Germany, 1983. doi: 10.1007/978-3-642-61798-0. Google Scholar [15] K. P. Hadeler, Structured populations with diffusion in state space, Mathematical Biosciences and Engineering, 7 (2010), 37-49. doi: 10.3934/mbe.2010.7.37. Google Scholar [16] A. Kolmogorov, I. Petrovskii and N. Piscunov, A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem, Byul. Moskovskogo Gos. Univ., 1 (1937), 1-25. Available from: https://biomath.usu.edu/files/2pd.pdf.Google Scholar [17] P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Springer-Verlag Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-78273-5. Google Scholar [18] I. Marek, Frobenius theory of positive operators: Comparison theorems and applications, SIAM Journal on Applied Mathematics, 19 (1970), 607-628. doi: 10.1137/0119060. Google Scholar [19] J. A. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, Springer Berlin Heidelberg, 68 (1986), 136-184. doi: 10.1007/978-3-662-13159-6_4. Google Scholar [20] M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory: New Aspects, vol. 46, World Scientific, 1997. doi: 10.1002/mma.497. Google Scholar [21] M. Mokhtar-Kharroubi, On the convex compactness property for the strong operator topology and related topics, Mathematical Methods in the Applied Sciences, 27 (2004), 687-701. doi: 10.1002/mma.497. Google Scholar [22] M. Mokhtar-Kharroubi, Spectral theory for neutron transport, in Evolutionary Equations with Applications in Natural Sciences (eds. J. Banasiak and M. Mokhtar-Kharroubi), Springer, 2126 (2015), 319-386. doi: 10.1007/978-3-319-11322-7_7. Google Scholar [23] J. D. Murray, Mathematical Biology, Biomathematics, Springer Verlag, Heiderberg, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar [24] R. Nagel, W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, F. Neubrander and U. Schlotterbeck, One-Parameter Semigroups of Positive Operators, vol. 1184, Springer-Verlag Berlin, 1986. doi: 10.1007/BFb0074922. Google Scholar [25] B. de Pagter, Irreducible compact operators, Mathematische Zeitschrift, 192 (1986), 149-153. doi: 10.1007/BF01162028. Google Scholar [26] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Springer New York, 1983. doi: 10.1007/978-1-4612-5561-1. Google Scholar [27] M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, 1984. doi: 10.1007/978-1-4612-5282-5. Google Scholar [28] Q. Richard, Work in progress.Google Scholar [29] G. Schlüchtermann, On weakly compact operators, Mathematische Annalen, 292 (1992), 263-266. doi: 10.1007/BF01444620. Google Scholar [30] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. doi: 10.2307/1934533. Google Scholar [31] J. G. Skellam, The formulation and interpretation of mathematical models of diffusionary processes in population biology, in The Mathematical Theory of the Dynamics of Biological Populations, New York, Academic press, 1973, 63-85.Google Scholar [32] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM Journal on Applied Mathematics, 70 (2009), 188-211. doi: 10.1137/080732870. Google Scholar [33] J. Voigt, On resolvent positive operators and positive $C_0$-semigroups on AL-spaces, Semigroup Forum, 38 (1989), 263-266. doi: 10.1007/BF02573236. Google Scholar [34] 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 [35] G. F. Webb, An operator-theoretic formulation of asynchronous exponential growth, Transactions of the American Mathematical Society, 303 (1987), 751-763. doi: 10.1090/S0002-9947-1987-0902796-7. Google Scholar [36] G. F. Webb, Population models structured by age, size, and spatial position, in Structured Population Models in Biology and Epidemiology, Lecture Notes in Math., 1936, Springer, Berlin, 2008, 1-49. doi: 10.1007/978-3-540-78273-5_1. Google Scholar [37] L. W. Weis, A generalization of the Vidav-Jorgens perturbation theorem for semigroups and its application to transport theory, Journal of Mathematical Analysis and Application, 129 (1988), 6-23. doi: 10.1016/0022-247X(88)90230-2. Google Scholar [38] A. D. Wentzell, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177. doi: 10.1137/1104014. Google Scholar
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