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  December 2018 Early access  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 and 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.

[2]

T. M. Apostol, Mathematical Analysis, Second Edition, Reading, Addison-Wesley Publishing Co, 1974.

[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.

[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.

[5]

A. Bobrowski, Convergence of One-Parameter Operator Semigroups, New Mathematical Monographs, 30, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316480663.

[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.

[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.

[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.

[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.

[10]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 63, Springer-Verlag, 2000. doi: 10.1007/b97696.

[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.

[12]

A. FaviniG. R. GoldsteinJ. 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.

[13]

W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  doi: 10.1090/S0002-9947-1954-0063607-6.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory: New Aspects, vol. 46, World Scientific, 1997. doi: 10.1002/mma.497.

[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.

[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.

[23]

J. D. Murray, Mathematical Biology, Biomathematics, Springer Verlag, Heiderberg, 1989. doi: 10.1007/978-3-662-08539-4.

[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.

[25]

B. de Pagter, Irreducible compact operators, Mathematische Zeitschrift, 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[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.

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[28]

Q. Richard, Work in progress.

[29]

G. Schlüchtermann, On weakly compact operators, Mathematische Annalen, 292 (1992), 263-266.  doi: 10.1007/BF01444620.

[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.

[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.

[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.

[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.

[34]

R. WaldstätterK. 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.

[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.

[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.

[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.

[38]

A. D. Wentzell, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177.  doi: 10.1137/1104014.

show all references

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.

[2]

T. M. Apostol, Mathematical Analysis, Second Edition, Reading, Addison-Wesley Publishing Co, 1974.

[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.

[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.

[5]

A. Bobrowski, Convergence of One-Parameter Operator Semigroups, New Mathematical Monographs, 30, Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316480663.

[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.

[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.

[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.

[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.

[10]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, vol. 63, Springer-Verlag, 2000. doi: 10.1007/b97696.

[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.

[12]

A. FaviniG. R. GoldsteinJ. 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.

[13]

W. Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc., 77 (1954), 1-31.  doi: 10.1090/S0002-9947-1954-0063607-6.

[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.

[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.

[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.

[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.

[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.

[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.

[20]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory: New Aspects, vol. 46, World Scientific, 1997. doi: 10.1002/mma.497.

[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.

[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.

[23]

J. D. Murray, Mathematical Biology, Biomathematics, Springer Verlag, Heiderberg, 1989. doi: 10.1007/978-3-662-08539-4.

[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.

[25]

B. de Pagter, Irreducible compact operators, Mathematische Zeitschrift, 192 (1986), 149-153.  doi: 10.1007/BF01162028.

[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.

[27]

M. H. Protter and H. F. Weinberger, Maximum Principles in Differential Equations, Springer Verlag, 1984. doi: 10.1007/978-1-4612-5282-5.

[28]

Q. Richard, Work in progress.

[29]

G. Schlüchtermann, On weakly compact operators, Mathematische Annalen, 292 (1992), 263-266.  doi: 10.1007/BF01444620.

[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.

[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.

[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.

[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.

[34]

R. WaldstätterK. 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.

[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.

[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.

[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.

[38]

A. D. Wentzell, On boundary conditions for multi-dimensional diffusion processes, Theor. Probability Appl., 4 (1959), 164-177.  doi: 10.1137/1104014.

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