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2016, 13(2): 261-279. doi: 10.3934/mbe.2015002

Structured populations with diffusion and Feller conditions

1. 

Faculty of Applied Physics and Mathematics, Gdańsk University of Technology, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland

2. 

Institute of Mathematics, University of Gdańsk, Wita Stwosza 57, 80-952 Gdańsk, Poland

Received  February 2015 Revised  September 2015 Published  November 2015

We prove a weak maximum principle for structured population models with dynamic boundary conditions. We establish existence and positivity of solutions of these models and investigate the asymptotic behaviour of solutions. In particular, we analyse so called size profile.
Citation: Agnieszka Bartłomiejczyk, Henryk Leszczyński. Structured populations with diffusion and Feller conditions. Mathematical Biosciences & Engineering, 2016, 13 (2) : 261-279. doi: 10.3934/mbe.2015002
References:
[1]

D. E. Apushkinskaya and N. I. Nazarov, A survey of results on nonlinear Wentzell problems, Appl. Math., 45 (2000), 69-80. doi: 10.1023/A:1022288717033.

[2]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhauser, 2014. doi: 10.1007/978-3-319-05140-6.

[3]

A. Bartłomiejczyk 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. Bartłomiejczyk and H. Leszczyński, Comparison principles for parabolic differential-functional initial-value problems, Nonlinear. Anal., 57 (2004), 63-84. doi: 10.1016/j.na.2003.11.005.

[5]

A. Bobrowski and K. Morawska, From a PDE model to an ODE model of dynamics of synaptic depression, Disc. Cont. Dyn. Sys. Series B, 17 (2012), 2313-2327. doi: 10.3934/dcdsb.2012.17.2313.

[6]

A. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equat., 12 (2012), 495-512. doi: 10.1007/s00028-012-0142-6.

[7]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.

[8]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.

[9]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. App., 328 (2007), 119-136. doi: 10.1016/j.jmaa.2006.05.032.

[10]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513. doi: 10.3934/mbe.2011.8.503.

[11]

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

[12]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J. 1964.

[13]

M. E. Gurtin and R. C. MacCamy, Diffusion models for age-structured populations, Math. Biosc., 54 (1981), 49-59. doi: 10.1016/0025-5564(81)90075-4.

[14]

K. P. Hadeler, Structured populations with diffusion in state space, Math. Biosci. Eng., 7 (2010), 37-49. doi: 10.3934/mbe.2010.7.37.

[15]

N. Kato, A general model of size-dependent population dynamics with nonlinear growth rate, J. Math. Anal. Appl., 297 (2004), 234-256. doi: 10.1016/j.jmaa.2004.05.004.

[16]

T. A. Kwembe and Z. Zhang, A semilinear equation with generalized Wentzell boundary condition, Non. Anal., 73 (2010), 3162-3170. doi: 10.1016/j.na.2010.06.068.

[17]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, (in Russian), Nauka, Moscow, 1967; (Translation of Mathematical Monographs, Vol. 23 Am. Math. Soc., Providence, R.I., 1968).

[18]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Vol.1936, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78273-5.

[19]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lect. Notes in Biomath. Vol. 68, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[20]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics series, Birkhäuser, Boston, 2007.

[21]

S. L. Tucker and S. O. Zimmermann, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables, SIAM J. Appl. Math., 48 (1988), 549-591. doi: 10.1137/0148032.

[22]

R. Waldstatter, 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.

[23]

W. Walter, Ordinary Differential Equations, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-1-4612-0601-9.

[24]

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

show all references

References:
[1]

D. E. Apushkinskaya and N. I. Nazarov, A survey of results on nonlinear Wentzell problems, Appl. Math., 45 (2000), 69-80. doi: 10.1023/A:1022288717033.

[2]

J. Banasiak and M. Lachowicz, Methods of Small Parameter in Mathematical Biology, Birkhauser, 2014. doi: 10.1007/978-3-319-05140-6.

[3]

A. Bartłomiejczyk 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. Bartłomiejczyk and H. Leszczyński, Comparison principles for parabolic differential-functional initial-value problems, Nonlinear. Anal., 57 (2004), 63-84. doi: 10.1016/j.na.2003.11.005.

[5]

A. Bobrowski and K. Morawska, From a PDE model to an ODE model of dynamics of synaptic depression, Disc. Cont. Dyn. Sys. Series B, 17 (2012), 2313-2327. doi: 10.3934/dcdsb.2012.17.2313.

[6]

A. Calsina and J. Z. Farkas, Steady states in a structured epidemic model with Wentzell boundary condition, J. Evol. Equat., 12 (2012), 495-512. doi: 10.1007/s00028-012-0142-6.

[7]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, 1998. doi: 10.1137/1.9781611970005.

[8]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.

[9]

J. Z. Farkas and T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. App., 328 (2007), 119-136. doi: 10.1016/j.jmaa.2006.05.032.

[10]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513. doi: 10.3934/mbe.2011.8.503.

[11]

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

[12]

A. Friedman, Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, N.J. 1964.

[13]

M. E. Gurtin and R. C. MacCamy, Diffusion models for age-structured populations, Math. Biosc., 54 (1981), 49-59. doi: 10.1016/0025-5564(81)90075-4.

[14]

K. P. Hadeler, Structured populations with diffusion in state space, Math. Biosci. Eng., 7 (2010), 37-49. doi: 10.3934/mbe.2010.7.37.

[15]

N. Kato, A general model of size-dependent population dynamics with nonlinear growth rate, J. Math. Anal. Appl., 297 (2004), 234-256. doi: 10.1016/j.jmaa.2004.05.004.

[16]

T. A. Kwembe and Z. Zhang, A semilinear equation with generalized Wentzell boundary condition, Non. Anal., 73 (2010), 3162-3170. doi: 10.1016/j.na.2010.06.068.

[17]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, Linear and Quasi-linear Equations of Parabolic Type, (in Russian), Nauka, Moscow, 1967; (Translation of Mathematical Monographs, Vol. 23 Am. Math. Soc., Providence, R.I., 1968).

[18]

P. Magal and S. Ruan, Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, Vol.1936, Springer-Verlag, Berlin, Heidelberg, 2008. doi: 10.1007/978-3-540-78273-5.

[19]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Lect. Notes in Biomath. Vol. 68, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.

[20]

B. Perthame, Transport Equations in Biology, Frontiers in Mathematics series, Birkhäuser, Boston, 2007.

[21]

S. L. Tucker and S. O. Zimmermann, A nonlinear model of population dynamics containing an arbitrary number of continuous structure variables, SIAM J. Appl. Math., 48 (1988), 549-591. doi: 10.1137/0148032.

[22]

R. Waldstatter, 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.

[23]

W. Walter, Ordinary Differential Equations, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-1-4612-0601-9.

[24]

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

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