×

Straightened characteristics of McKendrick-von Foerster equation. (English) Zbl 1500.35205

Summary: We study the McKendrick-von Foerster equation with renewal (that is the age-structured model, with total population dependent coefficient and nonlinearity). By using a change of variables, the model is then transformed to a standard age-structured model in which the total population dependent coefficient of the transport term reduces to a constant 1. We use this transformation to get existence, uniqueness of solutions of the problem in a semigroup setting. Since straight lines are more convenient in the exact and approximate solution of PDEs, we provide sufficient conditions of reducing more general equations. We give a difference scheme to find approximate solutions of the age-structured model. Finally, some numerical simulations are presented to demonstrate the convergence and stability of the difference scheme.

MSC:

35L50 Initial-boundary value problems for first-order hyperbolic systems
35R09 Integro-partial differential equations
92D25 Population dynamics (general)
47D06 One-parameter semigroups and linear evolution equations
49M15 Newton-type methods
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Abia, L. M.; Angulob, O.; López-Marcos, J. C.; López-Marcos, M. A., Numerical integration of a hierarchically size-structured population model with contest competition, J. Comput. Appl. Math., 258, 116-134 (2014) · Zbl 1323.92152
[2] Ackleh, A. S.; Deng, K., A monotone approximation for the nonautonomous size-structured population model, Q. Appl. Math., 57, 2, 261-267 (1999) · Zbl 1157.35491
[3] Ackleh, A. S.; Deng, K.; Wang, X., Competitive exclusion and coexistence for a quasilinear size-structured population model, Math. Biosci., 192, 177-192 (2004) · Zbl 1072.92046
[4] Ackleh, A. S.; Farkas, J. Z.; Li, X.; Ma, B., Finite difference approximations for a size-structured population model with distributed states in the recruitment, J. Biol. Dyn., 9, 2-31 (2015) · Zbl 1448.92159
[5] Ackleh, A. S.; Lyons, R.; Saintier, N., Finite difference schemes for a structured population model in the space of measures, Math. Biosci. Eng., 17, 1, 747-775 (2019) · Zbl 1470.92230
[6] Angulo, O.; Lopez-Marcos, J. C.; Milner, F. A., The application of an age-structured model with unbounded mortality to demography, Math. Biosci., 208, 495-520 (2007) · Zbl 1119.92049
[7] Bartłomiejczyk, A.; Leszczyński, H., Method of lines for physiologically structured models with diffusion, Appl. Numer. Math., 94, 140-148 (2015) · Zbl 1325.65132
[8] Bartłomiejczyk, A.; Leszczyński, H., Structured populations with diffusion and Feller conditions, Math. Biosci. Eng., 13, 2, 261-279 (2016) · Zbl 1329.35315
[9] Bartłomiejczyk, A.; Leszczyński, H.; Marciniak, A., Rothe’s method for physiologically structured models with diffusion, Math. Slovaca, 68, 211-224 (2018) · Zbl 1473.65156
[10] Bartłomiejczyk, A.; Leszczyński, H.; Zwierkowski, P., Existence and uniqueness of solutions for single-population McKendrick-von Foerster models with renewal, Rocky Mt. J. Math., 45, 401-426 (2015) · Zbl 1330.35463
[11] Bartłomiejczyk, A.; Wrzosek, M., Newton’s method for the McKendrick-von Foerster equation, (Banasiak, J.; Bobrowski, A.; Lachowicz, M.; Tomilov, Y., Semigroups of Operators - Theory and Applications. Semigroups of Operators - Theory and Applications, SOTA 2018. Semigroups of Operators - Theory and Applications. Semigroups of Operators - Theory and Applications, SOTA 2018, Springer Proceedings in Mathematics & Statistics, vol. 325 (2020), Springer: Springer Cham), 137-146 · Zbl 1494.92094
[12] Bernardelli, H., Population waves, J. Burma Res. Soc., 31, 1-18 (1941)
[13] Brokate, M., Pontryagin’s principle for control problems in age-dependent population dynamics, J. Math. Biol., 23, 75-101 (1985) · Zbl 0599.92017
[14] Calsina, A.; Saldana, J., A model of physiological structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33, 335-364 (1995) · Zbl 0828.92025
[15] Carrillo, J. A.; Colombo, R. M.; Gwiazda, P.; Ulikowska, A., Structured populations, cell growth and measure valued balance laws, J. Differ. Equ., 252, 3245-3277 (2012) · Zbl 1250.47080
[16] Carrillo, J. A.; Gwiazda, P.; Ulikowska, A., Splitting-particle methods for structured population models: convergence and applications, Math. Models Methods Appl. Sci., 24, 11, 2171-2197 (2014) · Zbl 1300.92072
[17] Cushing, J., An Introduction to Structured Population Dynamics (1998), SIAM: SIAM Philadelphia · Zbl 0939.92026
[18] Farkas, J. Z.; Gwiazda, P.; Marciniak-Czochra, A., Asymptotic behaviour of a structured population model on a space of measures
[19] Feichtinger, G.; Prskawetz, A.; Veliov, V. M., Age-structured optimal control in population economics, Theor. Popul. Biol., 65, 373-387 (2004) · Zbl 1110.92035
[20] Greenhalgh, D.; Györi, I.; Kovácsvölgyi, I., An age-dependent population dynamical model with a small parameter, Math. Comput. Model., 31, 63-72 (2000) · Zbl 1043.92521
[21] Gurtin, M. E.; McCamy, R. C., Non-linear age-dependent population dynamics, Arch. Ration. Mech. Anal., 54, 281-300 (1974) · Zbl 0286.92005
[22] Gwiazda, P.; Jabłoński, J.; Marciniak-Czochra, A.; Ulikowska, A., Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Numer. Methods Partial Differ. Equ., 30, 6, 1797-1820 (2014) · Zbl 1314.92128
[23] Inaba, H., Age-Structured Population Dynamics in Demography and Epidemiology (2017), Springer: Springer New York · Zbl 1370.92010
[24] Kakumani, B. K.; Tumuluri, S. K., A numerical scheme to the McKendrick-von Foerster equation with diffusion in age, Numer. Methods Partial Differ. Equ., 34, 6, 1-16 (2018) · Zbl 1407.65109
[25] Lasota, A.; Mackey, M. C.; Ważewska-Czyżewska, M., Minimizing therapeutically induced anemia, J. Math. Biol., 13, 149-158 (1981) · Zbl 0473.92003
[26] Leslie, P. H., The use of matrices in certain population mathematics, Biom. J., 33, 183-212 (1945) · Zbl 0060.31803
[27] Leszczyński, H.; Zwierkowski, P., Iterative methods for generalized von Foerster equations with functional dependences, J. Inequal. Appl., 14 (2007) · Zbl 1132.65063
[28] Lewis, E. G., On the generation and growth of a population, Sankhya, 6, 93-96 (1942)
[29] Magal, P.; Ruan, S., Theory and Applications of Abstract Semilinear Cauchy Problems (2018), Springer: Springer New York · Zbl 1447.34002
[30] McKendrick, A., Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44, 98-130 (1926) · JFM 52.0542.04
[31] (Metz, J. J.A. J.; Diekmann, O., The Dynamics of Physiologically Structured Populations. The Dynamics of Physiologically Structured Populations, Lecture Notes Biomath., vol. 68 (1986), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0614.92014
[32] Murray, J. D., Mathematical Biology. I. An Introduction (2002), Springer-Verlag: Springer-Verlag Berlin, Heidelberg
[33] Politikos, D. V.; Tzanetis, D. E.; Nikolopoulos, C. V.; Maravelias, C. D., The application of an age-structured model to the north Aegean anchovy fishery: an evaluation of different management measures, Math. Biosci., 237, 17-27 (2012) · Zbl 1241.92076
[34] Qin, Y., Integral and Discrete Inequalities and Their Applications, Volume I: Linear Inequalities (2016), Birkhäuser · Zbl 1359.26004
[35] von Foerster, H., Some remarks on changing populations, (Stohlman, F., The Kinetics of Cell Proliferation, Grune and Stratton. The Kinetics of Cell Proliferation, Grune and Stratton, New York (1959)), 382-407
[36] Webb, G. F., Population models structured by age, size, and spatial position, (Magal, P.; Ruan, S., Structured Population Models in Biology and Epidemiology. Structured Population Models in Biology and Epidemiology, Lecture Notes in Mathematics, vol. 1936 (2008), Springer: Springer Berlin, Heidelberg), 1-49 · Zbl 1138.92029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.