# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022128
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## Diffusive size-structured population model with time-varying diffusion rate

 School of Basic Sciences, Indian Institute of Technology Mandi, Kamand (H.P.) - 175005, India

* Corresponding author: Syed Abbas

Received  December 2021 Revised  May 2022 Early access July 2022

This work is devoted to the study of a size-structured population model with a time-varying diffusion rate. Due to the seasonal variation, it is natural to consider the time-varying diffusion rate. Moreover, introducing the time-varying diffusion rate makes the model more challenging and requires the results of evolution operators for the analysis. Under some assumptions on the time-dependent diffusion coefficient, the existence and uniqueness of mild solution is shown. We apply the method of characteristics and evolution operators to derive our results. Positivity and boundedness of mild solution is also shown. Some examples are provided to illustrate the theoretical findings. We also solve our model numerically to study the size and spatial dynamics of population density.

Citation: Manoj Kumar, Syed Abbas. Diffusive size-structured population model with time-varying diffusion rate. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022128
##### References:
 [1] B. P. Ayati and T. F. Dupont, Galerkin methods in age and space for a population model with nonlinear diffusion, SIAM J. Num. Anal., 40 (2002), 1064-1076.  doi: 10.1137/S0036142900379679. [2] G. I. Bell and E. C. Anderson, Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351. [3] S. Bentout, A. Tridane, S. Djilali and T. M. Touaoula, Age-structured modeling of COVID-19 epidemic in the USA, UAE and Algeria, Alexandria Eng. J., 60 (2021), 401-411. [4] A. Calsina and J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377. [5] C. Cusulin and L. Gerardo-Giorda, A numerical method for spatial diffusion in age-structured populations, Numer. Methods Partial Differential Equations, 26 (2010), 253-273.  doi: 10.1002/num.20425. [6] E. B. Davies, One-Parameter Semigroups, volume 15 of London Mathematical Society Monographs. Academic Press, Inc., London-New York, 1980. [7] A. Ducrot, Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections, J. Differential Equations, 250 (2011), 410-449.  doi: 10.1016/j.jde.2010.09.019. [8] J. Z. Farkas, Stability conditions for a non-linear size-structured model, Nonlinear Anal. Real World Appl., 6 (2005), 962-969.  doi: 10.1016/j.nonrwa.2004.06.002. [9] X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 109-131.  doi: 10.3934/dcdsb.2013.18.109. [10] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985. [11] M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.  doi: 10.1007/BF00250793. [12] K. P. Hadeler, Structured populations with diffusion in state space, Math. Biosci. Eng., 7 (2010), 37-49.  doi: 10.3934/mbe.2010.7.37. [13] N. Kato, Abstract linear partial differential equations related to size-structured population models with diffusion, J. Math. Anal. Appl., 436 (2016), 890-910.  doi: 10.1016/j.jmaa.2015.11.077. [14] M. Kumar and S. Abbas, Age-structured SIR model for the spread of infectious diseases through indirect contacts, Mediterr. J. Math., 19 (2022), Paper No. 14, 18 pp. doi: 10.1007/s00009-021-01925-z. [15] M. Langlais, Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion, J. Math. Biol., 26 (1988), 319-346.  doi: 10.1007/BF00277394. [16] Y. Liu and Z.-R. He, On the well-posedness of a nonlinear hierarchical size-structured population model, ANZIAM J., 58 (2017), 482-490. doi: 10.1017/S1446181117000025. [17] F. A. Milner, A numerical method for a model of population dynamics with spatial diffusion, Comp. Math Appl., 19 (1990), 31-43.  doi: 10.1016/0898-1221(90)90135-7. [18] A. G. M'Kendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [20] F. R. Sharpe and A. J. Lotka, A problem in age-distribution, Philos. Mag., 21 (1911), 435-438. [21] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. [22] G. F. Webb, Population models structured by age, size, and spatial position, In Structured Population Models in Biology and Epidemiology, volume 1936 of Lecture Notes in Math., Springer, Berlin, (2008), 1–49. doi: 10.1007/978-3-540-78273-5_1. [23] Y. Weitsman, Diffusion with time-varying diffusivity, with application to moisture-sorption in composites, J. Comp. Mat., 10 (1976), 193-204.

show all references

##### References:
 [1] B. P. Ayati and T. F. Dupont, Galerkin methods in age and space for a population model with nonlinear diffusion, SIAM J. Num. Anal., 40 (2002), 1064-1076.  doi: 10.1137/S0036142900379679. [2] G. I. Bell and E. C. Anderson, Cell growth and division. I. A mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351. [3] S. Bentout, A. Tridane, S. Djilali and T. M. Touaoula, Age-structured modeling of COVID-19 epidemic in the USA, UAE and Algeria, Alexandria Eng. J., 60 (2021), 401-411. [4] A. Calsina and J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol., 33 (1995), 335-364.  doi: 10.1007/BF00176377. [5] C. Cusulin and L. Gerardo-Giorda, A numerical method for spatial diffusion in age-structured populations, Numer. Methods Partial Differential Equations, 26 (2010), 253-273.  doi: 10.1002/num.20425. [6] E. B. Davies, One-Parameter Semigroups, volume 15 of London Mathematical Society Monographs. Academic Press, Inc., London-New York, 1980. [7] A. Ducrot, Travelling waves for a size and space structured model in population dynamics: Point to sustained oscillating solution connections, J. Differential Equations, 250 (2011), 410-449.  doi: 10.1016/j.jde.2010.09.019. [8] J. Z. Farkas, Stability conditions for a non-linear size-structured model, Nonlinear Anal. Real World Appl., 6 (2005), 962-969.  doi: 10.1016/j.nonrwa.2004.06.002. [9] X. Fu and D. Zhu, Stability results for a size-structured population model with delayed birth process, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 109-131.  doi: 10.3934/dcdsb.2013.18.109. [10] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985. [11] M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Arch. Rational Mech. Anal., 54 (1974), 281-300.  doi: 10.1007/BF00250793. [12] K. P. Hadeler, Structured populations with diffusion in state space, Math. Biosci. Eng., 7 (2010), 37-49.  doi: 10.3934/mbe.2010.7.37. [13] N. Kato, Abstract linear partial differential equations related to size-structured population models with diffusion, J. Math. Anal. Appl., 436 (2016), 890-910.  doi: 10.1016/j.jmaa.2015.11.077. [14] M. Kumar and S. Abbas, Age-structured SIR model for the spread of infectious diseases through indirect contacts, Mediterr. J. Math., 19 (2022), Paper No. 14, 18 pp. doi: 10.1007/s00009-021-01925-z. [15] M. Langlais, Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion, J. Math. Biol., 26 (1988), 319-346.  doi: 10.1007/BF00277394. [16] Y. Liu and Z.-R. He, On the well-posedness of a nonlinear hierarchical size-structured population model, ANZIAM J., 58 (2017), 482-490. doi: 10.1017/S1446181117000025. [17] F. A. Milner, A numerical method for a model of population dynamics with spatial diffusion, Comp. Math Appl., 19 (1990), 31-43.  doi: 10.1016/0898-1221(90)90135-7. [18] A. G. M'Kendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130. [19] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [20] F. R. Sharpe and A. J. Lotka, A problem in age-distribution, Philos. Mag., 21 (1911), 435-438. [21] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918. [22] G. F. Webb, Population models structured by age, size, and spatial position, In Structured Population Models in Biology and Epidemiology, volume 1936 of Lecture Notes in Math., Springer, Berlin, (2008), 1–49. doi: 10.1007/978-3-540-78273-5_1. [23] Y. Weitsman, Diffusion with time-varying diffusivity, with application to moisture-sorption in composites, J. Comp. Mat., 10 (1976), 193-204.
Figure shows the behaviour of diffusion coefficient for different values of $\alpha$
Diffusion coefficient for different values of $b$
Figure depicts size-space profile of population density at fixed time $t = 0$
Size-space profile of population density at fixed time $t = 0.4$
Dynamics of population at fixed time $t = 0.8$
Size-space profile of population density at fixed time $t = 1$
Size-space evolution of population at fixed time $t = 0$
Evolution of population at fixed time $t = 0.4$
Dynamics of population at fixed time $t = 0.8$
Size-space evolution of population at fixed time $t = 1$
Evolution of population at fixed spatial position $x = 1$
Dynamics of population at fixed spatial position $x = 2$
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