# American Institute of Mathematical Sciences

2015, 12(2): 233-258. doi: 10.3934/mbe.2015.12.233

## Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model

 1 Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, United States 2 Department of Mathematics, Box 8205, North Carolina State University, Raleigh, NC 27695-8205

Received  April 2014 Revised  September 2014 Published  December 2014

We study a quasilinear hierarchically size-structured population model presented in [4]. In this model the growth, mortality and reproduction rates are assumed to depend on a function of the population density. In [4] we showed that solutions to this model can become singular (measure-valued) in finite time even if all the individual parameters are smooth. Therefore, in this paper we develop a first order finite difference scheme to compute these measure-valued solutions. Convergence analysis for this method is provided. We also develop a high resolution second order scheme to compute the measure-valued solution of the model and perform a comparative study between the two schemes.
Citation: Azmy S. Ackleh, Vinodh K. Chellamuthu, Kazufumi Ito. Finite difference approximations for measure-valued solutions of a hierarchically size-structured population model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 233-258. doi: 10.3934/mbe.2015.12.233
##### References:
 [1] A. S. Ackleh, L. J. S. Allen and J. Carter, Establishing a beachhead: A stochastic population model with an allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300. doi: 10.1016/j.tpb.2006.12.006. [2] A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical size structured model: Well-posedness and approximation, Appl. Math. Optim., 51 (2005), 35-59. doi: 10.1007/s00245-004-0806-2. [3] A. S. Ackleh and K. Ito, A finite difference scheme for the nonlinear-size structured population model, J. Num. Funct. Anal. Optimization, 18 (1997), 865-884. doi: 10.1080/01630569708816798. [4] A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Differential Equations, 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013. [5] K. W. Blayneh, Hierarchical size-structured population model, Dynamic Systems Appl., 9 (2000), 527-539. [6] A. Calsina and J. Saldana, Asymptotic behaviour of a model of hierarchically structured population, J. Math. Biol., 35 (1997), 967-987. doi: 10.1007/s002850050085. [7] J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differetial Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003. [8] J. A. Carrillo, P. Gwiazda and A. Ulikowska, Splitting-particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014), 2171-2197. doi: 10.1142/S0218202514500183. [9] J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. [10] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models, II Nonlinear theory, J. Math. Biol., 43 (2001), 157-189. doi: 10.1007/s002850170002. [11] P. Gwiazda, J. Jablonski, A. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Num. Meth. Partial Diff. Eq., 30 (2014), 1797-1820. doi: 10.1002/num.21879. [12] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010. [13] S. M. Henson and J. M. Cushing, Hierarchical models of intra-specific competition: Scramble versus contest, J. Math. Biol., 34 (1996), 755-772. [14] E. A. Kraev, Existence and uniqueness for height structured hierarchical populations models, Natural Resource Modeling, 14 (2001), 45-70. doi: 10.1111/j.1939-7445.2001.tb00050.x. [15] S. Kruskov, First-order quasilinear equations in several independent variables, Mat. Sb., 123 (1970), 228-255; English transl. in Math. USSR Sb., 10 (1970), 217-273. [16] J. Shen, C.-W. Shu and M. Zhang, High resolution schemes for a hierarchical size-structured model, SIAM J. Numer. Anal., 45 (2007), 352-370. doi: 10.1137/050638126. [17] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. doi: 10.1016/0021-9991(88)90177-5. [18] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

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##### References:
 [1] A. S. Ackleh, L. J. S. Allen and J. Carter, Establishing a beachhead: A stochastic population model with an allee effect applied to species invasion, Theor. Popul. Biol., 71 (2007), 290-300. doi: 10.1016/j.tpb.2006.12.006. [2] A. S. Ackleh, K. Deng and S. Hu, A quasilinear hierarchical size structured model: Well-posedness and approximation, Appl. Math. Optim., 51 (2005), 35-59. doi: 10.1007/s00245-004-0806-2. [3] A. S. Ackleh and K. Ito, A finite difference scheme for the nonlinear-size structured population model, J. Num. Funct. Anal. Optimization, 18 (1997), 865-884. doi: 10.1080/01630569708816798. [4] A. S. Ackleh and K. Ito, Measure-valued solutions for a hierarchically size-structured population, J. Differential Equations, 217 (2005), 431-455. doi: 10.1016/j.jde.2004.12.013. [5] K. W. Blayneh, Hierarchical size-structured population model, Dynamic Systems Appl., 9 (2000), 527-539. [6] A. Calsina and J. Saldana, Asymptotic behaviour of a model of hierarchically structured population, J. Math. Biol., 35 (1997), 967-987. doi: 10.1007/s002850050085. [7] J. A. Carrillo, R. M. Colombo, P. Gwiazda and A. Ulikowska, Structured populations, cell growth and measure valued balance laws, J. Differetial Equations, 252 (2012), 3245-3277. doi: 10.1016/j.jde.2011.11.003. [8] J. A. Carrillo, P. Gwiazda and A. Ulikowska, Splitting-particle methods for structured population models: Convergence and applications, Math. Models Methods Appl. Sci., 24 (2014), 2171-2197. doi: 10.1142/S0218202514500183. [9] J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. [10] O. Diekmann, M. Gyllenberg, H. Huang, M. Kirkilionis, J. A. J. Metz and H. R. Thieme, On the formulation and analysis of general deterministic structured population models, II Nonlinear theory, J. Math. Biol., 43 (2001), 157-189. doi: 10.1007/s002850170002. [11] P. Gwiazda, J. Jablonski, A. Marciniak-Czochra and A. Ulikowska, Analysis of particle methods for structured population models with nonlocal boundary term in the framework of bounded Lipschitz distance, Num. Meth. Partial Diff. Eq., 30 (2014), 1797-1820. doi: 10.1002/num.21879. [12] P. Gwiazda, T. Lorenz and A. Marciniak-Czochra, A nonlinear structured population model: Lipschitz continuity of measure-valued solutions with respect to model ingredients, J. Differential Equations, 248 (2010), 2703-2735. doi: 10.1016/j.jde.2010.02.010. [13] S. M. Henson and J. M. Cushing, Hierarchical models of intra-specific competition: Scramble versus contest, J. Math. Biol., 34 (1996), 755-772. [14] E. A. Kraev, Existence and uniqueness for height structured hierarchical populations models, Natural Resource Modeling, 14 (2001), 45-70. doi: 10.1111/j.1939-7445.2001.tb00050.x. [15] S. Kruskov, First-order quasilinear equations in several independent variables, Mat. Sb., 123 (1970), 228-255; English transl. in Math. USSR Sb., 10 (1970), 217-273. [16] J. Shen, C.-W. Shu and M. Zhang, High resolution schemes for a hierarchical size-structured model, SIAM J. Numer. Anal., 45 (2007), 352-370. doi: 10.1137/050638126. [17] C.-W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77 (1988), 439-471. doi: 10.1016/0021-9991(88)90177-5. [18] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.
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