**Export file:**

**Format**

- RIS(for EndNote,Reference Manager,ProCite)
- BibTex
- Text

**Content**

- Citation Only
- Citation and Abstract

A numerical framework for computing steady states of structured population models and their stability

^{setArticleTag('','','','');},^{setArticleTag('','','1','dmbortz@colorado.edu');}

. Department of Applied Mathematics, University of Colorado, Boulder, CO, 80309-0526, United States

Received: , Accepted: , Published:

Structured population models are a class of general evolution equations which are widely used in the study of biological systems. Many theoretical methods are available for establishing existence and stability of steady states of general evolution equations. However, except for very special cases, finding an analytical form of stationary solutions for evolution equations is a challenging task. In the present paper, we develop a numerical framework for computing approximations to stationary solutions of general evolution equations, which can \emph{also} be used to produce approximate existence and stability regions for steady states. In particular, we use the Trotter-Kato Theorem to approximate the infinitesimal generator of an evolution equation on a finite dimensional space, which in turn reduces the evolution equation into a system of ordinary differential equations. Consequently, we approximate and study the asymptotic behavior of stationary solutions. We illustrate the convergence of our numerical framework by applying it to a linear Sinko-Streifer structured population model for which the exact form of the steady state is known. To further illustrate the utility of our approach, we apply our framework to nonlinear population balance equation, which is an extension of well-known Smoluchowski coagulation-fragmentation model to biological populations. We also demonstrate that our numerical framework can be used to gain insight about the theoretical stability of the stationary solutions of the evolution equations. Furthermore, the open source Python program that we have developed for our numerical simulations is freely available from our GitHub repository (*github.com*/*MathBioCU*).

# References

[1] A.S. Ackleh, Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal. Theory Methods Appl., 28 (1997): 837-854.

[2] A.S. Ackleh,B.G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, J. Math. Biol., 35 (1997): 480-502.

[3] V. I. Arnold, *Ordinary Differential Equations*, Second printing of the 1992 edition. Universitext. Springer-Verlag, Berlin, 2006.

[4] J. Banasiak, Blow-up of solutions to some coagulation and fragmentation equations with growth, Dyn. Syst., 1 (2011): 126-134.

[5] J. Banasiak,W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Syst. -Ser. B, 11 (2009): 563-585.

[6] H.T. Banks,F. Kappel, Transformation semigroups andL 1-approximation for size structured population models, Semigroup Forum, 38 (1989): 141-155.

[7] C. Biggs,P. Lant, Modelling activated sludge flocculation using population balances, Powder Technol., 124 (2002): 201-211.

[8] D. M. Bortz, Chapter 17: Modeling and simulation for nanomaterials in fluids: Nanoparticle self-assembly. In Tewary, V. and Zhang, Y., editors, *Modeling, characterization, and
production of nanomaterials: Electronics, Photonics and Energy Applications*, volume 73 of
*Woodhead Publishing Series in Electronic and Optical Materials*, (2015), 419–441. Woodhead
Publishing Ltd., Cambridge, UK.

[9] D.M. Bortz,T.L. Jackson,K.A. Taylor,A.P. Thompson,J.G. Younger, Klebsiella pneumoniae Flocculation Dynamics, Bull. Math. Biol., 70 (2008): 745-768.

[10] D. Breda, Solution operator approximations for characteristic roots of delay differential equations, Applied Numerical Mathematics, 56 (2006): 305-317.

[11] D. Breda,O. Diekmann,S. Maset,R. Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7 (2013): 4-20.

[12] D. Breda,S. Maset,R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J Numer Anal, 24 (2004): 1-19.

[13] D. Breda,S. Maset,R. Vermiglio, Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations, SIAM J. Sci. Comput., 27 (2005): 482-495.

[14] D. Breda,S. Maset,R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Applied Numerical Mathematics, 56 (2006): 318-331.

[15] D. Breda,S. Maset,R. Vermiglio, Numerical approximation of characteristic values of partial retarded functional differential equations, Numer. Math., 113 (2009): 181-242.

[16] E. Byrne, S. Dzul, M. Solomon, J. Younger and D. M. Bortz, Postfragmentation density function for bacterial aggregates in laminar flow, *Phys*. *Rev*. E, 83 (2011).

[17] V. Calvez,M. Doumic,P. Gabriel, Self-similarity in a general aggregation-fragmentation problem, Application to fitness analysis, Journal de Math{é}matiques Pures et Appliqu{é}es, 98 (2012): 1-27.

[18] V. Calvez,N. Lenuzza,M. Doumic,J.-P. Deslys,F. Mouthon,B. Perthame, Prion dynamics with size dependency-strain phenomena, J. Biol. Dyn., 4 (2010): 28-42.

[19] A.M. De Roos, Demographic analysis of continuous-time life-history models, Ecol. Lett., 11 (2008): 1-15.

[20] A. M. De Roos, PSPManalysis, 2014, https://staff.fnwi.uva.nl/a.m.deroos/PSPManalysis/index.html.

[21] A.M. de Roos,O. Diekmann,P. Getto,M.A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, Bulletin of Mathematical Biology, 72 (2010): 259-297.

[22] O. Diekmann,M. Gyllenberg,J. A.J. Metz, Steady-state analysis of structured population models, Theoretical Population Biology, 63 (2003): 309-338.

[23] C.A. Dorao,H.A. Jakobsen, Application of the least-squares method for solving population balance problems in Rd+1, Chemical Engineering Science, 61 (2006a): 5070-5081.

[24] C.A. Dorao,H.A. Jakobsen, A least squares method for the solution of population balance problems, Computers & Chemical Engineering, 30 (2006b): 535-547.

[25] C.A. Dorao,H.A. Jakobsen, Least-squares spectral method for solving advective population balance problems, Journal of Computational and Applied Mathematics, 201 (2007): 247-257.

[26] M. Doumic-Jauffret and P. Gabriel, Eigenelements of a general aggregation-fragmentation model, arXiv: 0907.5467, 2009.

[27] K. Engelborghs,T. Luzyanina,D. Roose, Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL, ACM Trans Math Softw, 28 (2002): 1-21.

[28] J.Z. Farkas,T. Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl., 328 (2007): 119-136.

[29] J.Z. Farkas,P. Hinow, Steady states in hierarchical structured populations with distributed states at birth, Discrete Contin. Dyn. Syst. -Ser. B, 17 (2012): 2671-2689.

[30] S. A. Gourley, R. Liu and J. Wu, Spatiotemporal Patterns of Disease Spread: Interaction of Physiological Structure, Spatial Movements, Disease Progression and Human Intervention, In Magal, P. and Ruan, S., editors, *Structured Population Models in Biology and Epidemiology*, number 1936 in Lecture Notes in Mathematics, (2008), 165–208. Springer Berlin Heidelberg.

[31] M.J. Hounslow, A discretized population balance for continuous systems at steady state, AIChE J., 36 (1990): 106-116.

[32] K. Ito,F. Kappel, The Trotter-Kato theorem and approximation of PDEs, Math Comp, 67 (1998): 21-44.

[33] F. Kappel,K. Kunisch, Spline Approximations for Neutral Functional Differential Equations, SIAM J. Numer. Anal., 18 (1981): 1058-1080.

[34] N. Kato, A Principle of Linearized Stability for Nonlinear Evolution Equations, Transactions of the American Mathematical Society, 347 (1995): 2851-2868.

[35] T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad., 35 (1959): 467-468.

[36] T. Kato, *Perturbation Theory for Linear Operators*, Classics in Mathematics. Springer Berlin Heidelberg, Berlin, Heidelberg, 1976.

[37] M.A. Kirkilionis,O. Diekmann,B. Lisser,M. Nool,B. Sommeijer,A.M. De Roos, Numerical continuation of equilibria of physiologically structured population models Ⅰ: Theory, Math. Models Methods Appl. Sci., 11 (2001): 1101-1127.

[38] P. Laurencot,C. Walker, Steady states for a coagulation-fragmentation equation with volume scattering, SIAM Journal on Mathematical Analysis, 37 (2005): 531-548.

[39] J. Makino,T. Fukushige,Y. Funato,E. Kokubo, On the mass distribution of planetesimals in the early runaway stage, New Astronomy, 3 (1998): 411-417.

[40] S.A. Matveev,A.P. Smirnov,E.E. Tyrtyshnikov, A fast numerical method for the Cauchy problem for the Smoluchowski equation, Journal of Computational Physics, 282 (2015): 23-32.

[41] G. Menon,R.L. Pego, Dynamical scaling in Smoluchowski's coagulation equations: Uniform convergence, SIAM Rev., 48 (2006): 745-768.

[42] I. Mirzaev, Steady state approximation, 2015. https://github.com/MathBioCU/SteadyStateApproximation.

[43] I. Mirzaev and D. M. Bortz, Criteria for linearized stability for a size-structured population model, arXiv: 1502.02754, 2015.

[44] I. Mirzaev and D. M. Bortz, Stability of steady states for a class of flocculation equations with growth and removal, arXiv: 1507.07127, 2015 (submitted).

[45] M. Nicmanis,M.J. Hounslow, Finite-element methods for steady-state population balance equations, AIChE J., 44 (1998): 2258-2272.

[46] M. Nicmanis,M.J. Hounslow, Error estimation and control for the steady state population balance equation: 1. An a posteriori error estimate, Chemical Engineering Science, 57 (2002): 2253-2264.

[47] H.-S. Niwa, School size statistics of fish, J. Theor. Biol., 195 (1998): 351-361.

[48] M. Powell, A hybrid method for nonlinear equations, Numerical Methods for Nonlinear Algebraic Equations, null (1970): 87-114, {Gordon & Breach}.

[49] H. R. Pruppacher and J. D. Klett, *Microphysics of Clouds and Precipitation: Reprinted 1980*, Springer Science & Business Media, 2012.

[50] D. Ramkrishna, *Population Balances*: *Theory and Applications to Particulate Systems in Engineering*, *Academic Press*, 2000.

[51] S.J. Schreiber,M.E. Ryan, Invasion speeds for structured populations in fluctuating environments, Theor. Ecol., 4 (2011): 423-434.

[52] J.W. Sinko,W. Streifer, A New Model For Age-Size Structure of a Population, Ecology, 48 (1967): 910-918.

[53] H.F. Trotter, Approximation of semi-groups of operators, Pacific J. Math., 8 (1958): 887-919.

[54] J.A. Wattis, An introduction to mathematical models of coagulation-fragmentation processes: A discrete deterministic mean-field approach, Phys. Nonlinear Phenom., 222 (2006): 1-20.

[55] G. F. Webb, *Theory of Nonlinear Age-Dependent Population Dynamics*, CRC Press, 1985.

[56] R.M. Ziff,G. Stell, Kinetics of polymer gelation, J. Chem. Phys., 73 (1980): 3492-3499.

Copyright Info: © 2017, David M. Bortz, et al., licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)