American Institute of Mathematical Sciences

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August  2017, 14(4): 933-952. doi: 10.3934/mbe.2017049

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

Inom Mirzaev and  David M. Bortz ,

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

* Corresponding author: dmbortz@colorado.edu

Received  January 2016 Accepted  January 2016 Published  February 2017

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).

Keywords: Stationary solutions, numerical stability analysis, nonlinear evolution, equations, population balance equations, size-structured population model, Trotter-Kato theorem.
Mathematics Subject Classification: Primary: 35B40, 92B05; Secondary: 65N40.
Citation: Inom Mirzaev, David M. Bortz. A numerical framework for computing steady states of structured population models and their stability. Mathematical Biosciences & Engineering, 2017, 14 (4) : 933-952. doi: 10.3934/mbe.2017049
References:
[1]

A.S. Ackleh, Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal. Theory Methods Appl., 28 (1997), 837-854.  doi: 10.1016/0362-546X(95)00195-2.

[2]

A.S. Ackleh and B.G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, J. Math. Biol., 35 (1997), 480-502.  doi: 10.1007/s002850050062.

[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 and W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Syst. -Ser. B, 11 (2009), 563-585.  doi: 10.3934/dcdsb.2009.11.563.

[6]

H.T. Banks and F. Kappel, Transformation semigroups andL 1-approximation for size structured population models, Semigroup Forum, 38 (1989), 141-155.  doi: 10.1007/BF02573227.

[7]

C. Biggs and 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 and J.G. Younger, Klebsiella pneumoniae Flocculation Dynamics, Bull. Math. Biol., 70 (2008), 745-768.  doi: 10.1007/s11538-007-9277-y.

[10]

D. Breda, Solution operator approximations for characteristic roots of delay differential equations, Applied Numerical Mathematics, 56 (2006), 305-317.  doi: 10.1016/j.apnum.2005.04.010.

[11]

D. Breda, O. Diekmann, S. Maset and R. Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7 (2013), 4-20.  doi: 10.1080/17513758.2013.789562.

[12]

D. Breda, S. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J Numer Anal, 24 (2004), 1-19.  doi: 10.1093/imanum/24.1.1.

[13]

D. Breda, S. Maset and R. Vermiglio, Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations, SIAM J. Sci. Comput., 27 (2005), 482-495.  doi: 10.1137/030601600.

[14]

D. Breda, S. Maset and R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Applied Numerical Mathematics, 56 (2006), 318-331.  doi: 10.1016/j.apnum.2005.04.011.

[15]

D. Breda, S. Maset and R. Vermiglio, Numerical approximation of characteristic values of partial retarded functional differential equations, Numer. Math., 113 (2009), 181-242.  doi: 10.1007/s00211-009-0233-7.

[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 and 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.  doi: 10.1016/j.matpur.2012.01.004.

[18]

V. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame, Prion dynamics with size dependency-strain phenomena, J. Biol. Dyn., 4 (2010), 28-42.  doi: 10.1080/17513750902935208.

[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 and M.A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, Bulletin of Mathematical Biology, 72 (2010), 259-297.  doi: 10.1007/s11538-009-9445-3.

[22]

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

[23]

C.A. Dorao and 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 and 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 and H.A. Jakobsen, Least-squares spectral method for solving advective population balance problems, Journal of Computational and Applied Mathematics, 201 (2007), 247-257.  doi: 10.1016/j.cam.2006.02.020.

[26]

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

[27]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL, ACM Trans Math Softw, 28 (2002), 1-21.  doi: 10.1145/513001.513002.

[28]

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

[29]

J.Z. Farkas and P. Hinow, Steady states in hierarchical structured populations with distributed states at birth, Discrete Contin. Dyn. Syst. -Ser. B, 17 (2012), 2671-2689.  doi: 10.3934/dcdsb.2012.17.2671.

[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. doi: 10.1007/978-3-540-78273-5_4.

[31]

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

[32]

K. Ito and F. Kappel, The Trotter-Kato theorem and approximation of PDEs, Math Comp, 67 (1998), 21-44.  doi: 10.1090/S0025-5718-98-00915-6.

[33]

F. Kappel and K. Kunisch, Spline Approximations for Neutral Functional Differential Equations, SIAM J. Numer. Anal., 18 (1981), 1058-1080.  doi: 10.1137/0718072.

[34]

N. Kato, A Principle of Linearized Stability for Nonlinear Evolution Equations, Transactions of the American Mathematical Society, 347 (1995), 2851-2868.  doi: 10.1090/S0002-9947-1995-1290722-8.

[35]

T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad., 35 (1959), 467-468.  doi: 10.3792/pja/1195524254.

[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 and A.M. De Roos, Numerical continuation of equilibria of physiologically structured population models Ⅰ: Theory, Math. Models Methods Appl. Sci., 11 (2001), 1101-1127.  doi: 10.1142/S0218202501001264.

[38]

P. Laurencot and C. Walker, Steady states for a coagulation-fragmentation equation with volume scattering, SIAM Journal on Mathematical Analysis, 37 (2005), 531-548.  doi: 10.1137/S0036141004444111.

[39]

J. Makino, T. Fukushige, Y. Funato and 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 and E.E. Tyrtyshnikov, A fast numerical method for the Cauchy problem for the Smoluchowski equation, Journal of Computational Physics, 282 (2015), 23-32.  doi: 10.1016/j.jcp.2014.11.003.

[41]

G. Menon and R.L. Pego, Dynamical scaling in Smoluchowski's coagulation equations: Uniform convergence, SIAM Rev., 48 (2006), 745-768.  doi: 10.1137/060662496.

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

[46]

M. Nicmanis and 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, (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 and M.E. Ryan, Invasion speeds for structured populations in fluctuating environments, Theor. Ecol., 4 (2011), 423-434. 

[52]

J.W. Sinko and 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.  doi: 10.2140/pjm.1958.8.887.

[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.  doi: 10.1016/j.physd.2006.07.024.

[55]

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

[56]

R.M. Ziff and G. Stell, Kinetics of polymer gelation, J. Chem. Phys., 73 (1980), 3492-3499.  doi: 10.1007/BF01012594.

show all references

References:
[1]

A.S. Ackleh, Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal. Theory Methods Appl., 28 (1997), 837-854.  doi: 10.1016/0362-546X(95)00195-2.

[2]

A.S. Ackleh and B.G. Fitzpatrick, Modeling aggregation and growth processes in an algal population model: analysis and computations, J. Math. Biol., 35 (1997), 480-502.  doi: 10.1007/s002850050062.

[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 and W. Lamb, Coagulation, fragmentation and growth processes in a size structured population, Discrete Contin. Dyn. Syst. -Ser. B, 11 (2009), 563-585.  doi: 10.3934/dcdsb.2009.11.563.

[6]

H.T. Banks and F. Kappel, Transformation semigroups andL 1-approximation for size structured population models, Semigroup Forum, 38 (1989), 141-155.  doi: 10.1007/BF02573227.

[7]

C. Biggs and 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 and J.G. Younger, Klebsiella pneumoniae Flocculation Dynamics, Bull. Math. Biol., 70 (2008), 745-768.  doi: 10.1007/s11538-007-9277-y.

[10]

D. Breda, Solution operator approximations for characteristic roots of delay differential equations, Applied Numerical Mathematics, 56 (2006), 305-317.  doi: 10.1016/j.apnum.2005.04.010.

[11]

D. Breda, O. Diekmann, S. Maset and R. Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, J. Biol. Dyn., 7 (2013), 4-20.  doi: 10.1080/17513758.2013.789562.

[12]

D. Breda, S. Maset and R. Vermiglio, Computing the characteristic roots for delay differential equations, IMA J Numer Anal, 24 (2004), 1-19.  doi: 10.1093/imanum/24.1.1.

[13]

D. Breda, S. Maset and R. Vermiglio, Pseudospectral Differencing Methods for Characteristic Roots of Delay Differential Equations, SIAM J. Sci. Comput., 27 (2005), 482-495.  doi: 10.1137/030601600.

[14]

D. Breda, S. Maset and R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Applied Numerical Mathematics, 56 (2006), 318-331.  doi: 10.1016/j.apnum.2005.04.011.

[15]

D. Breda, S. Maset and R. Vermiglio, Numerical approximation of characteristic values of partial retarded functional differential equations, Numer. Math., 113 (2009), 181-242.  doi: 10.1007/s00211-009-0233-7.

[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 and 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.  doi: 10.1016/j.matpur.2012.01.004.

[18]

V. Calvez, N. Lenuzza, M. Doumic, J.-P. Deslys, F. Mouthon and B. Perthame, Prion dynamics with size dependency-strain phenomena, J. Biol. Dyn., 4 (2010), 28-42.  doi: 10.1080/17513750902935208.

[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 and M.A. Kirkilionis, Numerical equilibrium analysis for structured consumer resource models, Bulletin of Mathematical Biology, 72 (2010), 259-297.  doi: 10.1007/s11538-009-9445-3.

[22]

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

[23]

C.A. Dorao and 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 and 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 and H.A. Jakobsen, Least-squares spectral method for solving advective population balance problems, Journal of Computational and Applied Mathematics, 201 (2007), 247-257.  doi: 10.1016/j.cam.2006.02.020.

[26]

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

[27]

K. Engelborghs, T. Luzyanina and D. Roose, Numerical Bifurcation Analysis of Delay Differential Equations Using DDE-BIFTOOL, ACM Trans Math Softw, 28 (2002), 1-21.  doi: 10.1145/513001.513002.

[28]

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

[29]

J.Z. Farkas and P. Hinow, Steady states in hierarchical structured populations with distributed states at birth, Discrete Contin. Dyn. Syst. -Ser. B, 17 (2012), 2671-2689.  doi: 10.3934/dcdsb.2012.17.2671.

[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. doi: 10.1007/978-3-540-78273-5_4.

[31]

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

[32]

K. Ito and F. Kappel, The Trotter-Kato theorem and approximation of PDEs, Math Comp, 67 (1998), 21-44.  doi: 10.1090/S0025-5718-98-00915-6.

[33]

F. Kappel and K. Kunisch, Spline Approximations for Neutral Functional Differential Equations, SIAM J. Numer. Anal., 18 (1981), 1058-1080.  doi: 10.1137/0718072.

[34]

N. Kato, A Principle of Linearized Stability for Nonlinear Evolution Equations, Transactions of the American Mathematical Society, 347 (1995), 2851-2868.  doi: 10.1090/S0002-9947-1995-1290722-8.

[35]

T. Kato, Remarks on pseudo-resolvents and infinitesimal generators of semi-groups, Proc. Japan Acad., 35 (1959), 467-468.  doi: 10.3792/pja/1195524254.

[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 and A.M. De Roos, Numerical continuation of equilibria of physiologically structured population models Ⅰ: Theory, Math. Models Methods Appl. Sci., 11 (2001), 1101-1127.  doi: 10.1142/S0218202501001264.

[38]

P. Laurencot and C. Walker, Steady states for a coagulation-fragmentation equation with volume scattering, SIAM Journal on Mathematical Analysis, 37 (2005), 531-548.  doi: 10.1137/S0036141004444111.

[39]

J. Makino, T. Fukushige, Y. Funato and 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 and E.E. Tyrtyshnikov, A fast numerical method for the Cauchy problem for the Smoluchowski equation, Journal of Computational Physics, 282 (2015), 23-32.  doi: 10.1016/j.jcp.2014.11.003.

[41]

G. Menon and R.L. Pego, Dynamical scaling in Smoluchowski's coagulation equations: Uniform convergence, SIAM Rev., 48 (2006), 745-768.  doi: 10.1137/060662496.

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

[46]

M. Nicmanis and 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, (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 and M.E. Ryan, Invasion speeds for structured populations in fluctuating environments, Theor. Ecol., 4 (2011), 423-434. 

[52]

J.W. Sinko and 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.  doi: 10.2140/pjm.1958.8.887.

[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.  doi: 10.1016/j.physd.2006.07.024.

[55]

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

[56]

R.M. Ziff and G. Stell, Kinetics of polymer gelation, J. Chem. Phys., 73 (1980), 3492-3499.  doi: 10.1007/BF01012594.

Figure 1.  Results-of-the Results of the numerical simulations. a) $a,\,b\,,c$ values satisfying the necessary condition (12), form a 3D surface (blue surface). Steady states of the Sinko-Streifer model only exist on the red line b) Comparison of exact stationary solution (for the point marked with red star in Figure 1a) with approximate stationary solution for $n=100$. c) Absolute error between exact stationary solution and approximate stationary solution decays linearly as the dimension of approximate subspaces $\mathcal{X}_{n}$ increase.
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Figure 2.  Existence and stability regions for the steady states of the PBE a) Existence region for the steady states of the PBE forms a wedge like shape. b) Stability region for $b=0.1$, $a\in[0,\,15]$ and $c\in[0,\,5]$. c) Stability region for $b=0.5$, $a\in[0,\,15]$ and $c\in[0,\,5]$. d) Stability region for $b=1.0$, $a\in[0,\,15]$ and $c\in[0,\,5]$. Color bar represents the real part of rightmost eigenvalue of the Jacobian matrix evaluated at each steady state. Yellow regions represents the region for which a positive steady state does not exists.
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Figure 3.  An example steady-state solution of the PBE for $b=0.5,\,a=c=1$. b) Steady states for increasing renewal rate and $b=c=1$
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Figure 4.  Time evolution of the flocculation model with arbitrary initial conditions. a) Four different initial conditions are chosen close to the steady state. b) Solution of the PBE for those initial conditions at $t=10$. c) Evolution of the total number $M_{0}(t)$ of the flocs for $t\in[0,\,10]$. d) Evolution of the total mass $M_{1}(t)$ of the flocs for $t\in[0,\,10]$.
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Figure 5.  Change in zeroth and first moments with increasing dimension of the approximate space $\mathcal{X}_{n}$. a) Change in the total number and the total mass of the flocs with respect to increasing dimension $n$. Dashed red lines and dotted green lines corresponds to the total number and the total mass of the flocs of the steady state for $n=1000$, respectively. b) Steady state solution for $n=100$ and $n=500$.
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Figure 6.  Eigenvalues of the Jacobian $J_{\mathcal{F}}(\alpha)$ multiplied by $\Delta x$ for the steady state illustrated in Figure 3a. a) Eigenvalues of the Jacobian plotted in the complex plane for $n=20$. b) Eigenvalues of the Jacobian plotted in the complex plane for $n=50$. c) Eigenvalues of the Jacobian plotted in the complex plane for $n=200$. d) Change in the rightmost eigenvalue for increasing $n$. Dashed black line corresponds to the rightmost eigenvalue of the Jacobian for $n=1000$.
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