# Propagation of Growth Uncertainty in a Physiologically Structured Population

Mathematical Modelling of Natural Phenomena (2012)

- Volume: 7, Issue: 5, page 7-23
- ISSN: 0973-5348

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topBanks, H.T., and Hu, S.. "Propagation of Growth Uncertainty in a Physiologically Structured Population." Mathematical Modelling of Natural Phenomena 7.5 (2012): 7-23. <http://eudml.org/doc/222433>.

@article{Banks2012,

abstract = {In this review paper we consider physiologically structured population models that have
been widely studied and employed in the literature to model the dynamics of a wide variety
of populations. However in a number of cases these have been found inadequate to describe
some phenomena arising in certain real-world applications such as dispersion in the
structure variables due to growth uncertainty/variability. Prompted by this, we described
two recent approaches that have been investigated in the literature to describe this
growth uncertainty/variability in a physiologically structured population. One involves
formulating growth as a Markov diffusion process while the other entails imposing a
probabilistic structure on the set of possible growth rates across the entire population.
Both approaches lead to physiologically structured population models with nontrivial
dispersion. Even though these two approaches are conceptually quite different, they were
found in [17] to have a close relationship: in some
cases with properly chosen parameters and coefficient functions, the resulting stochastic
processes have the same probability density function at each time.},

author = {Banks, H.T., Hu, S.},

journal = {Mathematical Modelling of Natural Phenomena},

keywords = {uncertainty; structured population models; Markov diffusion processes; Itô stochastic differential equations; Fokker-Planck equations},

language = {eng},

month = {10},

number = {5},

pages = {7-23},

publisher = {EDP Sciences},

title = {Propagation of Growth Uncertainty in a Physiologically Structured Population},

url = {http://eudml.org/doc/222433},

volume = {7},

year = {2012},

}

TY - JOUR

AU - Banks, H.T.

AU - Hu, S.

TI - Propagation of Growth Uncertainty in a Physiologically Structured Population

JO - Mathematical Modelling of Natural Phenomena

DA - 2012/10//

PB - EDP Sciences

VL - 7

IS - 5

SP - 7

EP - 23

AB - In this review paper we consider physiologically structured population models that have
been widely studied and employed in the literature to model the dynamics of a wide variety
of populations. However in a number of cases these have been found inadequate to describe
some phenomena arising in certain real-world applications such as dispersion in the
structure variables due to growth uncertainty/variability. Prompted by this, we described
two recent approaches that have been investigated in the literature to describe this
growth uncertainty/variability in a physiologically structured population. One involves
formulating growth as a Markov diffusion process while the other entails imposing a
probabilistic structure on the set of possible growth rates across the entire population.
Both approaches lead to physiologically structured population models with nontrivial
dispersion. Even though these two approaches are conceptually quite different, they were
found in [17] to have a close relationship: in some
cases with properly chosen parameters and coefficient functions, the resulting stochastic
processes have the same probability density function at each time.

LA - eng

KW - uncertainty; structured population models; Markov diffusion processes; Itô stochastic differential equations; Fokker-Planck equations

UR - http://eudml.org/doc/222433

ER -

## References

top- A.S. Ackleh, H.T. Banks, K. Deng. A finite difference approximation for a coupled system of nonlinear size-structured populations. Nonlinear Analysis, 50 (2002), 727–748. Zbl1002.65093
- A.S. Ackleh, K. Ito. An implicit finite-difference scheme for the nonlinear size-structured population model. Numer. Funct. Anal. Optim.18 (1997), 65–884. Zbl0897.35006
- A.S. Ackleh, K. Deng. A monotone approximation for the nonautonomous size-structured population model. Quart. Appl. Math., 57 (1999), 261–267. Zbl1157.35491
- O. Angulo, J.C.López-Marcos, Numerical integration of fully nonlinear size-structured population models. Applied Numerical Mathematics, 50 (2004), 291–327. Zbl1071.92033
- H.T. Banks. A Functional Analysis Framework for Modeling, Estimation and Control in Science and Engineering, CRC Press/Taylor and Frances Publishing, Boca Raton, FL, June, 2012, (258 pages).
- H.T. Banks, K.L. Bihari. Modelling and estimating uncertainty in parameter estimation. Inverse Problems, 17 (2001), 95–111. Zbl1054.35121
- H.T. Banks, V.A. Bokil, S. Hu, A.K. Dhar, R.A. Bullis, C.L. Browdy, F.C.T. Allnutt. Modeling shrimp biomass and viral infection for production of biological countermeasuresCRSC-TR05-45, NCSU, December, 2005 ; Mathematical Biosciences and Engineering, 3 (2006), 635–660. Zbl1113.92023
- H.T. Banks, L.W. Botsford, F. Kappel, C. Wang. Modeling and estimation in size structured population models. LCDS-CCS Report 87-13, Brown University ; Proceedings 2nd Course on Mathematical Ecology, (Trieste, December 8-12, 1986) World Press, Singapore, 1988, 521–541.
- H.T. Banks, F. Charles, M. Doumic, K.L. Sutton, W.C. Thompson. Label structured cell proliferation models. Appl. Math. Letters, 23 (2010), 1412–1415. Zbl1205.35321
- H.T. Banks, J.L. Davis. A comparison of approximation methods for the estimation of probability distributions on parameters. CRSC-TR05-38, October, 2005 ; Applied Numerical Mathematics, 57 (2007), 753–777. Zbl1115.65009
- H.T. Banks, J.L. Davis. Quantifying uncertainty in the estimation of probability distributions. CRSC-TR07-21, December, 2007 ; Math. Biosci. Engr., 5 (2008), 647–667. Zbl1160.35471
- H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich, A.K. Dhar, C.L. Browdy. A comparison of probabilistic and stochastic differential equations in modeling growth uncertainty and variability. CRSC-TR08-03, NCSU, February, 2008 ; Journal of Biological Dynamics, 3 (2009), 130–148.
- H.T. Banks, J.L. Davis, S.L. Ernstberger, S. Hu, E. Artimovich, A.K. Dhar. Experimental design and estimation of growth rate distributions in size-structured shrimp populations. CRSC-TR08-20, NCSU, November, 2008 ; Inverse Problems, 25 (2009), 095003(28pp). Zbl1173.35720
- H.T. Banks, J.L. Davis, S. Hu. A computational comparison of alternatives to including uncertainty in structured population models. CRSC-TR09-14, June, 2009 ; Three Decades of Progress in Systems and Control, X. Hu, U. Jonsson, B. Wahlberg, B. Ghosh (Eds.), Springer, 2010, 19–33.
- H.T. Banks, B.G. Fitzpatrick. Estimation of growth rate distributions in size structured population models. Quarterly of Applied Mathematics, 49 (1991), 215–235. Zbl0731.92021
- H.T. Banks, B.G. Fitzpatrick, L.K. Potter, Y. Zhang. Estimation of probability distributions for individual parameters using aggregate population data. CRSC-TR98-6, NCSU, January, 1998 ; Stochastic Analysis, Control, Optimization and Applications, (Edited by W. McEneaney, G. Yin and Q. Zhang), Birkhauser, Boston, 1998, 353–371. Zbl0922.92016
- H.T. Banks, S. Hu. Nonlinear stochastic Markov processes and modeling uncertainty in populations. CRSC-TR11-02, NCSU, January, 2011 ; Mathematical Bioscience and Engineering, 9 (2012), 1–25. Zbl1259.60090
- H.T. Banks, F. Kappel. Transformation semigroups and L1-approximation for size- structured population models. Semigroup Forum, 38 (1989), 141–155. Zbl0687.47033
- H.T. Banks, F. Kappel, C. Wang. Weak solutions and differentiability for size-structured population models. International Series of Numerical Mathematics, 100 (1991), 35–50. Zbl0766.92013
- H.T. Banks, G.A. Pinter. A probabilistic multiscale approach to hysteresis in shear wave propagation in biotissue, CRSC-TR04-03, January, 2004 ; SIAM J. Multiscale Modeling and Simulation, 3 (2005), 395–412. Zbl1140.93313
- H.T. Banks, K.L. Sutton, W.C. Thompson, G. Bocharov, M. Doumic, T. Schenkel, J. Argilaguet, S. Giest, C. Peligero, A. Meyerhans. A new model for the estimation of cell proliferation dynamics using CFSE data. Center for Research in Scientific Computation Technical Report CRSC-TR11-05, NCSU, July, 2011 ; J. Immunological Methods, 373 (2011), 143–160.
- H.T. Banks, K.L. Sutton, W.C. Thompson, G. Bocharov, D. Roose, T. Schenkel, A. Meyerhans. Estimation of cell proliferation dynamics using CFSE data, CRSC-TR09-17, NCSU, August, 2009 ; Bull. Math. Biol.70 (2011), 116–150; doi :. URI10.1007/s11538-010-9524-5
- H.T. Banks, W.C. Thompson. Mathematical models of dividing cell populations : Application to CFSE data. CRSC-TR12-10, N. C. State University, Raleigh, NC, April, 2012 ; Journal on Mathematical Modelling of Natural Phenomena, to appear. Zbl1260.76047
- H.T. Banks, W.C. Thompson. A division-dependent compartmental model with cyton and intracellular label dynamics. CRSC-TR12-12, N. C. State University, Raleigh, NC, May, 2012 ; Intl. J. of Pure and Applied Math., 77 (2012), 119–147. Zbl1247.92007
- H.T. Banks, W.C. Thompson, C. Peligero, S. Giest, J. Argilaguet, A. Meyerhans. A compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays, CRSC-TR12-03, NCSU, January, 2012 ; Math. Biosci. Engr., to appear. Zbl1259.92017
- H.T. Banks, H.T. Tran, D.E. Woodward. Estimation of variable coefficients in the Fokker-Planck equations using moving node finite elements. SIAM J. Numer. Anal., 30 (1993), 1574–1602. Zbl0791.65098
- G.I. Bell, E.C. Anderson. Cell growth and division I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures. Biophysical Journal, 7 (1967), 329–351.
- A. Calsina, J. Saldana. A model of physiologically structured population dynamics with a nonlinear individual growth rate. Journal of Mathematical Biology, 33 (1995), 335–364. Zbl0828.92025
- G. Casella, R.L. Berger. Statistical Inference. Duxbury, California, 2002.
- F.L. Castille, T.M. Samocha, A.L. Lawrence, H. He, P. Frelier, F. Jaenike. Variability in growth and survival of early postlarval shrimp (Penaeus vannamei Boone 1931)Aquaculture, 113 (1993), 65–81.
- J. Chu, A. Ducrot, P. Magal, S. Ruan. Hopf bifurcation in a size-structured population dynamic model with random growth. J. Differential Equations, 247 (2009), 956–1000. Zbl1172.35322
- J.M. Cushing. An Introduction to Structured Population Dynamics. CMB-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1998. Zbl0939.92026
- T.C. Gard. Introduction to Stochastic Differential Equations. Marcel Dekker, New York, 1988. Zbl0628.60064
- C.W. Gardiner. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, Berlin, 1983. Zbl0515.60002
- M. Gyllenberg, G.F. Webb. A nonlinear structured population model of tumor growth with quiescence. J. Math. Biol., 28 (1990), 671–694. Zbl0744.92026
- G.W. Harrison. Numerical solution of the Fokker-Planck equation using moving finite elements. Numerical Methods for Partial Differential Equations, 4 (1988), 219–232. Zbl0648.65096
- J. Hasenauer, D. Schittler, F. Allgöer. A computational model for proliferation dynamics of division- and label-structured populations. February, 2012, preprint.
- J. Hasenauer, S. Waldherr, M. Doszczak, P. Scheurich, N. Radde, F. Allgöer. Analysis of heterogeneous cell populations : a density-based modeling and identification framework. J. Process Control, 21 (2011), 1417–1425.
- K. Huang. Statistical Mechanics. J. Wiley & Sons, New York, NY, 1963.
- M. Iannelli. Mathematical Theory of Age-Structured Population Dynamics. Applied Math. Monographs, CNR, Giardini Editori e Stampatori, Pisa, 1995.
- M. Kimura. Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics, 39 (1954), 280–295.
- F. Klebaner. Introduction to Stochastic Calculus with Applications. 2nd ed., Imperial College Press, London, 2006. Zbl1077.60001
- I. Karatzas, S.E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer, New York, 1991. Zbl0734.60060
- T. Luzyanina, D. Roose, G. Bocharov. Distributed parameter identification for a label-structured cell population dynamics model using CFSE histogram time-series data. J. Math. Biol., 59 (2009), 581–603. Zbl1231.92027
- T. Luzyanina, M. Mrusek, J.T. Edwards, D. Roose, S. Ehl, G. Bocharov. Computational analysis of CFSE proliferation assay. J. Math. Biol., 54 (2007), 57–89. Zbl1113.92021
- T. Luzyanina, D. Roose, T. Schenkel, M. Sester, S. Ehl, A. Meyerhans, G. Bocharov. Numerical modelling of label-structured cell population growth using CFSE distribution data. Theoretical Biology and Medical Modelling, 4(2007), Published Online.
- A.G. McKendrick. Applications of mathematics to medical problems. Proceedings of the Edinburgh Mathematical Society, 44 (1926), 98–130.
- J.A.J. Metz, E.O. Diekmann, The Dynamics of Physiologically Structured Populations. Lecture Notes in Biomathematics, Vol. 68, Springer, Heidelberg, 1986. Zbl0614.92014
- J.E. Moyal. Stochastic processes and statistical physics. Journal of the Royal Statistical Society. Series B (Methodological), 11 (1949), 150–210. Zbl0039.21602
- Yu. V. Prohorov. Convergence of random processes and limit theorems in probability theory. Theor. Prob. Appl., 1 (1956), 157–214.
- B. Oksendal. Stochastic Differentail Equations. 5th edition, Springer, Berlin, 2000.
- A. Okubo. Diffusion and Ecological Problems : Mathematical Models. Biomathematics, 10 (1980), Springer-Verlag, Berlin. Zbl0422.92025
- G. Oster, Y. Takahashi. Models for age-specific interactions in a periodic environment. Ecological Monographs, 44 (1974), 483–501.
- B. Perthame. Transport Equations in Biology. Birkhauser Verlag, Basel, 2007. Zbl1185.92006
- R. Rudnicki. Models of population dynamics and genetics. From Genetics To Mathematics, (edited by M. Lachowicz and J. Miekisz), World Scientific, Singapore, 2009, 103–148. Zbl1286.92045
- H. Risken. The Fokker-Planck Equation : Methods of Solution and Applications. Springer, New York, 1996. Zbl0866.60071
- D. Schittler, J. Hasenauer, F. Allgöer. A generalized population model for cell proliferation : Integrating division numbers and label dynamics. Proceedings of Eighth International Workshop on Computational Systems Biology (WCSB 2011), June 2011, Zurich, Switzerland, 165–168.
- D. Schittler, J. Hasenauer, F. Allgöer. A model for proliferating cell populations that accounts for cell types. Proc. of 9th International Workshop on Computational Systems Biology, 2012, 84–87.
- J. Sinko, W. Streifer. A new model for age-size structure of a population. Ecology, 48 (1967), 910–918.
- T.T. Soong. Random Differential Equations in Science and Engineering. Academic Press, New York and London, 1973. Zbl0348.60081
- T.T. Soong, S.N. Chuang. Solutions of a class of random differential equations. SIAM J. Appl. Math., 24 (1973), 449–459. Zbl0237.60028
- W.C. Thompson. Partial Differential Equation Modeling of Flow Cytometry Data from CFSE-based Proliferation Assays. Ph.D. Dissertation, North Carolina State University, December, 2011.
- H. Von Foerster. Some remarks on changing populations. The Kinetics of Cellular Proliferation, F. Stohlman, Jr. (ed.), Grune and Stratton, New York, 1959.
- G.F. Webb. Theory of Nonlinear Age-dependent Population Dynamics. Marcel Dekker, New York, 1985. Zbl0555.92014
- A.Y. Weiße, R.H. Middleton, W. Huisinga. Quantifying uncertainty, variability and likelihood for ordinary differential equation models. BMC Syst. Bio., 4 (144), 2010.
- G.H. WeissEquation for the age structure of growing populations. Bull. Math. Biophy., 30 (1968), 427–435.
- . URIhttp://en.wikipedia.org/wiki/Probability_density_function

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