Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Sensitivity equations for a size-structured population model

Authors: H. T. Banks, Stacey L. Ernstberger and Shuhua Hu
Journal: Quart. Appl. Math. 67 (2009), 627-660
MSC (2000): Primary 93B35, 90C31, 92D25
Published electronically: May 5, 2009
MathSciNet review: 2588228
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we consider the classical Sinko-Streifer size-structured population model and derive sensitivity partial differential equations for the sensitivities of solutions with respect to initial conditions, growth rate, mortality rate and fecundity rate. Sample numerical results to illustrate the use of these equations are also presented.

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  • 1. H. M. Adelman and R.T. Haftka, Sensitivity analysis of discrete structural systems, A.I.A.A. Journal, 24 (1986), 823-832.
  • 2. Richard Bellman and Kenneth L. Cooke, Differential-difference equations, Academic Press, New York-London, 1963. MR 0147745
  • 3. H. T. Banks and F. Kappel, Transformation semigroups and 𝐿¹-approximation for size structured population models, Semigroup Forum 38 (1989), no. 2, 141–155. Semigroups and differential operators (Oberwolfach, 1988). MR 976199, 10.1007/BF02573227
  • 4. H. T. Banks, F. Kappel, and C. Wang, Weak solutions and differentiability for size structured population models, Estimation and control of distributed parameter systems (Vorau, 1990), Internat. Ser. Numer. Math., vol. 100, Birkhäuser, Basel, 1991, pp. 35–50. MR 1155635, 10.1007/978-3-0348-6418-3_2
  • 5. H. T. Banks, Fumio Kojima, and W. P. Winfree, Boundary estimation problems arising in thermal tomography, Inverse Problems 6 (1990), no. 6, 897–921. MR 1082231
  • 6. H. T. Banks and Hoan K. Nguyen, Sensitivity of dynamical systems to Banach space parameters, J. Math. Anal. Appl. 323 (2006), no. 1, 146–161. MR 2261157, 10.1016/j.jmaa.2005.09.084
  • 7. H. T. Banks, V. A. Bokil, Shuhua Hu, A. K. Dhar, R. A. Bullis, C. L. Browdy, and F. C. T. Allnutt, Modeling shrimp biomass and viral infection for production of biological countermeasures, Math. Biosci. Eng. 3 (2006), no. 4, 635–660. MR 2249893, 10.3934/mbe.2006.3.635
  • 8. H. T. Banks, M. Davidian and J.R. Samuels, Jr., An inverse problem statistical methodology summary, CRSC-TR07-14, August, 2007.
  • 9. H. T. Banks, S. Dediu, and H. K. Nguyen, Sensitivity of dynamical systems to parameters in a convex subset of a topological vector space, Math. Biosci. Eng. 4 (2007), no. 3, 403–430. MR 2328091, 10.3934/mbe.2007.4.403
  • 10. H. T. Banks, S. L. Ernstberger, and S. L. Grove, Standard errors and confidence intervals in inverse problems: sensitivity and associated pitfalls, J. Inverse Ill-Posed Probl. 15 (2007), no. 1, 1–18. MR 2313504, 10.1515/JIIP.2007.001
  • 11. Jerry J. Batzel, Franz Kappel, Daniel Schneditz, and Hien T. Tran, Cardiovascular and respiratory systems, Frontiers in Applied Mathematics, vol. 34, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. Modeling, analysis, and control. MR 2279586
  • 12. Kenneth P. Burnham and David R. Anderson, Model selection and multimodel inference, 2nd ed., Springer-Verlag, New York, 2002. A practical information-theoretic approach. MR 1919620
  • 13. J. A. Burns, T. Lin, and L. G. Stanley, A Petrov Galerkin finite-element method for interface problems arising in sensitivity computations, Comput. Math. Appl. 49 (2005), no. 11-12, 1889–1903. MR 2154692, 10.1016/j.camwa.2004.06.037
  • 14. J. A. Burns, D. Rubio, and M. I. Troparevsky, Sensitivity computations for elliptic equations with interfaces, Sixth International Conference on Mathematical Problems in Engineering and Aerospace Sciences, Camb. Sci. Publ., Cambridge, 2007, pp. 99–107. MR 2402293
  • 15. J. A. Burns and L. G. Stanley, A note on the use of transformations in sensitivity computations for elliptic systems, Math. Comput. Modelling 33 (2001), no. 1-3, 101–114. Computation and control, VI (Bozeman, MT, 1998). MR 1812544, 10.1016/S0895-7177(00)00232-6
  • 16. Jeff Borggaard and John Burns, A PDE sensitivity equation method for optimal aerodynamic design, J. Comput. Phys. 136 (1997), no. 2, 366–384. MR 1474410, 10.1006/jcph.1997.5743
  • 17. Jeff Borggaard and Arun Verma, On efficient solutions to the continuous sensitivity equation using automatic differentiation, SIAM J. Sci. Comput. 22 (2000), no. 1, 39–62. MR 1769486, 10.1137/S1064827599352136
  • 18. Àngel Calsina and Joan Saldaña, A model of physiologically structured population dynamics with a nonlinear individual growth rate, J. Math. Biol. 33 (1995), no. 4, 335–364. MR 1320428, 10.1007/BF00176377
  • 19. G. Casella and R. L. Berger, Statistical Inference, Duxbury, California, 2002.
  • 20. José B. Cruz Jr. (ed.), System sensitivity analysis, Dowden, Hutchinson and Ross, Inc., Stroudsburg, Pa., 1973. Benchmark Papers in Electrical Engineering and Computer Science. MR 0392045
  • 21. M. Davidian and D. M. Gilitan, Nonlinear Models for Repeated Measurement Data, Chapman & Hall, London, 1995.
  • 22. M. C. Delfour and J.-P. Zolésio, Anatomy of the shape Hessian, Ann. Mat. Pura Appl. (4) 159 (1991), 315–339 (English, with French summary). MR 1145103, 10.1007/BF01766307
  • 23. M. C. Delfour and J.-P. Zolésio, Shapes and geometries, Advances in Design and Control, vol. 4, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001. Analysis, differential calculus, and optimization. MR 1855817
  • 24. M. Eslami, Theory of sensitivity in dynamic systems, Springer-Verlag, Berlin, 1994. An introduction. MR 1318439
  • 25. Paul M. Frank, Introduction to system sensitivity theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0496987
  • 26. J. Haslinger and R. A. E. Mäkinen, Introduction to shape optimization, Advances in Design and Control, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2003. Theory, approximation, and computation. MR 1969772
  • 27. Edward J. Haug, Kyung K. Choi, and Vadim Komkov, Design sensitivity analysis of structural systems, Mathematics in Science and Engineering, vol. 177, Academic Press, Inc., Orlando, FL, 1986. MR 860040
  • 28. M. Kleiber, H. Antunez, T. D. Hien and P. Kowalczyk, Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations, John Wiley & Sons, New York, NY, 1997.
  • 29. K. Ito, F. Kappel, and G. Peichl, A fully discretized approximation scheme for size-structured population models, SIAM J. Numer. Anal. 28 (1991), no. 4, 923–954. MR 1111447, 10.1137/0728050
  • 30. D. Jiang, A. L. Lawrence, W. H. Neil and H. Gong, Effects of temperature and salinity on nitrogenous excretion by Litopenaeus vannamei juveniles, Journal of Experimental Marine Biology and Ecology, 253 (2000), 193-209.
  • 31. Mark Kot, Elements of mathematical ecology, Cambridge University Press, Cambridge, 2001. MR 2006645
  • 32. J. A. J. Metz and O. Diekmann (eds.), The dynamics of physiologically structured populations, Lecture Notes in Biomathematics, vol. 68, Springer-Verlag, Berlin, 1986. Papers from the colloquium held in Amsterdam, 1983. MR 860959
  • 33. A. G. McKendrick, Applications of mathematics to medical problems, Proceedings of the Edinburgh Mathematical Society, 40 (1926), 98-130.
  • 34. Olivier Pironneau, Optimal shape design for elliptic systems, Springer Series in Computational Physics, Springer-Verlag, New York, 1984. MR 725856
  • 35. Andrea Saltelli, Karen Chan, and E. Marian Scott (eds.), Sensitivity analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2000. MR 1886391
  • 36. G. A. F. Seber and C. J. Wild, Nonlinear regression, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1989. MR 986070
  • 37. J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology, 48 (1967), 910-918.
  • 38. Lisa G. Stanley and Dawn L. Stewart, Design sensitivity analysis, Frontiers in Applied Mathematics, vol. 25, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Computational issues of sensitivity equation methods. MR 1946496
  • 39. H. von Foerster, Some remarks on changing populations, in The Kinetics of Cellular Proliferation, F. Stohlman, Jr., ed., Grune & Stratton, New York, 1959, pp. 382-407.
  • 40. K. Thomaseth and C. Cobelli, Generalized sensitivity functions in physiological system identification, Annals of Biomedical Engineering, 27 (1999), 607-616.
  • 41. A. Wierzbicki, Models and Sensitivity of Control Systems, Studies in Automation and Control 5, Elsevier Science Publ. Co., Inc., New York, NY, 1984.
  • 42. P. V. Wyk and J. Scarpa, Water quality requirement and management, http://www.hboie.du/ aqua/downloads/pdf/ shrimpmanual_chapter8.pdf.

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Additional Information

H. T. Banks
Affiliation: Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205

Stacey L. Ernstberger
Affiliation: Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205

Shuhua Hu
Affiliation: Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205

DOI: https://doi.org/10.1090/S0033-569X-09-01105-1
Keywords: Size-structured population models, sensitivity equations, method of characteristics, renewal equations, finite difference schemes
Received by editor(s): October 11, 2007
Published electronically: May 5, 2009
Article copyright: © Copyright 2009 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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