Sensitivity analysis for a nonlinear size-structured population model
Authors:
Keng Deng and Yi Wang
Journal:
Quart. Appl. Math. 73 (2015), 401-417
MSC (2010):
Primary 35L60, 92D25, 93B35
DOI:
https://doi.org/10.1090/qam/1366
Published electronically:
June 11, 2015
MathSciNet review:
3400750
Full-text PDF Free Access
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Additional Information
Abstract: In this paper, we consider a nonlinear size-structured population model with vital rates depending on the total population. We derive sensitivity partial differential equations for the sensitivities of the solution with respect to the reproduction and mortality rates. We also present numerical results to illustrate the use of these sensitivity equations.
References
- Azmy S. Ackleh, Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal. 28 (1997), no. 5, 837–854. MR 1422189, DOI https://doi.org/10.1016/0362-546X%2895%2900195-2
- Azmy S. Ackleh, H. T. Banks, Keng Deng, and Shuhua Hu, Parameter estimation in a coupled system of nonlinear size-structured populations, Math. Biosci. Eng. 2 (2005), no. 2, 289–315. MR 2144241, DOI https://doi.org/10.3934/mbe.2005.2.289
- Azmy S. Ackleh and Keng Deng, Existence-uniqueness of solutions for a nonlinear nonautonomous size-structured population model: an upper-lower solution approach, Canad. Appl. Math. Quart. 8 (2000), no. 1, 1–15. MR 1825386, DOI https://doi.org/10.1216/camq/1008957332
- A. S. Ackleh and Keng Deng, Monotone scheme for nonlinear first-order hyperbolic initial-boundary value problems, Appl. Math. Lett. 13 (2000), no. 5, 111–119. MR 1760272, DOI https://doi.org/10.1016/S0893-9659%2800%2900042-2
- Azmy S. Ackleh and Kazufumi Ito, An implicit finite difference scheme for the nonlinear size-structured population model, Numer. Funct. Anal. Optim. 18 (1997), no. 9-10, 865–884. MR 1485984, DOI https://doi.org/10.1080/01630569708816798
- B. M. Adams, H. T. Banks, J. E. Banks, and J. D. Stark, Population dynamics models in plant-insect herbivore-pesticide interactions, Math. Biosci. 196 (2005), no. 1, 39–64. MR 2156608, DOI https://doi.org/10.1016/j.mbs.2004.09.001
- H. T. Banks, Jimena L. Davis, Stacey L. Ernstberger, Shuhua Hu, Elena Artimovich, and Arun K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, Inverse Problems 25 (2009), no. 9, 095003, 28. MR 2540883, DOI https://doi.org/10.1088/0266-5611/25/9/095003
- H. T. Banks, Stacey L. Ernstberger, and Shuhua Hu, Sensitivity equations for a size-structured population model, Quart. Appl. Math. 67 (2009), no. 4, 627–660. MR 2588228, DOI https://doi.org/10.1090/S0033-569X-09-01105-1
- H. T. Banks and F. Kappel, Transformation semigroups and $L^1$-approximation for size structured population models, Semigroup Forum 38 (1989), no. 2, 141–155. Semigroups and differential operators (Oberwolfach, 1988). MR 976199, DOI https://doi.org/10.1007/BF02573227
- H. T. Banks, F. Kappel, and C. Wang, A semigroup formulation of a nonlinear size-structured distributed rate population model, Control and estimation of distributed parameter systems: nonlinear phenomena (Vorau, 1993) Internat. Ser. Numer. Math., vol. 118, Birkhäuser, Basel, 1994, pp. 1–19. MR 1313507
- 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), no. 3, 745–768. MR 2393018, DOI https://doi.org/10.1007/s11538-007-9277-y
- 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, DOI https://doi.org/10.1007/BF00176377
- Àngel Calsina and Manuel Sanchón, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl. 286 (2003), no. 2, 435–452. MR 2008842, DOI https://doi.org/10.1016/S0022-247X%2803%2900464-5
- H. Caswell, Matrix Population Models, Construction Analysis and Interpretation, Sinauer Associates, Massachusetts, 2001.
- M. Davidian and D.M. Giltinan, Nonlinear models for repeated measurement data, Chapman and Hall/CRC, New York, 1995.
- Jozsef Z. Farkas and Thomas Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl. 328 (2007), no. 1, 119–136. MR 2285538, DOI https://doi.org/10.1016/j.jmaa.2006.05.032
- Ariane Verdy and Hal Caswell, Sensitivity analysis of reactive ecological dynamics, Bull. Math. Biol. 70 (2008), no. 6, 1634–1659. MR 2430320, DOI https://doi.org/10.1007/s11538-008-9312-7
References
- Azmy S. Ackleh, Parameter estimation in a structured algal coagulation-fragmentation model, Nonlinear Anal. 28 (1997), no. 5, 837–854. MR 1422189, DOI https://doi.org/10.1016/0362-546X%2895%2900195-2
- Azmy S. Ackleh, H. T. Banks, Keng Deng, and Shuhua Hu, Parameter estimation in a coupled system of nonlinear size-structured populations, Math. Biosci. Eng. 2 (2005), no. 2, 289–315. MR 2144241 (2006c:92015), DOI https://doi.org/10.3934/mbe.2005.2.289
- Azmy S. Ackleh and Keng Deng, Existence-uniqueness of solutions for a nonlinear nonautonomous size-structured population model: an upper-lower solution approach, Canad. Appl. Math. Quart. 8 (2000), no. 1, 1–15. MR 1825386 (2002c:35260), DOI https://doi.org/10.1216/camq/1008957332
- A. S. Ackleh and Keng Deng, Monotone scheme for nonlinear first-order hyperbolic initial-boundary value problems, Appl. Math. Lett. 13 (2000), no. 5, 111–119. MR 1760272 (2001a:35109), DOI https://doi.org/10.1016/S0893-9659%2800%2900042-2
- Azmy S. Ackleh and Kazufumi Ito, An implicit finite difference scheme for the nonlinear size-structured population model, Numer. Funct. Anal. Optim. 18 (1997), no. 9-10, 865–884. MR 1485984 (98m:65133), DOI https://doi.org/10.1080/01630569708816798
- B. M. Adams, H. T. Banks, J. E. Banks, and J. D. Stark, Population dynamics models in plant-insect herbivore-pesticide interactions, Math. Biosci. 196 (2005), no. 1, 39–64. MR 2156608 (2006c:92016), DOI https://doi.org/10.1016/j.mbs.2004.09.001
- H. T. Banks, Jimena L. Davis, Stacey L. Ernstberger, Shuhua Hu, Elena Artimovich, and Arun K. Dhar, Experimental design and estimation of growth rate distributions in size-structured shrimp populations, Inverse Problems 25 (2009), no. 9, 095003, 28. MR 2540883 (2010k:62077), DOI https://doi.org/10.1088/0266-5611/25/9/095003
- H. T. Banks, Stacey L. Ernstberger, and Shuhua Hu, Sensitivity equations for a size-structured population model, Quart. Appl. Math. 67 (2009), no. 4, 627–660. MR 2588228 (2011d:49038)
- H. T. Banks and F. Kappel, Transformation semigroups and $L^1$-approximation for size structured population models, Semigroup Forum 38 (1989), no. 2, 141–155. Semigroups and differential operators (Oberwolfach, 1988). MR 976199 (90b:92040), DOI https://doi.org/10.1007/BF02573227
- H. T. Banks, F. Kappel, and C. Wang, A semigroup formulation of a nonlinear size-structured distributed rate population model, Control and estimation of distributed parameter systems: nonlinear phenomena (Vorau, 1993) Internat. Ser. Numer. Math., vol. 118, Birkhäuser, Basel, 1994, pp. 1–19. MR 1313507 (96g:92008)
- 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), no. 3, 745–768. MR 2393018 (2009a:92026), DOI https://doi.org/10.1007/s11538-007-9277-y
- 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 (96i:92020), DOI https://doi.org/10.1007/BF00176377
- Àngel Calsina and Manuel Sanchón, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl. 286 (2003), no. 2, 435–452. MR 2008842 (2004g:92025), DOI https://doi.org/10.1016/S0022-247X%2803%2900464-5
- H. Caswell, Matrix Population Models, Construction Analysis and Interpretation, Sinauer Associates, Massachusetts, 2001.
- M. Davidian and D.M. Giltinan, Nonlinear models for repeated measurement data, Chapman and Hall/CRC, New York, 1995.
- Jozsef Z. Farkas and Thomas Hagen, Stability and regularity results for a size-structured population model, J. Math. Anal. Appl. 328 (2007), no. 1, 119–136. MR 2285538 (2008f:35038), DOI https://doi.org/10.1016/j.jmaa.2006.05.032
- Ariane Verdy and Hal Caswell, Sensitivity analysis of reactive ecological dynamics, Bull. Math. Biol. 70 (2008), no. 6, 1634–1659. MR 2430320 (2009h:92088), DOI https://doi.org/10.1007/s11538-008-9312-7
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Additional Information
Keng Deng
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
MR Author ID:
225222
Email:
deng@louisiana.edu
Yi Wang
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
Email:
yxw3828@louisiana.edu
Keywords:
Nonlinear size-structured population model,
sensitivity equations,
finite difference approximation
Received by editor(s):
February 6, 2013
Received by editor(s) in revised form:
April 11, 2013
Published electronically:
June 11, 2015
Article copyright:
© Copyright 2015
Brown University