Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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



A general nonlinear model for the interaction of a size-structured population and its environment: Well-posedness and approximation

Authors: Azmy S. Ackleh, Baoling Ma and Robert Miller
Journal: Quart. Appl. Math. 74 (2016), 671-704
MSC (2010): Primary 35Q92, 65M06
Published electronically: June 20, 2016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We present a general model for the interaction of a size-structured population with its environment. The vital rates of the individuals are assumed to depend on a number of variables including the total population and the environment. We develop a finite difference approximation for this general model and prove the convergence of the method to the unique weak solution of the nonlinear system of partial differential equations coupled with ordinary differential equations. Potential applications are presented in various fields ranging from blood cell population dynamics to the study of invasive species.

References [Enhancements On Off] (What's this?)

  • [1] Azmy S. Ackleh, H. T. Banks, and Keng Deng, A finite difference approximation for a coupled system of nonlinear size-structured populations, Nonlinear Anal. 50 (2002), no. 6, Ser. A: Theory Methods, 727–748. MR 1911742, 10.1016/S0362-546X(01)00780-5
  • [2] A. S. Ackleh, J. Carter, V .K. Chellamuthu, and B. Ma, A model for the interaction of frog population dynamics with Batrachochytrium dendrobaties, Janthinobacterium lividium and temperature and its implication for chytridiomycosis management, Ecol. Model. 320 (2016), 158-169.
  • [3] Azmy S. Ackleh, Jacoby Carter, Keng Deng, Qihua Huang, Nabendu Pal, and Xing Yang, Fitting a structured juvenile-adult model for green tree frogs to population estimates from capture-mark-recapture field data, Bull. Math. Biol. 74 (2012), no. 3, 641–665. MR 2889669, 10.1007/s11538-011-9682-0
  • [4] Azmy S. Ackleh and Keng Deng, A nonautonomous juvenile-adult model: well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math. 69 (2009), no. 6, 1644–1661. MR 2496711, 10.1137/080723673
  • [5] Azmy S. Ackleh, Keng Deng, Cammey E. Cole, and Hien T. Tran, Existence-uniqueness and monotone approximation for an erythropoiesis age-structured model, J. Math. Anal. Appl. 289 (2004), no. 2, 530–544. MR 2026923, 10.1016/j.jmaa.2003.08.037
  • [6] 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, 10.1080/01630569708816798
  • [7] Azmy S. Ackleh and Baoling Ma, A second-order high-resolution scheme for a juvenile-adult model of amphibians, Numer. Funct. Anal. Optim. 34 (2013), no. 4, 365–403. MR 3039409, 10.1080/01630563.2012.730595
  • [8] Bedr’eddine Ainseba, Delphine Picart, and Denis Thiéry, An innovative multistage, physiologically structured, population model to understand the European grapevine moth dynamics, J. Math. Anal. Appl. 382 (2011), no. 1, 34–46. MR 2805492, 10.1016/j.jmaa.2011.04.021
  • [9] H. T. Banks, Shuhua Hu, and W. Clayton Thompson, Modeling and inverse problems in the presence of uncertainty, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2014. MR 3203115
  • [10] F. Brauer and J. Noel, The Qualitative Theory of Ordinary Differential Equations, Dover, 1989.
  • [11] Cammey Elizabeth Cole, Benzene and its effect on erythropoiesis: Models, optimal controls, and analyses, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–North Carolina State University. MR 2702629
  • [12] M. J. Costello, Ecology of sea lice parasitic on farmed and wild fish, Trends in Parasitol. 22 (2006), 475-483.
  • [13] J. M. Cushing and Jia Li, Juvenile versus adult competition, J. Math. Biol. 29 (1991), no. 5, 457–473. MR 1110716, 10.1007/BF00160472
  • [14] M. M. Dekshenieks, E. E. Hofmann, J. M. Klinck, and E. N. Powell, Modeling the vertical distribution of oyster larvae in response to environmental conditions, Mar. Ecol. Prog. Ser. 136 (1996), 97-110.
  • [15] P. Domenici, C. Lefrancois, and A. Shingles, Hypoxia and the antipredator behaviours of fishes, Phil. Strans. R. Soc. B 362 (2007), 2105-2121.
  • [16] K. F. Gaines, D. E. Porter, T. Punshon, and I. L. Brisbin Jr., A spatially explicit model of the wild hog for ecological risk assessment activities at the Department of Energy's Savannah River site, Hum. Ecol. Risk Assess. 11 (2005), 567-589.
  • [17] R. S. Hestand, B. E. May, D. P. Schultz, and C. R. Walker, Ecological implications of water levels on plant growth in a shallow water reservoir. Cooperative Investigations of the Florida Game and Fresh Water Fish Commission and the Bureau of Sport Fisheries and Wildlife (1975).
  • [18] S. J. Jacquemin and M. Pyron, Effects of allometry, sex, and river location on morphological variation of freshwater drum Aplodinotus grunniens in the Wabash River, USA, Copeia 4 (2013), 740-749.
  • [19] Katumi Kamioka, Mathematical analysis of an age-structured population model with space-limited recruitment, Math. Biosci. 198 (2005), no. 1, 27–56. MR 2187067, 10.1016/j.mbs.2005.08.006
  • [20] Kosuke Kamizaki and Nobuyuki Kato, Size-structured population models having different nonlocal terms in vital rates, Mem. Grad. Sch. Sci. Eng. Shimane Univ. Ser. B Math. 46 (2013), 1–13. MR 3089227
  • [21] T. D. Mason, The influence of hydrilla infestation and drawdown on the food habits and growth of age-0 largemouth bass in the Atchafalaya River Basin, Louisiana. Louisiana State University and Agricultural and Mechanical College School of Renewable Resources Master's Thesis, 2002.
  • [22] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc. 44 (1926), 98-130.
  • [23] D. L. Nadler and I. G. Zurbenko, Developing a Weibull model extension to estimate cancer latency, ISRN Epidemiology (2011).
  • [24] Normandeau Associates Inc. Entrainment and impingement studies performed at Merrimack generating station from June 2005 through June 2007, Public Service of New Hampshire, 2007.
  • [25] C. S. Owens, R. M. Smart, D. R. Honnell and G. O. Dick, Effects of pH on growth of Salvinia molesta Mitchell, J. Aquat. Plant Manage. 43 (2005), 34-38.
  • [26] K. T. Paynter and M. E. Mallonee, Site-specific growth rates of oysters in Chesapeake Bay and impact of disease, New Perspectives in the Chesapeake System: A Research and Management Partnership. Proceedings of a conference 137 (1990).
  • [27] Delphine Picart and Bedr’eddine Ainseba, Parameter identification in multistage population dynamics model, Nonlinear Anal. Real World Appl. 12 (2011), no. 6, 3315–3328. MR 2832974, 10.1016/j.nonrwa.2011.05.030
  • [28] E. D. Prince and C. P. Goodyear, Hypoxia-based habitat compression of tropical pelagic fishes, Fish. Oceanogr. 15 (2006), 451-464.
  • [29] Michael C. Reed, Why is mathematical biology so hard?, Notices Amer. Math. Soc. 51 (2004), no. 3, 338–342. MR 2034240
  • [30] A. L. Rypel, D. R. Bayne, and J. B. Mitchell, Growth of freshwater drum from lotic and lentic habitats in Alabama, Trans. Am. Fish. Soc. 135 (2006), 987-997.
  • [31] J. Seales, K. Houston, J. Sibley, and E. Thames, Lake Bistineau Lake History & Management Issues. Office of Fisheries Inland Fisheries Section Part VI-B Water Management Plan Series (2013).
  • [32] J. W. Sinko and W. Streifer, A new model for age-size structure of a population, Ecology 48 (1967), 910-918.
  • [33] Joel Smoller, Shock waves and reaction-diffusion equations, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 258, Springer-Verlag, New York, 1994. MR 1301779
  • [34] J. H. Uphoff, M. McGinty, R. Lukacovic, J. Mowrer, and B. Pyle, Impervious surface, summer dissolved oxygen, and fish distribution in Chesapeake Bay subestuaries: linking watershed development, habitat conditions and fisheries management, N. Am. J. Fish. Manage. 31 (2011), 554-566.

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2010): 35Q92, 65M06

Retrieve articles in all journals with MSC (2010): 35Q92, 65M06

Additional Information

Azmy S. Ackleh
Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Lafayette, Louisiana 70504
Email: ackleh@louisiana.edu

Baoling Ma
Affiliation: Department of Mathematics, Millersville University of Pennsylvania, Millersville, Pennsylvania 17551
Email: baoling.ma@millersville.edu

Robert Miller
Affiliation: Engineering Department, C. H. Fenstermaker and Associates, LLC, Lafayette, Louisiana 70508
Email: robert@fenstermaker.com

DOI: https://doi.org/10.1090/qam/1439
Keywords: Structured population model, environment, finite difference approximations, convergence, well-posedness
Received by editor(s): August 2, 2015
Received by editor(s) in revised form: January 20, 2016
Published electronically: June 20, 2016
Additional Notes: This work is supported in part by the National Science Foundation under grant #DMS-1312963.
Article copyright: © Copyright 2016 Brown University

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2016 Brown University
Comments: qam-query@ams.org
AMS Website