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Mathematical Biosciences and Engineering

2016, Volume 13, Issue 4: 697-722. doi: 10.3934/mbe.2016015
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A toxin-mediated size-structured population model: Finite difference approximation and well-posedness

  • 1.
    Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1
  • 2.
    Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1
  • Received: 01 September 2015 Accepted: 29 June 2018 Published: 01 May 2016

  • MSC : 35L04, 92F99, 65Z99.

  • The question of the effects of environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. Mathematical models have been applied increasingly to predict the effects of toxins on a variety of ecological processes. Motivated by the fact that individuals with different sizes may have different sensitivities to toxins, we develop a toxin-mediated size-structured model which is given by a system of first order fully nonlinear partial differential equations (PDEs). It is very possible that this work represents the first derivation of a PDE model in the area of ecotoxicology. To solve the model, an explicit finite difference approximation to this PDE system is developed. Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. Numerical examples are provided by numerically solving the PDE model using the finite difference scheme.

    Citation: Qihua Huang, Hao Wang. A toxin-mediated size-structured population model: Finite difference approximation and well-posedness[J]. Mathematical Biosciences and Engineering, 2016, 13(4): 697-722. doi: 10.3934/mbe.2016015

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  • The question of the effects of environmental toxins on ecological communities is of great interest from both environmental and conservational points of view. Mathematical models have been applied increasingly to predict the effects of toxins on a variety of ecological processes. Motivated by the fact that individuals with different sizes may have different sensitivities to toxins, we develop a toxin-mediated size-structured model which is given by a system of first order fully nonlinear partial differential equations (PDEs). It is very possible that this work represents the first derivation of a PDE model in the area of ecotoxicology. To solve the model, an explicit finite difference approximation to this PDE system is developed. Existence-uniqueness of the weak solution to the model is established and convergence of the finite difference approximation to this unique solution is proved. Numerical examples are provided by numerically solving the PDE model using the finite difference scheme.


    [1] Nonlinear Analysis, 50 (2002), 727-748.
    [2] Applied Mathematics and Computation, 108 (2000), 103-113.
    [3] Quarterly of Applied Mathematics, 57 (1999), 261-267.
    [4] Journal of Biological Dynamics, 5 (2011), 64-83.
    [5] Nonlinear Analysis and Optimization, 513 (2010), 1-23.
    [6] Numerical Functional Analysis and Optimization, 18 (1997), 865-884.
    [7] Environmental Toxicology and Chemistry, 23 (2004), 2343-2355.
    [8] Journal of Mathematical Biology, 33 (1995), 335-364.
    [9] Mathematical of Computation, 34 (1980), 1-21.
    [10] Human and Ecological Risk Assessment, 9 (2003), 907-938.
    [11] Journal of Mathematical Biology, 30 (1991), 15-30.
    [12] Integrated Environmental Assessment and Management, 6 (2010), 338-360.
    [13] Journal of Mathematical Biology, 18 (1983), 25-37.
    [14] Ecological Modelling, 18 (1983), 291-304.
    [15] Ecological Modelling, 35 (1987), 249-273.
    [16] Journal of Theoretical Biology, 334 (2013), 71-79.
    [17] Nonlinear Analysis: Hybrid System, 18 (2015), 100-116.
    [18] Chaos, Solitions and Fractals, 45 (2012), 1541-1550.
    [19] Journal of Mathematical Biology, 30 (1991), 49-61.
    [20] Mathematical Biosciences, 101 (1990), 75-97.
    [21] Environmental Pollution, 110 (2000), 375-391.
    [22] Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986.
    [23] Journal of Mathematical Analysis and Applications, 252 (2000), 519-531.
    [24] Lewis Publishers, Boca Raton, FL, USA, 2001.
    [25] Human and Ecological Risk Assessment, 9 (2003), 939-972.
    [26] SIAM Journal on Numerical Analysis, 45 (2007), 352-370.
    [27] Springer, New York, 1994.
    [28] Princeton Series in Theoretical and Computational Biology, 2003.
    [29] Mathematical Biosciences, 133 (1996), 139-163.
    [30] Bulletin of Mathematical Biology, 77 (2015), 1285-1326.
    [31] IN: Canadian environmental quality guidelines, 1999. Canadian Council of Ministers of the Environment, Winnipeg, Manitoba. 6 pp. [Last accessed November 21, 2014], Available from: http://ceqg-rcqe.ccme.ca/download/en/191.
    [32] Available from: https://www.gpo.gov/fdsys/granule/CFR-2012-title40-vol30/CFR-2012-title40-vol30-part423-appA/content-detail.html.

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    2. Azmy S. Ackleh, Robert L. Miller, A numerical method for a nonlinear structured population model with an indefinite growth rate coupled with the environment, 2019, 35, 0749-159X, 2348, 10.1002/num.22418
    3. Azmy S. Ackleh, Robert L. Miller, Second-order finite difference approximation for a nonlinear size-structured population model with an indefinite growth rate coupled with the environment, 2021, 58, 0008-0624, 10.1007/s10092-021-00420-x
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