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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.


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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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