# American Institute of Mathematical Sciences

2016, 13(4): 697-722. doi: 10.3934/mbe.2016015

## 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, Canada 2 Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB T6G 2G1

Received  September 2015 Revised  January 2016 Published  May 2016

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. Mathematical Biosciences & Engineering, 2016, 13 (4) : 697-722. doi: 10.3934/mbe.2016015
##### References:
 [1] A. S. Ackleh, H. T. Banks and K. Deng, A finite difference approximation for a coupled system of nonlinear size-structured populations, Nonlinear Analysis, 50 (2002), 727-748. doi: 10.1016/S0362-546X(01)00780-5. [2] A. S. Ackleh and K. Deng, A monotone approximation for a nonlinear nonautonomous size-structured population model, Applied Mathematics and Computation, 108 (2000), 103-113. doi: 10.1016/S0096-3003(99)00002-8. [3] A. S. Ackleh and K. Deng, A monotone approximation for the nonautonomous size-structured population model, Quarterly of Applied Mathematics, 57 (1999), 261-267. [4] A. S. Ackleh, K. Deng and Q. Huang, Stochastic juvenile-adult models with application to a green tree frog population, Journal of Biological Dynamics, 5 (2011), 64-83. doi: 10.1080/17513758.2010.498924. [5] A. S. Ackleh, K. Deng and Q. Huang, Existence-uniqueness results and difference approximations for an amphibian juvenile-adult model, AMS Series in Contemporary Mathematics, Nonlinear Analysis and Optimization, 513 (2010), 1-23. doi: 10.1090/conm/513/10072. [6] A. S. Ackleh and K. Ito, An implicit finite difference scheme for the nonlinear size-structured population model, Numerical Functional Analysis and Optimization, 18 (1997), 865-884. doi: 10.1080/01630569708816798. [7] J. A. Arnot and F. A. Gobas, A food web bioaccumulation model for organic chemicals in aquatic ecosystems, Environmental Toxicology and Chemistry, 23 (2004), 2343-2355. doi: 10.1897/03-438. [8] A. Calsina and J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, Journal of Mathematical Biology, 33 (1995), 335-364. doi: 10.1007/BF00176377. [9] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Mathematical of Computation, 34 (1980), 1-21. doi: 10.1090/S0025-5718-1980-0551288-3. [10] S. M. Bartell, R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and C. Mackay, Realism and relevance of ecological models used in chemical risk assessment, Human and Ecological Risk Assessment, 9 (2003), 907-938. doi: 10.1080/713610016. [11] H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in single-species and predator-prey systems, Journal of Mathematical Biology, 30 (1991), 15-30. doi: 10.1007/BF00168004. [12] N. Galic, U. Hommen, J. H. Baveco and P. J. van den Brink, Potential application of population models in the European ecological risk assessment of chemical II: Review of models and their popential to address environmental protection aims, Integrated Environmental Assessment and Management, 6 (2010), 338-360. doi: 10.1002/ieam.68. [13] T. G. Hallam, C. E. Clark and G. S. Jordan, Effect of toxicants on populations: A qualitative approach. II. First order kinetics, Journal of Mathematical Biology, 18 (1983), 25-37. doi: 10.1007/BF00275908. [14] T. G. Hallam and C. E. Clark, Effect of toxicants on populations: A qualitative approach. I. Equilibrium environmental exposure, Ecological Modelling, 18 (1983), 291-304. doi: 10.1016/0304-3800(83)90019-4. [15] de J. T. Luna and T. G. Hallam, Effect of toxicants on populations: A qualitative approach. IV. Resource-consumer-toxiocant models, Ecological Modelling, 35 (1987), 249-273. [16] Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. A. Lewis, A model for the impact of contaminants on fish population dynamics, Journal of Theoretical Biology, 334 (2013), 71-79. doi: 10.1016/j.jtbi.2013.05.018. [17] M. Liu, Survival analysis of a cooperation system with random perturbations in a polluted environment, Nonlinear Analysis: Hybrid System, 18 (2015), 100-116. doi: 10.1016/j.nahs.2015.06.005. [18] M. Liu and K. Wang, Persistence and extincion of a single-species population system in a polluted environment with random perturbations and inpulsive toxicant input, Chaos, Solitions and Fractals, 45 (2012), 1541-1550. [19] H. Liu and Z. Ma, The threshold of survival for system of two species in a polluted environment, Journal of Mathematical Biology, 30 (1991), 49-61. doi: 10.1007/BF00168006. [20] Z. Ma, G. Cui and W. Wang, Persistence and extinction of a population in a polluted environment, Mathematical Biosciences, 101 (1990), 75-97. doi: 10.1016/0025-5564(90)90103-6. [21] D. Mackay and A. Fraser, Bioaccumulation of persistent organic chemicals: Mechanisms and models, Environmental Pollution, 110 (2000), 375-391. doi: 10.1016/S0269-7491(00)00162-7. [22] J. A. J. Metz and O. Diekmann, eds., The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. [23] J. Pan, Z. Jin and Z. Ma, Threshold of survival for a $n$-dimensional Volterra mutualistic system in a polluted environment, Journal of Mathematical Analysis and Applications, 252 (2000), 519-531. doi: 10.1006/jmaa.2000.6853. [24] R. A. Pastorok, S. M. Bartell, S. Ferson and L. R. Ginzburg, Ecological Modeling in Risk Assessment: Chemical Effects on Populations, Ecosystems, and Landscapes, Lewis Publishers, Boca Raton, FL, USA, 2001. doi: 10.1201/9781420032321. [25] R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and S. M. Bartell, Role of ecological modeling in risk assessment, Human and Ecological Risk Assessment, 9 (2003), 939-972. doi: 10.1080/713610017. [26] J. Shen, C. W. Shu and M. Zhang, High resolution schemes for a hierarchical size-structured model, SIAM Journal on Numerical Analysis, 45 (2007), 352-370. doi: 10.1137/050638126. [27] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [28] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, 2003. [29] D. M. Thomas, T. W. Snell and S. M. Jaffar, A control problem in a polluted environment, Mathematical Biosciences, 133 (1996), 139-163. doi: 10.1016/0025-5564(95)00091-7. [30] Y. Zhao, S, Yuan and J, Ma, Survival and stationary distribution analysis of a stochastic competition model of three species in a polluted environment, Bulletin of Mathematical Biology, 77 (2015), 1285-1326. doi: 10.1007/s11538-015-0086-4. [31] Canadian Council of Ministers of the Environment, 2003b., Canadian water quality guidelines for the protection of aquatic life: Inorganic mercury and methylmercury. 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] U. S. National Archives and Records Administration, Code of Federal Regulations, Title 40- Protection of Environment, Appendix A to part 423-126 priority pollutants, 2013,, Available from: , ().

show all references

##### References:
 [1] A. S. Ackleh, H. T. Banks and K. Deng, A finite difference approximation for a coupled system of nonlinear size-structured populations, Nonlinear Analysis, 50 (2002), 727-748. doi: 10.1016/S0362-546X(01)00780-5. [2] A. S. Ackleh and K. Deng, A monotone approximation for a nonlinear nonautonomous size-structured population model, Applied Mathematics and Computation, 108 (2000), 103-113. doi: 10.1016/S0096-3003(99)00002-8. [3] A. S. Ackleh and K. Deng, A monotone approximation for the nonautonomous size-structured population model, Quarterly of Applied Mathematics, 57 (1999), 261-267. [4] A. S. Ackleh, K. Deng and Q. Huang, Stochastic juvenile-adult models with application to a green tree frog population, Journal of Biological Dynamics, 5 (2011), 64-83. doi: 10.1080/17513758.2010.498924. [5] A. S. Ackleh, K. Deng and Q. Huang, Existence-uniqueness results and difference approximations for an amphibian juvenile-adult model, AMS Series in Contemporary Mathematics, Nonlinear Analysis and Optimization, 513 (2010), 1-23. doi: 10.1090/conm/513/10072. [6] A. S. Ackleh and K. Ito, An implicit finite difference scheme for the nonlinear size-structured population model, Numerical Functional Analysis and Optimization, 18 (1997), 865-884. doi: 10.1080/01630569708816798. [7] J. A. Arnot and F. A. Gobas, A food web bioaccumulation model for organic chemicals in aquatic ecosystems, Environmental Toxicology and Chemistry, 23 (2004), 2343-2355. doi: 10.1897/03-438. [8] A. Calsina and J. Saldana, A model of physiologically structured population dynamics with a nonlinear individual growth rate, Journal of Mathematical Biology, 33 (1995), 335-364. doi: 10.1007/BF00176377. [9] M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws, Mathematical of Computation, 34 (1980), 1-21. doi: 10.1090/S0025-5718-1980-0551288-3. [10] S. M. Bartell, R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and C. Mackay, Realism and relevance of ecological models used in chemical risk assessment, Human and Ecological Risk Assessment, 9 (2003), 907-938. doi: 10.1080/713610016. [11] H. I. Freedman and J. B. Shukla, Models for the effect of toxicant in single-species and predator-prey systems, Journal of Mathematical Biology, 30 (1991), 15-30. doi: 10.1007/BF00168004. [12] N. Galic, U. Hommen, J. H. Baveco and P. J. van den Brink, Potential application of population models in the European ecological risk assessment of chemical II: Review of models and their popential to address environmental protection aims, Integrated Environmental Assessment and Management, 6 (2010), 338-360. doi: 10.1002/ieam.68. [13] T. G. Hallam, C. E. Clark and G. S. Jordan, Effect of toxicants on populations: A qualitative approach. II. First order kinetics, Journal of Mathematical Biology, 18 (1983), 25-37. doi: 10.1007/BF00275908. [14] T. G. Hallam and C. E. Clark, Effect of toxicants on populations: A qualitative approach. I. Equilibrium environmental exposure, Ecological Modelling, 18 (1983), 291-304. doi: 10.1016/0304-3800(83)90019-4. [15] de J. T. Luna and T. G. Hallam, Effect of toxicants on populations: A qualitative approach. IV. Resource-consumer-toxiocant models, Ecological Modelling, 35 (1987), 249-273. [16] Q. Huang, L. Parshotam, H. Wang, C. Bampfylde and M. A. Lewis, A model for the impact of contaminants on fish population dynamics, Journal of Theoretical Biology, 334 (2013), 71-79. doi: 10.1016/j.jtbi.2013.05.018. [17] M. Liu, Survival analysis of a cooperation system with random perturbations in a polluted environment, Nonlinear Analysis: Hybrid System, 18 (2015), 100-116. doi: 10.1016/j.nahs.2015.06.005. [18] M. Liu and K. Wang, Persistence and extincion of a single-species population system in a polluted environment with random perturbations and inpulsive toxicant input, Chaos, Solitions and Fractals, 45 (2012), 1541-1550. [19] H. Liu and Z. Ma, The threshold of survival for system of two species in a polluted environment, Journal of Mathematical Biology, 30 (1991), 49-61. doi: 10.1007/BF00168006. [20] Z. Ma, G. Cui and W. Wang, Persistence and extinction of a population in a polluted environment, Mathematical Biosciences, 101 (1990), 75-97. doi: 10.1016/0025-5564(90)90103-6. [21] D. Mackay and A. Fraser, Bioaccumulation of persistent organic chemicals: Mechanisms and models, Environmental Pollution, 110 (2000), 375-391. doi: 10.1016/S0269-7491(00)00162-7. [22] J. A. J. Metz and O. Diekmann, eds., The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68, Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-662-13159-6. [23] J. Pan, Z. Jin and Z. Ma, Threshold of survival for a $n$-dimensional Volterra mutualistic system in a polluted environment, Journal of Mathematical Analysis and Applications, 252 (2000), 519-531. doi: 10.1006/jmaa.2000.6853. [24] R. A. Pastorok, S. M. Bartell, S. Ferson and L. R. Ginzburg, Ecological Modeling in Risk Assessment: Chemical Effects on Populations, Ecosystems, and Landscapes, Lewis Publishers, Boca Raton, FL, USA, 2001. doi: 10.1201/9781420032321. [25] R. A. Pastorok, H. R. Akcakaya, H. Regan, S. Ferson and S. M. Bartell, Role of ecological modeling in risk assessment, Human and Ecological Risk Assessment, 9 (2003), 939-972. doi: 10.1080/713610017. [26] J. Shen, C. W. Shu and M. Zhang, High resolution schemes for a hierarchical size-structured model, SIAM Journal on Numerical Analysis, 45 (2007), 352-370. doi: 10.1137/050638126. [27] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [28] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational Biology, 2003. [29] D. M. Thomas, T. W. Snell and S. M. Jaffar, A control problem in a polluted environment, Mathematical Biosciences, 133 (1996), 139-163. doi: 10.1016/0025-5564(95)00091-7. [30] Y. Zhao, S, Yuan and J, Ma, Survival and stationary distribution analysis of a stochastic competition model of three species in a polluted environment, Bulletin of Mathematical Biology, 77 (2015), 1285-1326. doi: 10.1007/s11538-015-0086-4. [31] Canadian Council of Ministers of the Environment, 2003b., Canadian water quality guidelines for the protection of aquatic life: Inorganic mercury and methylmercury. 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] U. S. National Archives and Records Administration, Code of Federal Regulations, Title 40- Protection of Environment, Appendix A to part 423-126 priority pollutants, 2013,, Available from: , ().
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