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.  doi: 10.1016/S0362-546X(01)00780-5.  Google Scholar

[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.  doi: 10.1016/S0096-3003(99)00002-8.  Google Scholar

[3]

A. S. Ackleh and K. Deng, A monotone approximation for the nonautonomous size-structured population model,, Quarterly of Applied Mathematics, 57 (1999), 261.   Google Scholar

[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.  doi: 10.1080/17513758.2010.498924.  Google Scholar

[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.  doi: 10.1090/conm/513/10072.  Google Scholar

[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.  doi: 10.1080/01630569708816798.  Google Scholar

[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.  doi: 10.1897/03-438.  Google Scholar

[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.  doi: 10.1007/BF00176377.  Google Scholar

[9]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws,, Mathematical of Computation, 34 (1980), 1.  doi: 10.1090/S0025-5718-1980-0551288-3.  Google Scholar

[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.  doi: 10.1080/713610016.  Google Scholar

[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.  doi: 10.1007/BF00168004.  Google Scholar

[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.  doi: 10.1002/ieam.68.  Google Scholar

[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.  doi: 10.1007/BF00275908.  Google Scholar

[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.  doi: 10.1016/0304-3800(83)90019-4.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2013.05.018.  Google Scholar

[17]

M. Liu, Survival analysis of a cooperation system with random perturbations in a polluted environment,, Nonlinear Analysis: Hybrid System, 18 (2015), 100.  doi: 10.1016/j.nahs.2015.06.005.  Google Scholar

[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, 45 (2012), 1541.   Google Scholar

[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.  doi: 10.1007/BF00168006.  Google Scholar

[20]

Z. Ma, G. Cui and W. Wang, Persistence and extinction of a population in a polluted environment,, Mathematical Biosciences, 101 (1990), 75.  doi: 10.1016/0025-5564(90)90103-6.  Google Scholar

[21]

D. Mackay and A. Fraser, Bioaccumulation of persistent organic chemicals: Mechanisms and models,, Environmental Pollution, 110 (2000), 375.  doi: 10.1016/S0269-7491(00)00162-7.  Google Scholar

[22]

J. A. J. Metz and O. Diekmann, eds., The Dynamics of Physiologically Structured Populations,, Lecture Notes in Biomathematics, (1986).  doi: 10.1007/978-3-662-13159-6.  Google Scholar

[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.  doi: 10.1006/jmaa.2000.6853.  Google Scholar

[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, (2001).  doi: 10.1201/9781420032321.  Google Scholar

[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.  doi: 10.1080/713610017.  Google Scholar

[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.  doi: 10.1137/050638126.  Google Scholar

[27]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[28]

H. R. Thieme, Mathematics in Population Biology,, Princeton Series in Theoretical and Computational Biology, (2003).   Google Scholar

[29]

D. M. Thomas, T. W. Snell and S. M. Jaffar, A control problem in a polluted environment,, Mathematical Biosciences, 133 (1996), 139.  doi: 10.1016/0025-5564(95)00091-7.  Google Scholar

[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.  doi: 10.1007/s11538-015-0086-4.  Google Scholar

[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).   Google Scholar

[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: , ().   Google Scholar

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.  doi: 10.1016/S0362-546X(01)00780-5.  Google Scholar

[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.  doi: 10.1016/S0096-3003(99)00002-8.  Google Scholar

[3]

A. S. Ackleh and K. Deng, A monotone approximation for the nonautonomous size-structured population model,, Quarterly of Applied Mathematics, 57 (1999), 261.   Google Scholar

[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.  doi: 10.1080/17513758.2010.498924.  Google Scholar

[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.  doi: 10.1090/conm/513/10072.  Google Scholar

[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.  doi: 10.1080/01630569708816798.  Google Scholar

[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.  doi: 10.1897/03-438.  Google Scholar

[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.  doi: 10.1007/BF00176377.  Google Scholar

[9]

M. G. Crandall and A. Majda, Monotone difference approximations for scalar conservation laws,, Mathematical of Computation, 34 (1980), 1.  doi: 10.1090/S0025-5718-1980-0551288-3.  Google Scholar

[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.  doi: 10.1080/713610016.  Google Scholar

[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.  doi: 10.1007/BF00168004.  Google Scholar

[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.  doi: 10.1002/ieam.68.  Google Scholar

[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.  doi: 10.1007/BF00275908.  Google Scholar

[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.  doi: 10.1016/0304-3800(83)90019-4.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2013.05.018.  Google Scholar

[17]

M. Liu, Survival analysis of a cooperation system with random perturbations in a polluted environment,, Nonlinear Analysis: Hybrid System, 18 (2015), 100.  doi: 10.1016/j.nahs.2015.06.005.  Google Scholar

[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, 45 (2012), 1541.   Google Scholar

[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.  doi: 10.1007/BF00168006.  Google Scholar

[20]

Z. Ma, G. Cui and W. Wang, Persistence and extinction of a population in a polluted environment,, Mathematical Biosciences, 101 (1990), 75.  doi: 10.1016/0025-5564(90)90103-6.  Google Scholar

[21]

D. Mackay and A. Fraser, Bioaccumulation of persistent organic chemicals: Mechanisms and models,, Environmental Pollution, 110 (2000), 375.  doi: 10.1016/S0269-7491(00)00162-7.  Google Scholar

[22]

J. A. J. Metz and O. Diekmann, eds., The Dynamics of Physiologically Structured Populations,, Lecture Notes in Biomathematics, (1986).  doi: 10.1007/978-3-662-13159-6.  Google Scholar

[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.  doi: 10.1006/jmaa.2000.6853.  Google Scholar

[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, (2001).  doi: 10.1201/9781420032321.  Google Scholar

[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.  doi: 10.1080/713610017.  Google Scholar

[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.  doi: 10.1137/050638126.  Google Scholar

[27]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, Springer, (1994).  doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[28]

H. R. Thieme, Mathematics in Population Biology,, Princeton Series in Theoretical and Computational Biology, (2003).   Google Scholar

[29]

D. M. Thomas, T. W. Snell and S. M. Jaffar, A control problem in a polluted environment,, Mathematical Biosciences, 133 (1996), 139.  doi: 10.1016/0025-5564(95)00091-7.  Google Scholar

[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.  doi: 10.1007/s11538-015-0086-4.  Google Scholar

[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).   Google Scholar

[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: , ().   Google Scholar

[1]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[2]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, 2021, 20 (1) : 359-368. doi: 10.3934/cpaa.2020270

[3]

Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320

[4]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[5]

Tien-Yu Lin, Bhaba R. Sarker, Chien-Jui Lin. An optimal setup cost reduction and lot size for economic production quantity model with imperfect quality and quantity discounts. Journal of Industrial & Management Optimization, 2021, 17 (1) : 467-484. doi: 10.3934/jimo.2020043

[6]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[7]

Karoline Disser. Global existence and uniqueness for a volume-surface reaction-nonlinear-diffusion system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 321-330. doi: 10.3934/dcdss.2020326

[8]

Helmut Abels, Johannes Kampmann. Existence of weak solutions for a sharp interface model for phase separation on biological membranes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 331-351. doi: 10.3934/dcdss.2020325

[9]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[10]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[11]

Manuel Friedrich, Martin Kružík, Jan Valdman. Numerical approximation of von Kármán viscoelastic plates. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 299-319. doi: 10.3934/dcdss.2020322

[12]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384

[13]

Cheng Peng, Zhaohui Tang, Weihua Gui, Qing Chen, Jing He. A bidirectional weighted boundary distance algorithm for time series similarity computation based on optimized sliding window size. Journal of Industrial & Management Optimization, 2021, 17 (1) : 205-220. doi: 10.3934/jimo.2019107

[14]

Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120

[15]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[16]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[17]

Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351

[18]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[19]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[20]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

2018 Impact Factor: 1.313

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]