September  2012, 32(9): 3081-3097. doi: 10.3934/dcds.2012.32.3081

Conservation laws in mathematical biology

1. 

The Ohio State University, Department of Mathematics, Columbus, OH 43210

Received  September 2011 Revised  March 2012 Published  April 2012

Many mathematical models in biology can be described by conservation laws of the form \begin{equation}\tag{0.1} \frac{\partial{\bf{u}}}{\partial t} + \rm{div}(V{\bf{u}})=F(t,{\bf{x}}, {\bf{u}})\quad ({\bf{x}}=(x_1,\dots, x_n)) \end{equation} where ${\bf{u}}={\bf{u}}(t,{\bf{x}})$ is a vector $(u_1,\dots,u_k)$, ${\bf{F}}$ is a vector $(F_1,\dots,F_k)$, $V$ is a matrix with elements $V_{ij}(t,{\bf{x}},{\bf{u}})$, and $F_i(t,{\bf{x}}, {\bf{u}})$, $V_{ij}(t,{\bf{x}}, {\bf{u}})$ are nonlinear and/or non-local functions of ${\bf{u}}$. From a mathematical point of view one would like to establish, first of all, the existence and uniqueness of solutions under some prescribed initial (and possibly also boundary) conditions. However, the more interesting questions relate to establishing properties of the solutions that are of biological interest.
    In this article we give examples of biological processes whose mathematical models are represented in the form (0.1). We describe results and present open problems.
Citation: Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081
References:
[1]

B. V. Bazaliy and A. Friedman, A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth,, Comm. in PDE, 28 (2003), 517.  doi: 10.1081/PDE-120020486.  Google Scholar

[2]

B. Bazaliy and A. Friedman, Global existence and stability for an elliptic-parabolic free boundary problem: An application to a model with tumor growth,, Indiana Univ. Math. J., 52 (2003), 1265.  doi: 10.1512/iumj.2003.52.2317.  Google Scholar

[3]

A. Brown, L. Wang and P. Jung, Stochastic simulation of neurofilament transport in axon: 'Stop and go' hypothesis,, Molec. Biol. Cell, 16 (2005), 4243.  doi: 10.1091/mbc.E05-02-0141.  Google Scholar

[4]

D. S. Burgess, Pharmacodynamic principles of antimicrobial therapy in the prevention of resistance,, Chest, 115 (1999).  doi: 10.1378/chest.115.suppl_1.19S.  Google Scholar

[5]

H. M. Byrne, M. A. J. Chaplain, D. L. Evans and I. Hopkinson, Mathematical modelling of angiogenesis in wound healing: Comparison of theory and experiment,, J. Theor. Med., 2 (2000), 175.  doi: 10.1080/10273660008833045.  Google Scholar

[6]

X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior,, Trans. Amer. Math. Soc., 357 (2005), 4771.  doi: 10.1090/S0002-9947-05-03784-0.  Google Scholar

[7]

X. Chen and A. Friedman, A free boundary problem for elliptic-hyperbolic system: An application to tumor growth,, SIAM J. Math. Anal., 35 (2003), 974.  doi: 10.1137/S0036141002418388.  Google Scholar

[8]

G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axon,, J. Theor. Biol., 237 (2005), 316.  doi: 10.1016/j.jtbi.2005.04.018.  Google Scholar

[9]

S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: An application to a model of tumor growth,, Trans. Amer. Math. Soc., 355 (2003), 3537.  doi: 10.1090/S0002-9947-03-03137-4.  Google Scholar

[10]

S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth,, Interfaces & Free Boundaries, 5 (2003), 159.  doi: 10.4171/IFB/76.  Google Scholar

[11]

E. M. C. D'Agata, M. A. Horn and G. F. Webb, The impact of persistent gastrointestinal colonization on the transmission dynamics of vancomycin-resistant enterococci,, The Journal of Infectious Diseases, 185 (2002), 766.  doi: 10.1086/339293.  Google Scholar

[12]

E. M. C. D'Agata, G. F. Webb and M. A. Horn, A mathematical model quantifying the impact of antibiotic exposure and other interventions on the endemic prevalence of vancomycin-resistant enterococci,, The Journal of Infectious Diseases, 192 (2005), 2004.  doi: 10.1086/498041.  Google Scholar

[13]

M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions,, Asymptotic Anal., 35 (2003), 187.   Google Scholar

[14]

A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth,, Interfaces & Free Boundaries, 8 (2006), 247.  doi: 10.4171/IFB/142.  Google Scholar

[15]

A. Friedman and G. Craciun, A model of intracellular transport of particles in an axon,, J. Math. Biol., 51 (2005), 217.  doi: 10.1007/s00285-004-0285-3.  Google Scholar

[16]

A. Friedman and G. Craciun, Approximate traveling waves in linear reaction-hyperbolic equations,, SIAM J. Math. Anal., 38 (2006), 741.  doi: 10.1137/050637947.  Google Scholar

[17]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model,, J. Diff. Eqs., 227 (2006), 598.  doi: 10.1016/j.jde.2005.09.008.  Google Scholar

[18]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model,, Arch Rat. Mech. Anal., 180 (2006), 293.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[19]

A. Friedman and B. Hu, Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems,, Indiana Univ. Math. J., 56 (2007), 2133.  doi: 10.1512/iumj.2007.56.3044.  Google Scholar

[20]

A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation,, SIAM J. Math. Anal., 39 (2007), 174.  doi: 10.1137/060656292.  Google Scholar

[21]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation,, J. Math. Anal. Appl., 327 (2007), 643.  doi: 10.1016/j.jmaa.2006.04.034.  Google Scholar

[22]

A. Friedman and B. Hu, Stability and instability of Liapounov-Schmidt and Hopf bifurcations for a free boundary problem arising in a tumor model,, Trans. Amer. Math. Soc., 360 (2008), 5291.  doi: 10.1090/S0002-9947-08-04468-1.  Google Scholar

[23]

A. Friedman, B. Hu and J. P. Keener, The diffusion approximation for linear non-autonomous reaction-hyperbolic equations,, submitted., ().   Google Scholar

[24]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds,, SIAM J. Math. Anal., 42 (2010), 2013.  doi: 10.1137/090772630.  Google Scholar

[25]

A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation,, Discrete and Continuous Dynamical Systems Series B, ().   Google Scholar

[26]

A. Friedman, C.-Y. Kao and C.-W. Shih, Asymptotic phases in a cell differentiation model,, J. Diff. Eqs., 247 (2009), 736.  doi: 10.1016/j.jde.2009.03.033.  Google Scholar

[27]

A. Friedman, C.-Y. Kao and C.-W. Shih, Transcriptional control in cell differentiation: Asymptotic limits,, submitted., ().   Google Scholar

[28]

A. Friedman and F. Reitich, Analysis of a mathematical model for growth of tumors,, J. Math. Biol., 38 (1999), 262.  doi: 10.1007/s002850050149.  Google Scholar

[29]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth,, Trans. Amer. Math. Soc., 353 (2001), 1587.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[30]

A. Friedman and C. Xue, A mathematical model for chronic wounds,, Mathematical Biosciences and Engineering, 8 (2011), 253.  doi: 10.3934/mbe.2011.8.253.  Google Scholar

[31]

A. Friedman, N. Ziyadi and K. Boushaba, A model of drug resistance with infection by health care workers,, Math. Biosciences and Engineering, 7 (2010), 779.  doi: 10.3934/mbe.2010.7.779.  Google Scholar

[32]

L. Mariani, M. Lohning, A. Radbruch and T. Hofer, Transcriptional control networks of cell differentiation: Insights from helper T lymphocytes,, Biophys. Mol. Biol., 86 (2004), 45.  doi: 10.1016/j.pbiomolbio.2004.02.007.  Google Scholar

[33]

L. R. Peterson, Squeezing the antibiotic balloon: The impact of antimicrobial classes on emerging resistance,, Clin. Microbiol. Infect. 11 Suppl., 5 (2005), 4.   Google Scholar

[34]

G. J. Pettet, H. M. Byrne, D. L. S. McElwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue,, Mathematical Biosciences, 136 (1996), 35.  doi: 10.1016/0025-5564(96)00044-2.  Google Scholar

[35]

G. Pettet, M. A. J. Chaplain, D. L. S. McElwain and H. M. Byrne, On the role of angiogenesis in wound healing,, Proc. R. Soc. Lond. B, 263 (1996), 1487.  doi: 10.1098/rspb.1996.0217.  Google Scholar

[36]

G. J. Pettet, C. P. Please, M. J. Tindall and D. L. S. McElwain, The migration of cells in multicell tumor spheroids,, Bull. Math. Biol., 63 (2001), 231.  doi: 10.1006/bulm.2000.0217.  Google Scholar

[37]

M. C. Reed, S. Venakides and J. J. Blum, Approximate traveling waves in linear reaction-hyperbolic equations,, SIAM J. Appl. Math., 50 (1990), 167.  doi: 10.1137/0150011.  Google Scholar

[38]

F. Salvarani and J. L. Vazquez, The diffusive limit for Carleman-type kinetic models,, Nonlinearity, 18 (2005), 1223.  doi: 10.1088/0951-7715/18/3/015.  Google Scholar

[39]

R. C. Schugart, A. Friedman, R. Zhao and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model,, PNAS, 105 (2008), 2628.  doi: 10.1073/pnas.0711642105.  Google Scholar

[40]

G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic-resistant bacterial epidemics in hospitals,, PNAS, 102 (2005), 13343.  doi: 10.1073/pnas.0504053102.  Google Scholar

[41]

J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues,, SIAM J. Math. Anal., 41 (2009), 391.  doi: 10.1137/080726550.  Google Scholar

[42]

C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds,, PNAS, 106 (2009), 16782.  doi: 10.1073/pnas.0909115106.  Google Scholar

[43]

A. Yates, R. Callard and J. Stark, Combining cytokine signaling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: A model for cellular decision making,, J. Theor. Biol., 231 (2004), 181.  doi: 10.1016/j.jtbi.2004.06.013.  Google Scholar

show all references

References:
[1]

B. V. Bazaliy and A. Friedman, A free boundary problem for an elliptic-parabolic system: Application to a model of tumor growth,, Comm. in PDE, 28 (2003), 517.  doi: 10.1081/PDE-120020486.  Google Scholar

[2]

B. Bazaliy and A. Friedman, Global existence and stability for an elliptic-parabolic free boundary problem: An application to a model with tumor growth,, Indiana Univ. Math. J., 52 (2003), 1265.  doi: 10.1512/iumj.2003.52.2317.  Google Scholar

[3]

A. Brown, L. Wang and P. Jung, Stochastic simulation of neurofilament transport in axon: 'Stop and go' hypothesis,, Molec. Biol. Cell, 16 (2005), 4243.  doi: 10.1091/mbc.E05-02-0141.  Google Scholar

[4]

D. S. Burgess, Pharmacodynamic principles of antimicrobial therapy in the prevention of resistance,, Chest, 115 (1999).  doi: 10.1378/chest.115.suppl_1.19S.  Google Scholar

[5]

H. M. Byrne, M. A. J. Chaplain, D. L. Evans and I. Hopkinson, Mathematical modelling of angiogenesis in wound healing: Comparison of theory and experiment,, J. Theor. Med., 2 (2000), 175.  doi: 10.1080/10273660008833045.  Google Scholar

[6]

X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior,, Trans. Amer. Math. Soc., 357 (2005), 4771.  doi: 10.1090/S0002-9947-05-03784-0.  Google Scholar

[7]

X. Chen and A. Friedman, A free boundary problem for elliptic-hyperbolic system: An application to tumor growth,, SIAM J. Math. Anal., 35 (2003), 974.  doi: 10.1137/S0036141002418388.  Google Scholar

[8]

G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axon,, J. Theor. Biol., 237 (2005), 316.  doi: 10.1016/j.jtbi.2005.04.018.  Google Scholar

[9]

S. Cui and A. Friedman, A free boundary problem for a singular system of differential equations: An application to a model of tumor growth,, Trans. Amer. Math. Soc., 355 (2003), 3537.  doi: 10.1090/S0002-9947-03-03137-4.  Google Scholar

[10]

S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth,, Interfaces & Free Boundaries, 5 (2003), 159.  doi: 10.4171/IFB/76.  Google Scholar

[11]

E. M. C. D'Agata, M. A. Horn and G. F. Webb, The impact of persistent gastrointestinal colonization on the transmission dynamics of vancomycin-resistant enterococci,, The Journal of Infectious Diseases, 185 (2002), 766.  doi: 10.1086/339293.  Google Scholar

[12]

E. M. C. D'Agata, G. F. Webb and M. A. Horn, A mathematical model quantifying the impact of antibiotic exposure and other interventions on the endemic prevalence of vancomycin-resistant enterococci,, The Journal of Infectious Diseases, 192 (2005), 2004.  doi: 10.1086/498041.  Google Scholar

[13]

M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions,, Asymptotic Anal., 35 (2003), 187.   Google Scholar

[14]

A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth,, Interfaces & Free Boundaries, 8 (2006), 247.  doi: 10.4171/IFB/142.  Google Scholar

[15]

A. Friedman and G. Craciun, A model of intracellular transport of particles in an axon,, J. Math. Biol., 51 (2005), 217.  doi: 10.1007/s00285-004-0285-3.  Google Scholar

[16]

A. Friedman and G. Craciun, Approximate traveling waves in linear reaction-hyperbolic equations,, SIAM J. Math. Anal., 38 (2006), 741.  doi: 10.1137/050637947.  Google Scholar

[17]

A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model,, J. Diff. Eqs., 227 (2006), 598.  doi: 10.1016/j.jde.2005.09.008.  Google Scholar

[18]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in a tumor model,, Arch Rat. Mech. Anal., 180 (2006), 293.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[19]

A. Friedman and B. Hu, Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems,, Indiana Univ. Math. J., 56 (2007), 2133.  doi: 10.1512/iumj.2007.56.3044.  Google Scholar

[20]

A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation,, SIAM J. Math. Anal., 39 (2007), 174.  doi: 10.1137/060656292.  Google Scholar

[21]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation,, J. Math. Anal. Appl., 327 (2007), 643.  doi: 10.1016/j.jmaa.2006.04.034.  Google Scholar

[22]

A. Friedman and B. Hu, Stability and instability of Liapounov-Schmidt and Hopf bifurcations for a free boundary problem arising in a tumor model,, Trans. Amer. Math. Soc., 360 (2008), 5291.  doi: 10.1090/S0002-9947-08-04468-1.  Google Scholar

[23]

A. Friedman, B. Hu and J. P. Keener, The diffusion approximation for linear non-autonomous reaction-hyperbolic equations,, submitted., ().   Google Scholar

[24]

A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds,, SIAM J. Math. Anal., 42 (2010), 2013.  doi: 10.1137/090772630.  Google Scholar

[25]

A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation,, Discrete and Continuous Dynamical Systems Series B, ().   Google Scholar

[26]

A. Friedman, C.-Y. Kao and C.-W. Shih, Asymptotic phases in a cell differentiation model,, J. Diff. Eqs., 247 (2009), 736.  doi: 10.1016/j.jde.2009.03.033.  Google Scholar

[27]

A. Friedman, C.-Y. Kao and C.-W. Shih, Transcriptional control in cell differentiation: Asymptotic limits,, submitted., ().   Google Scholar

[28]

A. Friedman and F. Reitich, Analysis of a mathematical model for growth of tumors,, J. Math. Biol., 38 (1999), 262.  doi: 10.1007/s002850050149.  Google Scholar

[29]

A. Friedman and F. Reitich, Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth,, Trans. Amer. Math. Soc., 353 (2001), 1587.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[30]

A. Friedman and C. Xue, A mathematical model for chronic wounds,, Mathematical Biosciences and Engineering, 8 (2011), 253.  doi: 10.3934/mbe.2011.8.253.  Google Scholar

[31]

A. Friedman, N. Ziyadi and K. Boushaba, A model of drug resistance with infection by health care workers,, Math. Biosciences and Engineering, 7 (2010), 779.  doi: 10.3934/mbe.2010.7.779.  Google Scholar

[32]

L. Mariani, M. Lohning, A. Radbruch and T. Hofer, Transcriptional control networks of cell differentiation: Insights from helper T lymphocytes,, Biophys. Mol. Biol., 86 (2004), 45.  doi: 10.1016/j.pbiomolbio.2004.02.007.  Google Scholar

[33]

L. R. Peterson, Squeezing the antibiotic balloon: The impact of antimicrobial classes on emerging resistance,, Clin. Microbiol. Infect. 11 Suppl., 5 (2005), 4.   Google Scholar

[34]

G. J. Pettet, H. M. Byrne, D. L. S. McElwain and J. Norbury, A model of wound-healing angiogenesis in soft tissue,, Mathematical Biosciences, 136 (1996), 35.  doi: 10.1016/0025-5564(96)00044-2.  Google Scholar

[35]

G. Pettet, M. A. J. Chaplain, D. L. S. McElwain and H. M. Byrne, On the role of angiogenesis in wound healing,, Proc. R. Soc. Lond. B, 263 (1996), 1487.  doi: 10.1098/rspb.1996.0217.  Google Scholar

[36]

G. J. Pettet, C. P. Please, M. J. Tindall and D. L. S. McElwain, The migration of cells in multicell tumor spheroids,, Bull. Math. Biol., 63 (2001), 231.  doi: 10.1006/bulm.2000.0217.  Google Scholar

[37]

M. C. Reed, S. Venakides and J. J. Blum, Approximate traveling waves in linear reaction-hyperbolic equations,, SIAM J. Appl. Math., 50 (1990), 167.  doi: 10.1137/0150011.  Google Scholar

[38]

F. Salvarani and J. L. Vazquez, The diffusive limit for Carleman-type kinetic models,, Nonlinearity, 18 (2005), 1223.  doi: 10.1088/0951-7715/18/3/015.  Google Scholar

[39]

R. C. Schugart, A. Friedman, R. Zhao and C. K. Sen, Wound angiogenesis as a function of tissue oxygen tension: A mathematical model,, PNAS, 105 (2008), 2628.  doi: 10.1073/pnas.0711642105.  Google Scholar

[40]

G. F. Webb, E. M. C. D'Agata, P. Magal and S. Ruan, A model of antibiotic-resistant bacterial epidemics in hospitals,, PNAS, 102 (2005), 13343.  doi: 10.1073/pnas.0504053102.  Google Scholar

[41]

J. Wu and S. Cui, Asymptotic stability of stationary solutions of a free boundary problem modeling the growth of tumors with fluid tissues,, SIAM J. Math. Anal., 41 (2009), 391.  doi: 10.1137/080726550.  Google Scholar

[42]

C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds,, PNAS, 106 (2009), 16782.  doi: 10.1073/pnas.0909115106.  Google Scholar

[43]

A. Yates, R. Callard and J. Stark, Combining cytokine signaling with T-bet and GATA-3 regulation in Th1 and Th2 differentiation: A model for cellular decision making,, J. Theor. Biol., 231 (2004), 181.  doi: 10.1016/j.jtbi.2004.06.013.  Google Scholar

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