# American Institute of Mathematical Sciences

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 $$\tag{0.1} \frac{\partial{\bf{u}}}{\partial t} + \rm{div}(V{\bf{u}})=F(t,{\bf{x}}, {\bf{u}})\quad ({\bf{x}}=(x_1,\dots, x_n))$$ 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 and Continuous Dynamical Systems, 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), 517-560. doi: 10.1081/PDE-120020486. [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), 1265-1304. doi: 10.1512/iumj.2003.52.2317. [3] A. Brown, L. Wang and P. Jung, Stochastic simulation of neurofilament transport in axon: 'Stop and go' hypothesis, Molec. Biol. Cell, 16 (2005), 4243-4255. doi: 10.1091/mbc.E05-02-0141. [4] D. S. Burgess, Pharmacodynamic principles of antimicrobial therapy in the prevention of resistance, Chest, 115 (1999), 19S-23S. doi: 10.1378/chest.115.suppl_1.19S. [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), 175-197. doi: 10.1080/10273660008833045. [6] X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Amer. Math. Soc., 357 (2005), 4771-4804. doi: 10.1090/S0002-9947-05-03784-0. [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), 974-986. doi: 10.1137/S0036141002418388. [8] G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axon, J. Theor. Biol., 237 (2005), 316-322. doi: 10.1016/j.jtbi.2005.04.018. [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), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4. [10] S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces & Free Boundaries, 5 (2003), 159-181. doi: 10.4171/IFB/76. [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), 766-773. doi: 10.1086/339293. [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), 2004-2011. doi: 10.1086/498041. [13] M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Anal., 35 (2003), 187-206. [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), 247-261. doi: 10.4171/IFB/142. [15] A. Friedman and G. Craciun, A model of intracellular transport of particles in an axon, J. Math. Biol., 51 (2005), 217-246. doi: 10.1007/s00285-004-0285-3. [16] A. Friedman and G. Craciun, Approximate traveling waves in linear reaction-hyperbolic equations, SIAM J. Math. Anal., 38 (2006), 741-758. doi: 10.1137/050637947. [17] A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Diff. Eqs., 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008. [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), 293-330. doi: 10.1007/s00205-005-0408-z. [19] A. Friedman and B. Hu, Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems, Indiana Univ. Math. J., 56 (2007), 2133-2158. doi: 10.1512/iumj.2007.56.3044. [20] A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292. [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), 643-664. doi: 10.1016/j.jmaa.2006.04.034. [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), 5291-5342. doi: 10.1090/S0002-9947-08-04468-1. [23] A. Friedman, B. Hu and J. P. Keener, The diffusion approximation for linear non-autonomous reaction-hyperbolic equations,, submitted., (). [24] A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040. doi: 10.1137/090772630. [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, (). [26] A. Friedman, C.-Y. Kao and C.-W. Shih, Asymptotic phases in a cell differentiation model, J. Diff. Eqs., 247 (2009), 736-769. doi: 10.1016/j.jde.2009.03.033. [27] A. Friedman, C.-Y. Kao and C.-W. Shih, Transcriptional control in cell differentiation: Asymptotic limits,, submitted., (). [28] A. Friedman and F. Reitich, Analysis of a mathematical model for growth of tumors, J. Math. Biol., 38 (1999), 262-284. doi: 10.1007/s002850050149. [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), 1587-1634. doi: 10.1090/S0002-9947-00-02715-X. [30] A. Friedman and C. Xue, A mathematical model for chronic wounds, Mathematical Biosciences and Engineering, 8 (2011), 253-261. doi: 10.3934/mbe.2011.8.253. [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), 779-792. doi: 10.3934/mbe.2010.7.779. [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), 45-76. doi: 10.1016/j.pbiomolbio.2004.02.007. [33] L. R. Peterson, Squeezing the antibiotic balloon: The impact of antimicrobial classes on emerging resistance, Clin. Microbiol. Infect. 11 Suppl., 5 (2005), 4-16. [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), 35-63. doi: 10.1016/0025-5564(96)00044-2. [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), 1487-1493. doi: 10.1098/rspb.1996.0217. [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), 231-257. doi: 10.1006/bulm.2000.0217. [37] M. C. Reed, S. Venakides and J. J. Blum, Approximate traveling waves in linear reaction-hyperbolic equations, SIAM J. Appl. Math., 50 (1990), 167-180. doi: 10.1137/0150011. [38] F. Salvarani and J. L. Vazquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity, 18 (2005), 1223-1248. doi: 10.1088/0951-7715/18/3/015. [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), 2628-2633. doi: 10.1073/pnas.0711642105. [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), 13343-13348. doi: 10.1073/pnas.0504053102. [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), 391-414. doi: 10.1137/080726550. [42] C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787. doi: 10.1073/pnas.0909115106. [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), 181-196. doi: 10.1016/j.jtbi.2004.06.013.

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), 517-560. doi: 10.1081/PDE-120020486. [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), 1265-1304. doi: 10.1512/iumj.2003.52.2317. [3] A. Brown, L. Wang and P. Jung, Stochastic simulation of neurofilament transport in axon: 'Stop and go' hypothesis, Molec. Biol. Cell, 16 (2005), 4243-4255. doi: 10.1091/mbc.E05-02-0141. [4] D. S. Burgess, Pharmacodynamic principles of antimicrobial therapy in the prevention of resistance, Chest, 115 (1999), 19S-23S. doi: 10.1378/chest.115.suppl_1.19S. [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), 175-197. doi: 10.1080/10273660008833045. [6] X. Chen, S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth: Asymptotic behavior, Trans. Amer. Math. Soc., 357 (2005), 4771-4804. doi: 10.1090/S0002-9947-05-03784-0. [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), 974-986. doi: 10.1137/S0036141002418388. [8] G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axon, J. Theor. Biol., 237 (2005), 316-322. doi: 10.1016/j.jtbi.2005.04.018. [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), 3537-3590. doi: 10.1090/S0002-9947-03-03137-4. [10] S. Cui and A. Friedman, A hyperbolic free boundary problem modeling tumor growth, Interfaces & Free Boundaries, 5 (2003), 159-181. doi: 10.4171/IFB/76. [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), 766-773. doi: 10.1086/339293. [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), 2004-2011. doi: 10.1086/498041. [13] M. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Anal., 35 (2003), 187-206. [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), 247-261. doi: 10.4171/IFB/142. [15] A. Friedman and G. Craciun, A model of intracellular transport of particles in an axon, J. Math. Biol., 51 (2005), 217-246. doi: 10.1007/s00285-004-0285-3. [16] A. Friedman and G. Craciun, Approximate traveling waves in linear reaction-hyperbolic equations, SIAM J. Math. Anal., 38 (2006), 741-758. doi: 10.1137/050637947. [17] A. Friedman and B. Hu, Asymptotic stability for a free boundary problem arising in a tumor model, J. Diff. Eqs., 227 (2006), 598-639. doi: 10.1016/j.jde.2005.09.008. [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), 293-330. doi: 10.1007/s00205-005-0408-z. [19] A. Friedman and B. Hu, Uniform convergence for approximate traveling waves in linear reaction-hyperbolic systems, Indiana Univ. Math. J., 56 (2007), 2133-2158. doi: 10.1512/iumj.2007.56.3044. [20] A. Friedman and B. Hu, Bifurcation for a free boundary problem modeling tumor growth by Stokes equation, SIAM J. Math. Anal., 39 (2007), 174-194. doi: 10.1137/060656292. [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), 643-664. doi: 10.1016/j.jmaa.2006.04.034. [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), 5291-5342. doi: 10.1090/S0002-9947-08-04468-1. [23] A. Friedman, B. Hu and J. P. Keener, The diffusion approximation for linear non-autonomous reaction-hyperbolic equations,, submitted., (). [24] A. Friedman, B. Hu and C. Xue, Analysis of a mathematical model of ischemic cutaneous wounds, SIAM J. Math. Anal., 42 (2010), 2013-2040. doi: 10.1137/090772630. [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, (). [26] A. Friedman, C.-Y. Kao and C.-W. Shih, Asymptotic phases in a cell differentiation model, J. Diff. Eqs., 247 (2009), 736-769. doi: 10.1016/j.jde.2009.03.033. [27] A. Friedman, C.-Y. Kao and C.-W. Shih, Transcriptional control in cell differentiation: Asymptotic limits,, submitted., (). [28] A. Friedman and F. Reitich, Analysis of a mathematical model for growth of tumors, J. Math. Biol., 38 (1999), 262-284. doi: 10.1007/s002850050149. [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), 1587-1634. doi: 10.1090/S0002-9947-00-02715-X. [30] A. Friedman and C. Xue, A mathematical model for chronic wounds, Mathematical Biosciences and Engineering, 8 (2011), 253-261. doi: 10.3934/mbe.2011.8.253. [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), 779-792. doi: 10.3934/mbe.2010.7.779. [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), 45-76. doi: 10.1016/j.pbiomolbio.2004.02.007. [33] L. R. Peterson, Squeezing the antibiotic balloon: The impact of antimicrobial classes on emerging resistance, Clin. Microbiol. Infect. 11 Suppl., 5 (2005), 4-16. [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), 35-63. doi: 10.1016/0025-5564(96)00044-2. [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), 1487-1493. doi: 10.1098/rspb.1996.0217. [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), 231-257. doi: 10.1006/bulm.2000.0217. [37] M. C. Reed, S. Venakides and J. J. Blum, Approximate traveling waves in linear reaction-hyperbolic equations, SIAM J. Appl. Math., 50 (1990), 167-180. doi: 10.1137/0150011. [38] F. Salvarani and J. L. Vazquez, The diffusive limit for Carleman-type kinetic models, Nonlinearity, 18 (2005), 1223-1248. doi: 10.1088/0951-7715/18/3/015. [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), 2628-2633. doi: 10.1073/pnas.0711642105. [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), 13343-13348. doi: 10.1073/pnas.0504053102. [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), 391-414. doi: 10.1137/080726550. [42] C. Xue, A. Friedman and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787. doi: 10.1073/pnas.0909115106. [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), 181-196. doi: 10.1016/j.jtbi.2004.06.013.
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