December  2012, 7(4): 691-703. doi: 10.3934/nhm.2012.7.691

PDE problems arising in mathematical biology

1. 

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

Received  March 2012 Revised  September 2012 Published  December 2012

This article reviews biological processes that can be modeled by PDEs, it describes mathematical results, and suggests open problems. The first topic deals with tumor growth. This is modeled as a free boundary problem for a coupled system of elliptic, hyperbolic and parabolic equations. Existence theorems, stability of radially symmetric stationary solutions, and symmetry-breaking bifurcation results are stated. Next, a free boundary problem for wound healing is described, again involving a coupled system of PDEs. Other topics include movement of molecules in a neuron, modeled as a system of reaction-hyperbolic equations, and competition for resources, modeled as a system of reaction-diffusion equations.
Citation: Avner Friedman. PDE problems arising in mathematical biology. Networks and Heterogeneous Media, 2012, 7 (4) : 691-703. doi: 10.3934/nhm.2012.7.691
References:
[1]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh, 137A (2007), 497-518. doi: 10.1017/S0308210506000047.

[2]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math Biosc. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17.

[3]

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.

[4]

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.

[5]

X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. doi: 10.1007/s00285-008-0166-2.

[6]

G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322. doi: 10.1016/j.jtbi.2005.04.018.

[7]

S. Cui, Existence of a stationary solution for the modified Ward-King tumor growth model, Adv. in Appl. Math., 36 (2006), 421-446. doi: 10.1016/j.aam.2005.04.002.

[8]

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.

[9]

J. Dockery, V. Huston, K. Mischaikow and M. Pernarowsky, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.

[10]

M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Anal., 35 (2003), 187-206.

[11]

A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces & Free Bound., 8 (2006), 247-261. doi: 10.4171/IFB/142.

[12]

A. Friedman, A multiscale tumor model, Interfaces & Free Bound., 10 (2008), 245-262. doi: 10.4171/IFB/188.

[13]

A. Friedman, Free boundary value problems associated with multiscale tumor models, Mathematical Modeling of Natural Phenomena, 4 (2009), 134-155. doi: 10.1051/mmnp/20094306.

[14]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in tumor model, Arch. Rat. Mech. Anal, 180 (2006), 293-330. doi: 10.1007/s00205-005-0408-z.

[15]

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.

[16]

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.

[17]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, Math. Anal & Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.

[18]

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.

[19]

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.

[20]

A. Friedman and B. Hu, The role of oxygen in tissue maintenance: Mathematical modeling and qualitative analysis, Math. Mod. Meth. Appl. Sci, 18 (2008), 1-33. doi: 10.1142/S021820250800308X.

[21]

A. Friedman, B. Hu and C-Y Kao, Cell cycle control at the first restriction point and its effect on tissue growth, J. Math. Biol., 60 (2010), 881-907. doi: 10.1007/s00285-009-0290-7.

[22]

A. Friedman, B. Hu and J. P. Keener, The diffusion approximation for linear non-autonomous reaction-hyperbolic equations, to appear.

[23]

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.

[24]

A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation, Disc. Cont. Dynam. Syst., to appear.

[25]

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.

[26]

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.

[27]

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.

[28]

C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal, Disc. Cont. Dynam. Sys. Series A., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.

[29]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y-L Chuang, X. Li, P. Macklin, S. M. Wise and V. Christini, Nonlinear modeling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), 1-91. doi: 10.1088/0951-7715/23/1/001.

[30]

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.

[31]

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.

[32]

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.

[33]

C. Xue, A. Friedmand and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.

show all references

References:
[1]

R. S. Cantrell, C. Cosner and Y. Lou, Advection-mediated coexistence of competing species, Proc. Roy. Soc. Edinburgh, 137A (2007), 497-518. doi: 10.1017/S0308210506000047.

[2]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and ideal free distribution, Math Biosc. Eng., 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17.

[3]

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.

[4]

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.

[5]

X. Chen, R. Hambrock and Y. Lou, Evolution of conditional dispersal: A reaction-diffusion-advection model, J. Math. Biol., 57 (2008), 361-386. doi: 10.1007/s00285-008-0166-2.

[6]

G. Craciun, A. Brown and A. Friedman, A dynamical system model of neurofilament transport in axons, J. Theoret. Biol., 237 (2005), 316-322. doi: 10.1016/j.jtbi.2005.04.018.

[7]

S. Cui, Existence of a stationary solution for the modified Ward-King tumor growth model, Adv. in Appl. Math., 36 (2006), 421-446. doi: 10.1016/j.aam.2005.04.002.

[8]

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.

[9]

J. Dockery, V. Huston, K. Mischaikow and M. Pernarowsky, The evolution of slow dispersal rates: A reaction-diffusion model, J. Math. Biol., 37 (1998), 61-83. doi: 10.1007/s002850050120.

[10]

M. A. Fontelos and A. Friedman, Symmetry-breaking bifurcations of free boundary problems in three dimensions, Asymptotic Anal., 35 (2003), 187-206.

[11]

A. Friedman, A free boundary problem for a coupled system of elliptic, hyperbolic, and Stokes equations modeling tumor growth, Interfaces & Free Bound., 8 (2006), 247-261. doi: 10.4171/IFB/142.

[12]

A. Friedman, A multiscale tumor model, Interfaces & Free Bound., 10 (2008), 245-262. doi: 10.4171/IFB/188.

[13]

A. Friedman, Free boundary value problems associated with multiscale tumor models, Mathematical Modeling of Natural Phenomena, 4 (2009), 134-155. doi: 10.1051/mmnp/20094306.

[14]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem arising in tumor model, Arch. Rat. Mech. Anal, 180 (2006), 293-330. doi: 10.1007/s00205-005-0408-z.

[15]

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.

[16]

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.

[17]

A. Friedman and B. Hu, Bifurcation from stability to instability for a free boundary problem modeling tumor growth by Stokes equation, Math. Anal & Appl., 327 (2007), 643-664. doi: 10.1016/j.jmaa.2006.04.034.

[18]

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.

[19]

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.

[20]

A. Friedman and B. Hu, The role of oxygen in tissue maintenance: Mathematical modeling and qualitative analysis, Math. Mod. Meth. Appl. Sci, 18 (2008), 1-33. doi: 10.1142/S021820250800308X.

[21]

A. Friedman, B. Hu and C-Y Kao, Cell cycle control at the first restriction point and its effect on tissue growth, J. Math. Biol., 60 (2010), 881-907. doi: 10.1007/s00285-009-0290-7.

[22]

A. Friedman, B. Hu and J. P. Keener, The diffusion approximation for linear non-autonomous reaction-hyperbolic equations, to appear.

[23]

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.

[24]

A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation, Disc. Cont. Dynam. Syst., to appear.

[25]

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.

[26]

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.

[27]

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.

[28]

C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs. non-local dispersal, Disc. Cont. Dynam. Sys. Series A., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.

[29]

J. S. Lowengrub, H. B. Frieboes, F. Jin, Y-L Chuang, X. Li, P. Macklin, S. M. Wise and V. Christini, Nonlinear modeling of cancer: Bridging the gap between cells and tumors, Nonlinearity, 23 (2010), 1-91. doi: 10.1088/0951-7715/23/1/001.

[30]

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.

[31]

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.

[32]

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.

[33]

C. Xue, A. Friedmand and C. K. Sen, A mathematical model of ischemic cutaneous wounds, PNAS, 106 (2009), 16782-16787.

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