Article Contents
Article Contents

# PDE problems arising in mathematical biology

• 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.
Mathematics Subject Classification: Primary: 49J10, 35Q92, 35R35, 92C50; Secondary: 35J47, 35K57, 35L40, 35M30.

 Citation:

•  [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.