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 & 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.  doi: 10.1017/S0308210506000047.  Google Scholar

[2]

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

[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.  doi: 10.1090/S0002-9947-05-03784-0.  Google Scholar

[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.  doi: 10.1137/S0036141002418388.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1090/S0002-9947-03-03137-4.  Google Scholar

[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.  doi: 10.1007/s002850050120.  Google Scholar

[10]

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

[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.  doi: 10.4171/IFB/142.  Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[15]

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

[16]

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

[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.  doi: 10.1016/j.jmaa.2006.04.034.  Google Scholar

[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.  doi: 10.1137/060656292.  Google Scholar

[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.  doi: 10.1090/S0002-9947-08-04468-1.  Google Scholar

[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.  doi: 10.1142/S021820250800308X.  Google Scholar

[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.  doi: 10.1007/s00285-009-0290-7.  Google Scholar

[22]

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

[23]

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

[24]

A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation,, Disc. Cont. Dynam. Syst., ().   Google Scholar

[25]

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

[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.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[27]

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

[28]

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

[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.  doi: 10.1088/0951-7715/23/1/001.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/0150011.  Google Scholar

[32]

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

[33]

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

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.  doi: 10.1017/S0308210506000047.  Google Scholar

[2]

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

[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.  doi: 10.1090/S0002-9947-05-03784-0.  Google Scholar

[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.  doi: 10.1137/S0036141002418388.  Google Scholar

[5]

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

[6]

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

[7]

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

[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.  doi: 10.1090/S0002-9947-03-03137-4.  Google Scholar

[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.  doi: 10.1007/s002850050120.  Google Scholar

[10]

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

[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.  doi: 10.4171/IFB/142.  Google Scholar

[12]

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

[13]

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

[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.  doi: 10.1007/s00205-005-0408-z.  Google Scholar

[15]

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

[16]

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

[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.  doi: 10.1016/j.jmaa.2006.04.034.  Google Scholar

[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.  doi: 10.1137/060656292.  Google Scholar

[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.  doi: 10.1090/S0002-9947-08-04468-1.  Google Scholar

[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.  doi: 10.1142/S021820250800308X.  Google Scholar

[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.  doi: 10.1007/s00285-009-0290-7.  Google Scholar

[22]

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

[23]

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

[24]

A. Friedman, B. Hu and C. Xue, A three dimensional model of wound healing: Analysis and computation,, Disc. Cont. Dynam. Syst., ().   Google Scholar

[25]

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

[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.  doi: 10.1090/S0002-9947-00-02715-X.  Google Scholar

[27]

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

[28]

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

[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.  doi: 10.1088/0951-7715/23/1/001.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1137/0150011.  Google Scholar

[32]

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

[33]

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

[1]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039

[2]

Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128

[3]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[4]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020033

[5]

Piermarco Cannarsa, Giuseppe Da Prato. Invariance for stochastic reaction-diffusion equations. Evolution Equations & Control Theory, 2012, 1 (1) : 43-56. doi: 10.3934/eect.2012.1.43

[6]

Martino Prizzi. A remark on reaction-diffusion equations in unbounded domains. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 281-286. doi: 10.3934/dcds.2003.9.281

[7]

Angelo Favini, Atsushi Yagi. Global existence for Laplace reaction-diffusion equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020083

[8]

Peter E. Kloeden, Thomas Lorenz, Meihua Yang. Reaction-diffusion equations with a switched--off reaction zone. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1907-1933. doi: 10.3934/cpaa.2014.13.1907

[9]

Ciprian G. Gal, Mahamadi Warma. Reaction-diffusion equations with fractional diffusion on non-smooth domains with various boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (3) : 1279-1319. doi: 10.3934/dcds.2016.36.1279

[10]

Marek Fila, Hirokazu Ninomiya, Juan-Luis Vázquez. Dirichlet boundary conditions can prevent blow-up in reaction-diffusion equations and systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (1) : 63-74. doi: 10.3934/dcds.2006.14.63

[11]

Jong-Shenq Guo, Yoshihisa Morita. Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 193-212. doi: 10.3934/dcds.2005.12.193

[12]

Jacson Simsen, Mariza Stefanello Simsen, Marcos Roberto Teixeira Primo. Reaction-Diffusion equations with spatially variable exponents and large diffusion. Communications on Pure & Applied Analysis, 2016, 15 (2) : 495-506. doi: 10.3934/cpaa.2016.15.495

[13]

Ming Mei. Stability of traveling wavefronts for time-delayed reaction-diffusion equations. Conference Publications, 2009, 2009 (Special) : 526-535. doi: 10.3934/proc.2009.2009.526

[14]

Antoine Mellet, Jean-Michel Roquejoffre, Yannick Sire. Generalized fronts for one-dimensional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 303-312. doi: 10.3934/dcds.2010.26.303

[15]

Matthieu Alfaro, Thomas Giletti. Varying the direction of propagation in reaction-diffusion equations in periodic media. Networks & Heterogeneous Media, 2016, 11 (3) : 369-393. doi: 10.3934/nhm.2016001

[16]

Wei Wang, Anthony Roberts. Macroscopic discrete modelling of stochastic reaction-diffusion equations on a periodic domain. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 253-273. doi: 10.3934/dcds.2011.31.253

[17]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581

[18]

Ivan Gentil, Bogusław Zegarlinski. Asymptotic behaviour of reversible chemical reaction-diffusion equations. Kinetic & Related Models, 2010, 3 (3) : 427-444. doi: 10.3934/krm.2010.3.427

[19]

Masaharu Taniguchi. Multi-dimensional traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (3) : 1011-1046. doi: 10.3934/dcds.2012.32.1011

[20]

Filipa Caetano, Martin J. Gander, Laurence Halpern, Jérémie Szeftel. Schwarz waveform relaxation algorithms for semilinear reaction-diffusion equations. Networks & Heterogeneous Media, 2010, 5 (3) : 487-505. doi: 10.3934/nhm.2010.5.487

2018 Impact Factor: 0.871

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]