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]

Lin Shi, Xuemin Wang, Dingshi Li. Limiting behavior of non-autonomous stochastic reaction-diffusion equations with colored noise on unbounded thin domains. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5367-5386. doi: 10.3934/cpaa.2020242

[2]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[3]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[4]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[5]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

[6]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[7]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

[8]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[9]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[10]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020267

[11]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[12]

Hua Qiu, Zheng-An Yao. The regularized Boussinesq equations with partial dissipations in dimension two. Electronic Research Archive, 2020, 28 (4) : 1375-1393. doi: 10.3934/era.2020073

[13]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[14]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[15]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[16]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[17]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[18]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[19]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[20]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

2019 Impact Factor: 1.053

Metrics

  • PDF downloads (98)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]