# American Institute of Mathematical Sciences

January  2012, 11(1): 229-241. doi: 10.3934/cpaa.2012.11.229

## Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations

 1 Department of Mechanics and Mathematics, Franko Lviv National University, Lviv 79000, Ukraine 2 University of Heidelberg, Interdisciplinary Center for Scientific Computing (IWR), Institute of Applied Mathematics and BIOQUANT, Im Neuenheimer Feld 267, 69120 Heidelberg, Germany 3 Department of Mathematics I, RWTH Aachen University, Wüllnerstr. 5b, 52056 Aachen, Germany

Received  March 2010 Revised  January 2011 Published  September 2011

In this paper we study pattern formation arising in a system of a single reaction-diffusion equation coupled with subsystem of ordinary differential equations, describing spatially-distributed growth of clonal populations of precancerous cells, whose proliferation is controlled by growth factors diffusing in the extracellular medium and binding to the cell surface. We extend the results on the existence of nonhomogenous stationary solutions obtained in [9] to a general Hill-type production function and full parameter set. Using spectral analysis and perturbation theory we derive conditions for the linearized stability of such spatial patterns.
Citation: Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229
##### References:
 [1] V. I. Arnold, "Ordinary Differential Equations,", MIT Press, (1978).   Google Scholar [2] K. I. Chueh, C. Conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations,, Ind. Univ. Math. J., 26 (1977), 373.  doi: 10.1512/iumj.1977.26.26029.  Google Scholar [3] A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach,, Phys. D, 122 (1998), 1.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [4] A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Indiana Univ. Math. J., 50 (2001), 443.  doi: 10.1512/iumj.2001.50.1873.  Google Scholar [5] D. Henry, "Geomertic Theory of Semilinear Parabolic Equations,", Springer-Verlag, (1981).   Google Scholar [6] T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1966).   Google Scholar [7] A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells,, Math. Mod. Meth. Appl. Sci., 17 (2007), 1693.  doi: 10.1142/S0218202507002443.  Google Scholar [8] A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signalling along linear and surface structures in very early tumours,, Comp. Math. Meth. Med., 7 (2006), 189.  doi: 10.1080/10273660600969091.  Google Scholar [9] A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells,, {Math. Model. Nat.Phenom.}, 3 (2008), 90.  doi: 10.1051/mmnp:2008043.  Google Scholar [10] J. D. Murray, "Mathematical Biology,", Springer-Verlag, (2003).   Google Scholar [11] P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology,", John Wiley & Sons, (1999).   Google Scholar [12] F. Rothe, "Global Solutions of Reaction-Diffusion Systems,", Springer-Verlag, (1994).   Google Scholar [13] J. Smoller, "Shock-Waves and Reaction-Diffusion Equations,", Springer-Verlag, (1994).   Google Scholar [14] A. M. Turing, The chemical basis of morphogenesis,, {Phil. Trans. Roy. Soc. B}, 237 (1952), 37.  doi: 10.1098/rstb.1952.0012.  Google Scholar [15] J. Wei, On the interior spike layer solutions for some singular perturbation problems,, { Proc. Royal Soc. Edinb.}, 128A (1998), 849.   Google Scholar

show all references

##### References:
 [1] V. I. Arnold, "Ordinary Differential Equations,", MIT Press, (1978).   Google Scholar [2] K. I. Chueh, C. Conley and J. Smoller, Positively invariant regions for systems of nonlinear diffusion equations,, Ind. Univ. Math. J., 26 (1977), 373.  doi: 10.1512/iumj.1977.26.26029.  Google Scholar [3] A. Doelman, R. A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: A matched asymptotics approach,, Phys. D, 122 (1998), 1.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar [4] A. Doelman, R. A. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, Indiana Univ. Math. J., 50 (2001), 443.  doi: 10.1512/iumj.2001.50.1873.  Google Scholar [5] D. Henry, "Geomertic Theory of Semilinear Parabolic Equations,", Springer-Verlag, (1981).   Google Scholar [6] T. Kato, "Perturbation Theory for Linear Operators,", Springer-Verlag, (1966).   Google Scholar [7] A. Marciniak-Czochra and M. Kimmel, Modelling of early lung cancer progression: Influence of growth factor production and cooperation between partially transformed cells,, Math. Mod. Meth. Appl. Sci., 17 (2007), 1693.  doi: 10.1142/S0218202507002443.  Google Scholar [8] A. Marciniak-Czochra and M. Kimmel, Dynamics of growth and signalling along linear and surface structures in very early tumours,, Comp. Math. Meth. Med., 7 (2006), 189.  doi: 10.1080/10273660600969091.  Google Scholar [9] A. Marciniak-Czochra and M. Kimmel, Reaction-diffusion model of early carcinogenesis: The effects of influx of mutated cells,, {Math. Model. Nat.Phenom.}, 3 (2008), 90.  doi: 10.1051/mmnp:2008043.  Google Scholar [10] J. D. Murray, "Mathematical Biology,", Springer-Verlag, (2003).   Google Scholar [11] P. K. Maini, In "On Growth and Form. Spatio-Temporal Pattern Formation in Biology,", John Wiley & Sons, (1999).   Google Scholar [12] F. Rothe, "Global Solutions of Reaction-Diffusion Systems,", Springer-Verlag, (1994).   Google Scholar [13] J. Smoller, "Shock-Waves and Reaction-Diffusion Equations,", Springer-Verlag, (1994).   Google Scholar [14] A. M. Turing, The chemical basis of morphogenesis,, {Phil. Trans. Roy. Soc. B}, 237 (1952), 37.  doi: 10.1098/rstb.1952.0012.  Google Scholar [15] J. Wei, On the interior spike layer solutions for some singular perturbation problems,, { Proc. Royal Soc. Edinb.}, 128A (1998), 849.   Google Scholar
 [1] Cheng-Hsiung Hsu, Jian-Jhong Lin. Stability analysis of traveling wave solutions for lattice reaction-diffusion equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020001 [2] Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255 [3] Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031 [4] 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 [5] Hongyan Zhang, Siyu Liu, Yue Zhang. Dynamics and spatiotemporal pattern formations of a homogeneous reaction-diffusion Thomas model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1149-1164. doi: 10.3934/dcdss.2017062 [6] Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133 [7] L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630 [8] Costică Moroşanu. Stability and errors analysis of two iterative schemes of fractional steps type associated to a nonlinear reaction-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-21. doi: 10.3934/dcdss.2020089 [9] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 [10] Zhao-Xing Yang, Guo-Bao Zhang, Ge Tian, Zhaosheng Feng. Stability of non-monotone non-critical traveling waves in discrete reaction-diffusion equations with time delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 581-603. doi: 10.3934/dcdss.2017029 [11] Yicheng Jiang, Kaijun Zhang. Stability of traveling waves for nonlocal time-delayed reaction-diffusion equations. Kinetic & Related Models, 2018, 11 (5) : 1235-1253. doi: 10.3934/krm.2018048 [12] Wei-Jie Sheng, Wan-Tong Li. Multidimensional stability of time-periodic planar traveling fronts in bistable reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2681-2704. doi: 10.3934/dcds.2017115 [13] Shi-Liang Wu, Tong-Chang Niu, Cheng-Hsiung Hsu. Global asymptotic stability of pushed traveling fronts for monostable delayed reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3467-3486. doi: 10.3934/dcds.2017147 [14] Xiaojie Hou, Yi Li. Local stability of traveling-wave solutions of nonlinear reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 681-701. doi: 10.3934/dcds.2006.15.681 [15] Abraham Solar. Stability of non-monotone and backward waves for delay non-local reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5799-5823. doi: 10.3934/dcds.2019255 [16] 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 [17] 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 [18] 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 [19] 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 [20] Tomás Caraballo, José A. Langa, James C. Robinson. Stability and random attractors for a reaction-diffusion equation with multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 875-892. doi: 10.3934/dcds.2000.6.875

2018 Impact Factor: 0.925

## Metrics

• PDF downloads (16)
• HTML views (0)
• Cited by (1)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]