The continuum reaction-diffusion limit of a stochastic cellular growth model

Stephan Luckhaus; Livio Triolo

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2004)

  • Volume: 15, Issue: 3-4, page 215-223
  • ISSN: 1120-6330

Abstract

top
A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered. The asymptotic behavior of the system can be partially described through the analysis of the stationary wave which connects different equilibria.

How to cite

top

Luckhaus, Stephan, and Triolo, Livio. "The continuum reaction-diffusion limit of a stochastic cellular growth model." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 15.3-4 (2004): 215-223. <http://eudml.org/doc/252410>.

@article{Luckhaus2004,
abstract = {A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered. The asymptotic behavior of the system can be partially described through the analysis of the stationary wave which connects different equilibria.},
author = {Luckhaus, Stephan, Triolo, Livio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Tumor growth model; Hydrodynamic limits; Degenerate Reaction-Diffusion system},
language = {eng},
month = {12},
number = {3-4},
pages = {215-223},
publisher = {Accademia Nazionale dei Lincei},
title = {The continuum reaction-diffusion limit of a stochastic cellular growth model},
url = {http://eudml.org/doc/252410},
volume = {15},
year = {2004},
}

TY - JOUR
AU - Luckhaus, Stephan
AU - Triolo, Livio
TI - The continuum reaction-diffusion limit of a stochastic cellular growth model
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2004/12//
PB - Accademia Nazionale dei Lincei
VL - 15
IS - 3-4
SP - 215
EP - 223
AB - A competition-diffusion system, where populations of healthy and malignant cells compete and move on a neutral matrix, is analyzed. A coupled system of degenerate nonlinear parabolic equations is derived through a scaling procedure from the microscopic, Markovian dynamics. The healthy cells move much slower than the malignant ones, such that no diffusion for their density survives in the limit. The malignant cells may locally accumulate, while for the healthy ones an exclusion rule is considered. The asymptotic behavior of the system can be partially described through the analysis of the stationary wave which connects different equilibria.
LA - eng
KW - Tumor growth model; Hydrodynamic limits; Degenerate Reaction-Diffusion system
UR - http://eudml.org/doc/252410
ER -

References

top
  1. CHEN, M.F., From Markov Chains to Non-equilibrium Particle Systems. World Scientific, Singapore1992. Zbl0753.60055MR2091955DOI10.1142/9789812562456
  2. DE MASI, A. - PRESUTTI, E., Mathematical methods for hydrodynamical limits. Lecture Notes in Mathematics, 1501, Springer-Verlag, Berlin-Heidelberg-New York1991. Zbl0754.60122MR1175626
  3. DUNBAR, S., Traveling wave solutions of diffusive Lotka-Volterra equations: a heteroclinic connection in R 4 . Trans. Am. Math. Soc., 286, 1984, 557-594. Zbl0556.35078MR760975DOI10.2307/1999810
  4. DURRETT, R. - LEVIN, S., The importance of being discrete (and spatial). Theor. Population Biol., 46, 1994, 363-394. Zbl0846.92027
  5. DURRETT, R. - NEUHAUSER, C., Particle systems and reaction diffusion equations. Ann. Probab., 22, 1994, 289-333. Zbl0799.60093MR1258879
  6. FIFE, P., Mathematical aspects of reacting and diffusing systems. Lectures Notes in Biomath., 28, Springer-Verlag, Berlin-Heidelberg-New York1978. Zbl0403.92004MR527914
  7. GATENBY, R.A. - GAWLINSKI, E.T., A reaction-diffusion model of cancer invasion. Cancer Res., 56, 1996, 5745-5753. 
  8. GOBRON, T. - SAADA, E. - TRIOLO, L., The competition-diffusion limit of a stochastic growth model. Math. and Comp. Modelling, 37, 2003, 1153-1161. Zbl1046.92027
  9. KENNEDY, C.R. - ARIS, R., Traveling waves in a simple population model involving growth and death. Bull. Math. Biol., 42, 1980, 397-429. Zbl0431.92021MR661329DOI10.1016/S0092-8240(80)80057-7
  10. KIPNIS, C. - LANDIM, C., Scaling limits for interacting particle systems. Springer-Verlag, Berlin-Heidelberg-New York1999. Zbl0927.60002MR1707314
  11. KLAASEN, G.A. - TROY, W.C., The stability of traveling wave front solutions of a reaction-diffusion system. SIAM J. Appl. Math., 41, 1981, 145-167. Zbl0467.35011MR622879DOI10.1137/0141011
  12. LIGGETT, T.M., Interacting Particle Systems. Springer-Verlag, Berlin-Heidelberg-New York1985. Zbl1103.82016MR776231DOI10.1007/978-1-4613-8542-4
  13. MARCHANT, B.P. - NORBURY, J. - PERUMPANANI, A.J., Traveling shock waves arising in a model of malignant invasion. SIAM J. Appl. Math., 60, 2000, 463-476. Zbl0944.34021MR1740255DOI10.1137/S0036139998328034
  14. PERRUT, A., Hydrodynamic limits for a two-species reaction-diffusion process. Annals of Appl. Probab., 10, 2000, 163-191. Zbl1171.60394MR1765207DOI10.1214/aoap/1019737668
  15. PERUMPANANI, A.J. - SHERRATT, J.A. - NORBURY, J. - BYRNE, H.M., A two parameter family of travelling waves with a singular barrier arising from the modelling of extracellular matrix mediated cellular invasion. Physica D, 126, 1999, 145-159. Zbl1001.92523

NotesEmbed ?

top

You must be logged in to post comments.