Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-28T17:57:31.907Z Has data issue: false hasContentIssue false
  • English
  • Français

The asymptotic behaviour of an invasion process

Published online by Cambridge University Press:  14 July 2016

F. P. Kelly
F. P. Kelly*
Affiliation:
University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

Black and white cells are positioned at the vertices of a rectangular lattice. When a cell division occurs, the daughter cells are of the same colour as the parent cell; one of them replaces an adjacent cell and the other remains in the position of the parent cell. In one variant of the model it is assumed that whenever a white cell appears at the origin it is transformed into a black cell; apart from this the black and white cells are equally competitive and in particular they divide at the same rate. Initially, only the cell at the origin is black. The asymptotic behaviour of the black clone is investigated.

Keywords

SPATIAL PROCESSINVASIONTUMOUR GROWTHRANDOM WALK
Type
Short Communications
Information
Journal of Applied Probability , Volume 14 , Issue 3 , September 1977 , pp. 584 - 590
Copyright
Copyright © Applied Probability Trust 1977 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athreya, K. and Ney, P. (1972) Branching Processes. Springer-Verlag, New York.CrossRefGoogle Scholar
Carslaw, H. S. and Jaeger, J. C. (1959) Conduction of Heat in Solids , 2nd edn. Clarendon Press, Oxford.Google Scholar
Clifford, P. and Sudbury, A. (1973) A model for spatial conflict. Biometrika 60, 581–588.CrossRefGoogle Scholar
Downham, D. Y. and Morgan, R. K. B. (1973a) Growth of abnormal cells. Nature 242, 528–530.CrossRefGoogle ScholarPubMed
Downham, D. Y. and Morgan, R. K. B. (1973b) A stochastic model for two-dimensional growth on a square lattice. Bull. Inst. Int. Statist. 45, 324–329.Google Scholar
Erdös, P. and Dvoretsky, A. (1951) Some problems on random walk in space. Proc. 2nd Berkeley Symp. Math. Statist. Prob. , 353–367.Google Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications , Vol. I, 3rd edn. Wiley, New York.Google Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications , Vol. II, 2nd edn. Wiley, New York.Google Scholar
Kelly, F. P. (1976) Stochastic models with electrical analogues. Math. Proc. Camb. Phil. Soc. 80, 145–151.CrossRefGoogle Scholar
Mollison, D. (1972) Conjecture on the spread of infection in two dimensions disproved. Nature 240, 467–468.CrossRefGoogle ScholarPubMed
Mollison, D. (1974) Percolation processes and tumour growth. Adv. Appl. Prob. 6, 233–235.CrossRefGoogle Scholar
Mollison, D. (1977) Spatial contact models for ecological and epidemiological spread. J. R. Statist. Soc. B 39.Google Scholar
Richardson, D. (1973) Random growth in a tesselation. Proc. Camb. Phil. Soc. 74, 515–528.CrossRefGoogle Scholar
Schürger, K. and Tautu, P. (1976) A Markovian configuration model for carcinogenesis. In Mathematical Models in Medicine, Lecture Notes in Biomathematics , 11, Springer-Verlag, New York. 92–108.CrossRefGoogle Scholar
Sudbury, A. (1976) The size of the region occupied by one type in an invasion process. J. Appl. Prob. 13, 355–356.CrossRefGoogle Scholar
Williams, T. and Bjerknes, R. (1971) Hyperplasia: the spread of abnormal cells through a plane lattice. Adv. Appl. Prob. 3, 210–211.CrossRefGoogle Scholar
Williams, T. and Bjerknes, R. (1972) Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 19–21.CrossRefGoogle ScholarPubMed