Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-24T19:44:07.850Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic models of grouping changes

Published online by Cambridge University Press:  01 July 2016

Byron J. T. Morgan
Byron J. T. Morgan*
Affiliation:
University of Kent
Get access Rights & Permissions [Opens in a new window]

Abstract

Several simple stochastic models are given for a finite closed system of individuals existing in clusters which may come together to form larger clusters which may in turn split up. Some of these models are analysed and compared in equilibrium. Several of the models fit into a general framework established and investigated by Whittle (1965a); it is shown that these models have an equilibrium solution of a particularly simple form, deduced by Whittle, if and only if the models are stochastically reversible. A normal approximation to two of the models in equilibrium is found to give the same mean value as a deterministic approximation.

Keywords

CLUSTERING-SPLITTING PROCESSSTOCHASTIC REVERSIBILITYNORMAL APPROXIMATION
Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 1 , March 1976 , pp. 30 - 57
Copyright
Copyright © Applied Probability Trust 1976 

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

Bailey, N. T. J. (1963) The simple stochastic epidemic: a complete solution in terms of known functions. Biometrika 50, 235–240.Google Scholar
Barton, J. A. (1973) Mathematical models of group forming behaviour. , University of Kent.Google Scholar
Bithell, J. F. (1969) A stochastic model for the breaking of molecular segments. J. Appl. Prob. 6, 59–73.Google Scholar
Boulière, F. (1963) Observations on the ecology of some large African mammals. African Ecology and Human Evolution 36, 43–73.Google Scholar
Bundy, B. D. (1970) The growth of elephant herds. Math. Gazette 54, 38–40.Google Scholar
Buss, I. O. (1961) Some observations on food habits and behaviour of the African elephant. J. Wildlife Mgmt. 25, 131–148.Google Scholar
Charlesby, A. (1954) Molecular-weight changes in the degradation of long-chain polymers. Proc. Roy. Soc. A 224, 120–128.Google Scholar
Cohen, J. E. (1971) Casual Groups of Monkeys and Men. Harvard University Press, Cambridge, Mass.Google Scholar
Cohen, J. E. (1972) Markov population processes as models of primate societies and population dynamics. Theoret. Pop. Biol. 3, 119–134.Google Scholar
Cohen, M. H. and Robertson, A. (1971) Wave propagation in the early stages of aggregation of cellular slime moulds. J. Theoret. Biol. 31, 101–118.Google Scholar
Coleman, J. S. and James, J. (1961) The equilibrium size distribution of freely forming groups. Sociometry 24, 36–45.Google Scholar
Cox, D. R. (1962) Renewal Theory. Methuen, London.Google Scholar
Erdös, P., Guy, R. K. and Moon, J. W. (1975) On refining partitions. J. Lond. Math. Soc. 9, 565–570.Google Scholar
Fielding, G. T. (1970) A further note on elephant herds. Math. Gazette 54, 297–298.CrossRefGoogle Scholar
Flory, P. J. (1953) Principles of Polymer Chemistry. Cornell University Press.Google Scholar
Hillaby, J. (1961) Elephants as a pest control problem. New Scientist 12, 736–738.Google Scholar
Holgate, P. (1967) The size of elephant herds. Math. Gazette 51, 302–304.Google Scholar
James, J. (1953) The distribution of free-forming small group size. Amer. Soc. Rev. 18, 569–570.Google Scholar
Keller, E. F. and Segel, L. A. (1970) Initiation of slime mold aggregation viewed as an instability. J. Theoret. Biol. 26, 399–415.Google Scholar
Kendall, D. G. (1959) Unitary dilations of one-parameter semi-groups of Markov transition operators and the corresponding representations for Markov processes with a countable infinity of states. Proc. Lond. Math. Soc. 7, 417–431.Google Scholar
Kingman, J. F. C. (1969) Markov population processes. J. Appl. Prob. 6, 1–18.CrossRefGoogle Scholar
Montroll, E. W. (1950) Markov chains and excluded volume effect in polymer chains. J. Chem. Phys. 18, 734–743.Google Scholar
Montroll, E. W. and Simha, R. (1940) Theory of depolymerization of long chain molecules. J. Chem. Phys. 8, 721–727.Google Scholar
Morgan, B. J. T. (1974) On the distribution of inanimate marks over a linear birth-and-death process. J. Appl. Prob. 11, 423–436.Google Scholar
McQuarrie, D. A. (1967) Stochastic approach to chemical kinetics. J. Appl. Prob. 4, 413–478.Google Scholar
Newell, G. F. and Montroll, E. W. (1963) On the theory of the Ising model of ferromagnetism. Rev. Mod. Phys. 25, 353–389.CrossRefGoogle Scholar
Prendiville, B. J. (1949) Discussion of Symposium on Stochastic Processes. J. R. Statist. Soc. B 11, 273.Google Scholar
Reich, E. (1957) Waiting times when queues are in tandem. Ann. Math. Statist. 28, 768–773.Google Scholar
Rushton, S. and Lang, E. D. (1954) Tables of the confluent hypergeometric function. Sankhyā 13, 377–411.Google Scholar
Sagan, H. (1961) Boundary and Eigenvalue Problems in Mathematical Physics. Wiley, New York.Google Scholar
Slater, L. J. (1960) Confluent Hypergeometric Functions. Cambridge University Press.Google Scholar
Stewart, D. R. M. and Stewart, J. (1963) The distribution of some large mammals in Kenya. J. East. Afr. Nat. Hist. Soc. 24, 1–52.Google Scholar
Struhsaker, T. T. (1967) Social structure among Vervet monkeys. Behaviour 29, 83–121.Google Scholar
Takashima, M. (1956) Note on evolutionary processes. Bull. Math. Statist. 7, 18–24.CrossRefGoogle Scholar
Van Lawick-Goodall, J. (1968) The behaviour of free-living chimpanzees in the Gombe Stream Reserve. Animal Behaviour Monographs 1, 161–301.Google Scholar
Wagner, S. S. and Altmann, S. A. (1973) What time do baboons come down from the trees? (An estimation problem) Biometrics 29, 623–636.Google Scholar
White, H. C. (1962) Chance models of systems of casual groups. Sociometry 25, 153–172.Google Scholar
Whittaker, E. T. and Watson, G. N. (1920) A Course of Modern Analysis. Cambridge University Press.Google Scholar
Whittle, P. (1957) On the use of the normal approximation in the treatment of stochastic processes. J. R. Statist. Soc. B 19, 268–281.Google Scholar
Whittle, P. (1965a) Statistical processes of aggregation and polymerisation. Proc. Camb. Phil. Soc. 61, 475–495.Google Scholar
Whittle, P. (1965b) The equilibrium statistics of a clustering process in the uncondensed phase. Proc. Roy. Soc. A 285, 501–519.Google Scholar