A Model for Spatial Conflict

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY Two species compete for territory along their mutual boundary. The species are fairly matched and the result of conflict is the invasion by one of the species of territory held by the other. A simple stochastic model for this process is described and rules are given for the calculation, as a function of time, of the probabilities that individual territories and groups of territories are held by a single species. Asymptotic results are given for the pattern of territories held by a single species, and here a remarkable distinction is seen between the cases of one-dimensional and two-dimensional competition and the case of three-dimensional competition. The process of invasion is contrasted with two relatively benign processes, that in which an exchange of territories is arranged and that in which a spatial alternation of species may be beneficial to the community. The implications of the long-term behaviour of such processes are discussed. It is suggested further that, under certain ideal conditions, it may be possible to estimate the duration of a spatial struggle for life by observing at some time the configuration of territories held by each species.

Social structure and the effects of conformity

“…This model has been used in mathematics(Durrett and Steif 1993;Goles and Olivos 1980;Granville 1991;Holley and Liggett 1975;Moran 1994a Moran ,b, 1995Mustafa and Pekev 2004;Steif 1994), computer science(Flocchini et al 1998;Hassin and Peleg 2001;Královic 2001;Nakata et al 1999Nakata et al , 2000Peleg 1998Peleg , 2002, biology(Agur 1987;Agur et al 1988;Clifford and Sudbury 1973), and social psychology(Latané and Nowak 1997;Poljak and Sura 1983).…”

Opinion Dynamics and Learning in Social Networks

“…Most work in this area relies on classical models of interacting particle systems whose motivation comes from statistical mechanics (see [65] and references therein). In these models, opinions are represented by either finitely many discrete values (as in the Ising model, introduced by Glauber [48] and the voter model, introduced independently by Clifford and Sudbury [23] and by Holley and Liggett [60]), or continuous values (as in the DeGroot model [28] and the bounded confidence models of Krause [63] (further developed by Hegselmann and Krause [59], Deffuant, Neau, Amblard and Weisbuch [27] and Weisbuch, Kirman and Herreiner [93])). The opinions of the agent evolve dynamically over time as a function of their neighbors' opinions.…”

Isolation, insertion, and reconstruction: Three strategies to intervene in rumor spread based on supernetwork model

“…Spins present agents with different opinions in Ising model. Based on the Ising model, a number of classic public opinion evolution models start to emerge, that include voter model [4], Sznajd model [5], majority rule model [6][7][8], social impact theory [9], bounded confidence models [10][11][12], CODA model [13][14][15][16], and the gambling model [17,18]. These models simulate the process and final results of public opinion evolution.…”