title: Analytic solution of Hubbell's model of local community dynamics.
abstract: Recent theoretical approaches to community structure and dynamics reveal that many large-scale features of community structure (such as species-rank distributions and species-area relations) can be explained by a so-called neutral model. Using this approach, species are taken to be equivalent and trophic relations are not taken into account explicitly. Here we provide a general analytic solution to the local community model of Hubbell's neutral theory of biodiversity by recasting it as an urn model, i.e. a Markovian description of states and their transitions. Both stationary and time-dependent distributions are analysed. The stationary distribution-also called the zero-sum multinomial-is given in closed form. An approximate form for the time-dependence is obtained by using an expansion of the master equation. The temporal evolution of the approximate distribution is shown to be a good representation for the true temporal evolution for a large range of parameter values.
authors.affiliation: Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK. alan.mckane@man.av.uk
authors.forename: Alan J
authors.initials: AJ
authors.lastname: McKane
authors.forename: David
authors.initials: D
authors.lastname: Alonso
authors.forename: Ricard V
authors.initials: RV
authors.lastname: Solé
mesh_terms: D044822:Biodiversity; D006801:Humans; D008390:Markov Chains; D008962:Models, Theoretical; D011157:Population Dynamics; D012111:Residence Characteristics / Q000706:statistics & numerical data*; D013996:Time and Motion Studies