A dynamic and spatial model with migration generating the log-Gaussian ®eld of population densities Steinar Engen * Institute for Mathematical Sciences, Norwegian University of Science and Technology, Glùshaugen, N-7491 Trondheim, Norway Received 17 October 2000; received in revised form 15 June 2001; accepted 1 August 2001 Abstract A spatial version of a dynamic population model leading to the lognormal distribution is de®ned. The model establishes relations between the joint spatial and temporal autocorrelation and biological concepts like environmental stochasticity, migration and strength of local density-regulation. The model is gener- alized to describe communities of species leading to a dynamic and spatial lognormal species abundance model with migration. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Spatial population model; Spatial dynamics; Migration; Environmental stochasticity; Population ¯uctu- ations; Lognormal distribution; Stochastic abundance models; Lognormal species abundance distribution 1. Introduction There is an increasing interest in spatial aspects of population dynamics [1±4]. Populations can usually not be considered as isolated, but interact with neighboring populations through migra- tion of individuals. Although various types of island models or metapopulation models are often used in theoretical analysis [5±8], real populations are usually not split into distinct islands, but are spread over large areas with an almost continuous and often large variation in density. Methods from spatial statistics [9±11] are often used to analyze spatial and temporal variations in the densities of such widespread populations by estimating the spatial autocorrelation for the pop- ulation sizes or the between years changes [12,13]. Moran [14] showed, in the case of no migration, that if the same linear time series model is valid for the log of population sizes at two locations, then the spatial correlation between the log population sizes or the between years changes in these, www.elsevier.com/locate/mbs Mathematical Biosciences 173 2001) 85±102 * Fax: +47-73 591 038. E-mail address: steinaen@math.ntnu.no S. Engen). 0025-5564/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII:S0025-556401)00077-3