Periodic dynamical systems in unidirectional metapopulation models JOHN E. FRANKE† and ABDUL-AZIZ YAKUBU‡ * †Department of Mathematics, North Carolina State University, Raleigh, N.C. 27695, USA ‡Department of Mathematics, Howard University, Washington, D.C. 20059, USA (Received 7 June 2004; in final form) In periodically varying environments, population models generate periodic dynamical systems. To understand the effects of unidirectional dispersal on local patch dynamics in fluctuating environments, dynamical systems theory is used to study the resulting periodic dynamical systems. In particular, a unidirectional dispersal linked two patch nonautonomous metapopulation model is constructed and used to explain the qualitative dynamics of linked versus unlinked independent patches. As in single- patch, single-species population models, unidirectional nonautonomous models support multiple attractors where local population models support single attractors. Keywords: Metapopulation; Multiple attractors; Nonautonomous models; Periodic dynamical systems; Unidirectional dispersal 1. Introduction Populations are often subdivided in space and spread among independent patches that are connected via dispersal or migration [1–6,8–10,15,17,18,20–28,33–34,40–42]. The dynamics of discretely reproducing populations can be complex, especially when vital rates are density-dependent and are subject to environmental stochasticity [13,16,29,31,35 – 37,40]. Many scientists have used simple autonomous and nonautonomous nonlinear difference equations to model single-species discretely reproducing closed populations [2,6,7,16,25,32,35 – 42]. Others have used autonomous nonlinear systems of difference equations to model spatially-explicit metapopulations [1,3 – 5,15,22 – 24,33,42]. The focus of this paper is on the effects of unidirectional dispersal on local dynamics in periodically varying environments. In particular, a unidirectional dispersal linked two patch nonautonomous metapopulation model is used to study the qualitative dynamics of linked versus unlinked independent patches. Section 2, the preliminaries section, introduces a very general single-patch single-species nonautonomous population model without dispersal [16]. In Section 3, we introduce the main model, a two-patch unidirectional dispersal linked nonautonomous metapopulation model. Such nonautonomous discrete-time models generate periodically forced dynamical systems [11,12,14,16]. In Section 4, we use dynamical systems theory to understand the relation between local patch dynamics and metapopulation dynamics. In Section 5, we test Journal of Difference Equations and Applications ISSN 1023-6198 print/ISSN 1563-5120 online q 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/10236190412331334563 *Corresponding author. Email: ayakubu@howard.edu Journal of Difference Equations and Applications, Vol. 11, No. 7, June 2005, 687–700