Eur. Phys. J. B 38, 261–268 (2004) DOI: 10.1140/epjb/e2004-00118-9 T HE EUROPEAN P HYSICAL JOURNAL B Networks in metapopulation dynamics V. Vuorinen 1 , M. Peltom¨ aki 1 , M. Rost 2 , and M.J. Alava 1, a 1 Laboratory of Physics, Helsinki University of Technology, 02015 Espoo, Finland 2 Abteilung Theoretische Biologie, IZMB, Universit¨at Bonn, 53012 Bonn, Germany Received 24 October 2003 / Received in final form 8 December 2003 Published online 14 May 2004 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2004 Abstract. The behavior of spatially inhomogeneous populations in networks of habitats provides exam- ples of dynamical systems on random graphs with structure. A particular example is a butterfly species inhabiting the ˚ Aland archipelago. A metapopulation description of the patch occupancies is here mapped to a quenched graph, using the empirical ecology-based incidence function description as a starting point. Such graphs are shown to have interesting features that both reflect the probably “self-organized” nature of a metapopulation that can survive and the geographical details of the landscape. Simulations of the Susceptible-Infected-Susceptible model, to mimick the time-dependent population dynamics relate to the graph features: lack of a typical scale, large connectivity per vertex, and the existence of independent subgraphs. Finally, ideas related to the application and extension of scale-free graphs to metapopulations are discussed. PACS. 87.23.Cc Population-dynamics and ecological pattern formation – 89.75.Hc Networks and genealogical trees – 89.75.Fb Structures and organization in complex systems 1 Introduction Population dynamics or ecology offer numerous appli- cations for stochastic processes like directed percolation and other related models studied by statistical mechan- ics. Their characteristics are determined by the structure of the underlying habitat or landscape. A very common class are ensembles of habitat patches, where each sin- gle patch is too small to carry a stable population but the population can survive by constantly colonizing empty patches [1]. In the biological literature such systems are called metapopulations [2,3]. Metapopulations appear in many different organisms, e.g., the Spotted Owl in South- ern California [4] or the European nuthatch [5], the land snail Arianta arbustorum in northern Switzerland [6], wa- ter voles in Scotland [7] or the American pika [8], even spatial aspects of HIV in lymphoid tissue can be cap- tured by a metapopulation model [9]. The perhaps best studied cases are insects, such as the Granville fritillary butterfly Melitaea cinxia on the archipelago of ˚ Aland in the Baltic Sea between Sweden and Finland (60 ◦ north- ern latitude, 20 ◦ eastern longitude) [10]. Its larvae feed on host plants, Plantago lanceolata (Plantaginaceae) and Veronica spicata (Scrophulariaceae) growing on well dis- tinct meadows which form the butterfly’s habitat patches. In the plants’ centre the larvae form small nests around which they spin a white web, which is the easiest to rec- ognize indicator that a habitat patch is occupied. a e-mail: mja@fyslab.hut.fi In modeling population dynamics certain features are isolated or highlighted and then compared with field ob- servations. Thus one hopes to find and understand the key mechanisms behind the observed phenomena. Models exist on different levels of abstraction. The state of a pop- ulation could, e.g., be rendered by an exact list of locations of each individual, or by a mere number of the total pop- ulation size. In the case of the butterfly metapopulation considered here it makes sense to consider each patch of habitat as either populated or empty, thus introducing a state variable ρ i = 1 or 0 for each patch i. Insects lay their eggs on host plants which are so dense in each patch that after a short while the entire patch will be colonized, typi- cally after a few weeks. In the same way, extinction events hit the patch as a whole. On the other hand patches are sufficiently isolated by their mutual distance which is sub- stantially larger than their respective sizes, so colonization of one patch does not immediately imply a new population on a neighbouring patch. This process is much slower and it may take years before an empty patch gets colonized. This discrete state space (ρ i =1/0) is different from the original approach of Levins [1] where population dynam- ics is rendered by a system of coupled ordinary differential equations (ODEs) for the population sizes in each patch. Migration couples the population in different patches, and the coupling strength depends on their mutual dis- tance and the geography of the landscape between the patches. Ecologists commonly use the term incidence for the intensity of immigrant arrival to a patch from all other