Theoretical Population Biology 62, 121–128 (2002) doi:10.1006/tpbi.2002.1584 Population Dynamics with a Refuge: Fractal Basins and the Suppression of Chaos T. J. Newman Department of Physics and Department of Biology, University of Virginia, Charlottesville, Virginia 22903 and J. Antonovics and H. M. Wilbur Department of Biology, University of Virginia, Charlottesville, Virginia 22903 Received January 20, 2001 We consider the effect of coupling an otherwise chaotic population to a refuge. A rich set of dynamical phenomena is uncovered. We consider two forms of density dependence in the active population: logistic and exponential. In the former case, the basin of attraction for stable population growth becomes fractal, and the bifurcation diagrams for the active and refuge populations are chaotic over a wide range of parameter space. In the case of exponential density dependence, the dynamics are unconditionally stable (in that the population size is always positive and finite), and chaotic behavior is completely eradicated for modest amounts of dispersal. We argue that the use of exponential density dependence is more appropriate, theoretically as well as empirically, in a model of refuge dynamics. & 2002 Elsevier Science (USA) Key Words: seed bank; dormancy; chaos; dispersal; spatial ecology; logistic map; exponential map 1. INTRODUCTION Non-linear maps have contributed greatly to our understanding of population dynamics over the past 25 years (May, 1976). The famous bifurcation diagram of the logistic map (i.e., the discrete time analog of the logistic model) encapsulates the related ideas of period doublings in steady-state populations as the reproduc- tive rate is increased, and fully chaotic dynamics as the reproductive rate exceeds a critical value. In more recent years, workers have utilized non-linear maps in a spatially explicit context by constructing coupled map lattices (CML) (Kaneko, 1992; Sol ! e et al., 1992). In these models, a spatial system is represented by a grid on each site of which is placed a non-linear map. The maps are coupled together to represent spatial dispersal. On varying the dispersal parameter, along with the reproductive rate, a wide range of dynamical behavior is observed (see for example, Hassell et al., 1991). Given the complexity of CML, it is difficult to understand these different dynamical phases even qualitatively. In an attempt to unravel the complexity of spatially explicit models, we introduce here the simplest spatial generalization of a non-linear map. This is a two-site system, with density-dependent regulation at one site, and density-independent dynamics at the other. The second site is coupled to the first via dispersal. To our knowledge this model has not been studied before. There have been many studies of two-site systems, but in all cases the populations of both sites are updated under density-dependent regulation (see for example, Vandermeer (1997), and in the chaos literature Maistrenko et al. (1999), and references therein). A particular class of these two-site systems has been intensively studied in the context of source–sink dynamics (Hanski, 1999). A continuum version was 121 0040-5809/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.