Metapopulation theory for fragmented landscapes
Introduction
A large body of theory has been developed in recent years for the spatial dynamics of populations (Hanski and Gilpin, 1997; Tilman and Kareiva, 1997; Bascompte and Solé, 1998; Hanski, 1999; Keeling, 1999; Dieckmann et al., 2000). A large fraction of that theory assumes homogeneous space, and the primary aim of the research has been to dissect the population dynamic processes that may generate spatial heterogeneity in population densities in the absence of spatial heterogeneity in the environment. Unfortunately, the assumption that space is homogeneous makes it difficult to test theoretical predictions with empirical data, as real populations typically occur in heterogeneous space. Further progress in spatial ecology calls for theoretical approaches that allow one to relax the assumption of homogeneous space. For instance, Murrell and Law (2000) have used the method of moments to model the dynamics of carabid beetles in heterogeneous landscapes with three different classes of land type, woodland, agricultural land and urban areas, and Keeling (2000) has applied the method of moments to single-species and predator–prey dynamics in coupled local populations in a metapopulation.
One well-established approach to spatial ecology that assumes heterogeneous space is based on stochastic patch occupancy models (SPOM) (e.g. Day and Possingham, 1995; Frank and Wissel, 1998; Moilanen, 1999; Hanski and Ovaskainen, 2000). These models assume that the habitat occurs in discrete patches surrounded by unsuitable matrix, in the spirit of the dynamic theory of island biogeography (MacArthur and Wilson, 1967) and island models in population genetics (Wright (1931), Wright (1932); Barton and Whitlock, 1997). The major simplification, and a potential shortcoming, of SPOMs is that only the presence or absence of the focal species in the habitat patches is modelled. This is an acceptable simplification for highly fragmented landscapes, in which the focal habitat occurs as small or relatively small and discrete patches. Such landscapes are not uncommon.
Mathematically, SPOMs are formulated as Markov chains or Markov processes with 2n possible states in a network of n habitat patches. The familiar Levins (1969) metapopulation model is a deterministic approximation of a simple SPOM. The Levins model assumes an infinite number of identical patches, in which case the state of the deterministic system is fully described by the scalar variable of the fraction of occupied patches. A stochastic version of the Levins model with a finite number of patches, which has been analysed in the disguise of the stochastic logistic model, is an example of homogeneous SPOM, with identical and equally connected patches.
In this paper we review recent research on heterogeneous SPOMs, which assume a finite number of dissimilar habitat patches. In these models, the transition probabilities for individual patches from the occupied to the empty state, and vice versa, are different for different patches. In nature, such differences among patches are commonplace, as there are always differences in the areas, qualities, and relative spatial positions of patches in a patch network, and these differences are likely to influence the patch-specific transition rates. It is indeed tempting to construct models that make structural assumptions as to how the transition probabilities of patch occupancy depend on the physical attributes of the landscape. The combination of heterogeneous SPOMs with such assumptions leads to what we have called the spatially realistic metapopulation theory (SMT) (Hanski, 2001). The incidence function model (IFM) (Hanski, 1994b) was the first model of this type; subsequently we and others have contributed to the mathematical theory (as reviewed here) and to methods of parameter estimation of these models (Moilanen (1999), Moilanen (2000); O’Hara et al., 2002; ter Braak and Etienne, 2003).
SMT has two advantages in comparison with other modelling approaches to spatial ecology. First, the model structure—the assumption that the habitat consists of a network of discrete and dissimilar patches—is a good approximation of the landscape structure for many species, and the models can often be rigorously parameterized with data that are commonly available. Second, SMT allows a rigorous mathematical analysis, in contrast to simulation approaches that are commonly employed when the researcher wishes to incorporate a realistic description of the landscape structure in the model. The focus of this review is in theory, and we only mention research that has been done on parameter estimation and application of the models to real metapopulations.
Section snippets
Stochastic patch occupancy models
SPOMs assume a network of n discrete habitat patches, each of which has two possible states, occupied by the focal species or empty (we do not consider multispecies SPOMs, which are reviewed by e.g. Nee et al., 1997; Holt, 1997). In discrete-time SPOMs, the transition probabilities are assumed to depend on the state of the system (pattern of patch occupancy) in the previous time step, defining the model as a Markov chain. In continuous time, SPOMs are defined as Markov processes, with the
The deterministic core model
The Levins model is a deterministic approximation of a homogeneous SPOM for an infinite number of identical patches. The dynamic variable in the Levins model is the fraction of patches that are occupied, p, and the model is given bywhere c and e are colonization and extinction rate parameters. Given that the patches are identical and their dynamics are independent, the rate of change in the probability of any one patch being occupied is also given by the same equation.
Description of landscape structure
Straightforward ecological considerations suggest that the colonization (Ci(p)) and extinction (Ei(p)) rates in Eq. (3) are generally related to the physical attributes of the patch network in which the metapopulation occurs (Hanski, 1994b). Other things being equal, the expected size of local populations scales with the area of the respective patch or with an effective area corrected for habitat quality. As extinction rate generally decreases with increasing population size (Diamond, 1984;
Deterministic analysis
In this section we continue with the example of the SRLM. The key point is that the essential behaviour of the original n-dimensional model can be well approximated by a one-dimensional equation. In the case of SRLM, the one-dimensional model is structurally identical with the original Levins model (Eq. (2)), but with a new interpretation of the dynamic variable and the model parameters (Ovaskainen and Hanski (2001), Ovaskainen and Hanski (2002)). Specifically, metapopulation size is now
Stochastic analysis
We have already emphasized that the mathematical analysis of stochastic and heterogeneous SPOMs for networks of many habitat patches is hampered by the huge size of the state space. Some approximation is therefore called for. In this section we outline an approach to the study of heterogeneous SPOMs that is based on the same idea as the concept of effective population size in population genetics: construct a homogeneous SPOM which behaves in the same manner as the heterogeneous SPOM with
Contributions to the unification of research in population biology
The spatially realistic metapopulation theory (SMT) contributes to the unification of population ecology and related disciplines in several ways. First, SMT unites the classic metapopulation theory (CMT) based on the pioneering models by Levins (1969), Levins (1970) and the dynamic theory of island biogeography (DTIB) of MacArthur and Wilson (1963), MacArthur and Wilson (1967) (Hanski, 2001). CMT and DTIB are obviously related, because the expected number of species on an island or in a habitat
Acknowledgements
We thank Karin Frank for comments on the manuscript. This study was supported by the Academy of Finland (grant no. 50165 and the Finnish Centre of Excellence Programme 2000-2005, grant no. 44887).
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