TWO RELATED NOTIONS about natural populations featured prominently in the writings of several ecologists in the 1950s. These authors realized that populations have a spatial structure, in the sense that a “population” in the wider landscape often consists of more or less distinct local populations. And secondly, these local populations may have more or less independent demographic fates, which has consequences for the dynamics of the regional population as a whole. Explaining their ideas at length in The Distribution and Abundance of Animals (1954), the Australian ecologists H. G. (Herbert) Andrewartha and L. Charles Birch put an especially strong emphasis on small-scale spatial structure of populations. They argued that local populations are often characterized by high rates of extinction and reestablishment, a viewpoint that contrasted with the then prevailing paradigm of stable populations regulated by density-dependent processes (reviewed by, e.g., Sinclair 1989). John Curtis, Professor of Botany in the University of Wisconsin, understood clearly the consequences of human-caused habitat loss and fragmentation on population processes and the spatial distribution of species. He wrote:
Within the remnant forest stands, a number of changes of possible importance may take place. The small size and increased isolation of the stands tend to prevent the easy exchange of members from one stand to another. Various accidental happenings in any given stand over a period of years may eliminate one or more species from the community. Such a local catastrophe under natural conditions would be quickly healed by migration of new individuals from adjacent unaffected areas. . . . In the isolated stands, however, opportunities for inward migration are small or nonexistent. As a result, the stands gradually lose some of their species, and those remaining achieve unusual positions of relative abundance. (Curtis 1956, p. 729)
Not only does this paragraph describe the processes of local extinction and recolonization, but it also contains a vision of the extinction threshold. In the next paragraph on the same page Curtis commented on microevolutionary changes that are likely to take place in response to changing population structure due to habitat fragmentation. Quite a page! Carl Huffaker (1958), building upon the earlier theoretical work of the Australian Alexander J. Nicholson (1933), investigated in a fascinating experimental study the consequences of small-scale spatial structure of habitat for the dynamics and stability of predator-prey interaction. Mention should also be made of the “island model” in theoretical population genetics, already established by Sewall Wright in 1940.
The theories of island biogeography and metapopulation dynamics were introduced, respectively, by Robert MacArthur and Edward O. Wilson (1963, 1967) and by Richard Levins (1969, 1970) in the 1960s. From our present perspective, it is surprising that the island theory and metapopulation theory appear to have had their own independent origins, and origins that were independent of the work done on spatial population structures in the 1950s and earlier. In the case of MacArthur and Wilson (1963), the origin was their attempt to explain why large islands tend to have more species than small ones, while Levins’s (1969, 1970) primary concerns were some demographic and evolutionary consequences of extinction-colonization dynamics. Of the papers and books that I cited in the first paragraph, MacArthur and Wilson (1967) referred only to Curtis (1956), by reproducing a figure illustrating the human-caused reduction in the total area and increase in the degree of fragmentation of woodland in the Cadiz Township in Wisconsin from 1831 until 1950 (reproduced here as figure 7.1). It is curious that, having included Curtis’s fragmentation maps as the very first illustration in their book, MacArthur and Wilson made no real attempt to apply their model of island biogeography to fragmented landscapes without a mainland. I say more about this in the following sections; here it suffices to recapitulate that the written papers, chapters, and books suggest that there were several independent origins in the middle of the last century for the general idea that natural populations in larger regions consist of discrete local populations, and that this spatial structure of regional populations may have important consequences for their dynamics and long-term viability.
In the following two sections, my purpose is to show that MacArthur and Wilson’s model of island biogeography and Levins’s model of metapopulation dynamics are in fact special cases of a more general model, which can also accommodate the earlier descriptions of spatial population structure by Andrewartha and Birch and by Curtis. In this framework, the island model is a straightforward extension of the single-species metapopulation model to many co-occurring but dynamically independent species. The reasons for laboring this point, which is rather obvious when you come to think about it, are twofold. It is of historical interest to ponder why the connection was not made explicitly early on. And secondly, the unified model, bringing together the key innovations in the respective models of MacArthur and Wilson and of Levins, has substantial power to predict the distribution of species in fragmented landscapes, and it leads to new insights about familiar patterns in the large-scale occurrence of species. Concerning the latter, I examine in this chapter how the species-area relationship, the feature of island communities that so much stimulated the work of MacArthur and Wilson (see the introduction to their 1963 paper), can be derived from the single-species metapopulation model, and I point out how intimately the species-area relationship is related to another well-established pattern in the occurrence of species, the distribution-abundance relationship.
Figure 7.1. Reduction in the area and fragmentation of the woodland in the Cadiz township in Wisconsin from 1831 until 1950 (Curtis 1956). This figure was reproduced in MacArthur and Wilson (1967), p. 4. Curtis (1956) pioneered landscape ecology by calculating for the four maps the total area of woodland, the number of separate woodlots, the average size of woodlots, the length of the woodland periphery, and the periphery/area ratio. Incidentally, a look at this area today, with the help of GoogleEarth, reveals that some further fragmentation has occurred in the past 50 years, though the bigger woodland fragments in 1950 are still there (42°32′54.45″N, 89°45′52.06″W).
Before moving on, I add a personal note. I am one of the many ecologists whose research has been greatly influenced by the works of MacArthur, Wilson, and Levins; it has been a privilege and source of enjoyment to write this chapter. I have taken the liberty of addressing selectively a few topics that stem directly from the classic models of island biogeography and metapopulation dynamics and to which I have attempted to make contributions over a prolonged period of time. This chapter is not a review of the literature, partly for lack of space but also because my particular purpose is to focus on the core concepts of MacArthur and Wilson and of Levins, and to highlight their role in the subsequent development of metapopulation models and theory. The simple MacArthur-Wilson island model and the Levins metapopulation model are by now largely history and replaced by many more specific models, and by a range of more general models of spatial dynamics, but these simple models splendidly exemplify the motto of this chapter: science marches forward, but the legacy of good ideas lasts for a long time.
As is well known, the setting of MacArthur and Wilson’s island model involves a large mainland area, which is true mainland in the case of islands off the mainland but more generally a very large expanse of habitat, where P species have stable populations. Outside the mainland, there are islands, or more generally fragments of habitat, with dissimilar areas and with dissimilar distances (isolation) from the mainland. Migrants that originate from the mainland may establish new populations on the islands, and the island populations have a smaller or greater risk of local extinction. Migration among the islands is ignored; hence the essential dynamics of the model can be understood by considering just the mainland and one island. The MacArthur-Wilson model, in spite of its simplicity, is potentially a good description of the long-term dynamics of species occurring on true islands that are rather sparsely distributed off the mainland, making migration among the islands unlikely.
The core idea of the model is formulated in the following differential equation:
which specifies the rate of change in S, the number of species present on a particular island. The number of species increases due to colonizations: each species in the mainland pool of P species that is not yet on the island (there are P−S such species) has the same probability of colonization, which translates into a constant colonization rate parameter I in the continuous-time model. The number of species on the island decreases due to extinctions: all species have the same extinction risk, and hence the total extinction rate is given by the constant extinction rate parameter E times the current number of species. At equilibrium,
Turning to Levins’s metapopulation model, it is appropriate for highly fragmented landscapes such as shown in figure 7.1d: a large network of small or relatively small habitat fragments (patches) without any large expanse of habitat (mainland). To construct his model, Levins made the simplifying assumption that all patches are of the same size and that migration is global, equally likely among any pair of populations and patches (this is the island model assumption made in Sewall Wright’s 1940 model, which, however, assumed stable populations). The set of local populations inhabiting the network of patches is called the metapopulation, a term that Levins (1970) coined, the size of which is given by the fraction of patches occupied, denoted by p.
Levins formulated the core idea of classic metapopulation dynamics with the following differential equation:
Here c and e are the colonization and extinction rate parameters, describing the colonization capacity and the extinction-proneness of the species. Because colonization rate is proportional to just the fraction of occupied patches, which are the sources of migration, the Levins model does not contain any description of the landscape structure and it best applies to species for which the spatial configuration of habitat makes little difference due to frequent long-range migration.
It is of interest to ask why MacArthur and Wilson and Levins did not refer to each other’s work in their respective publications, to say nothing about why they did not explore the conceptual and theoretical similarities in their models. The reason is not that they did not know about each other. They did, they met (see figure 1.3 in Wilson, this volume), and they even coauthored papers in the mid 1960s on the coexistence of competitors and maintenance of genetic polymorphism in heterogeneous environments (MacArthur and Levins 1964, Levins and MacArthur 1996). And as a matter of fact, Levins actually derived in a little-known paper published in 1963 (Levins and Heatwole 1963) the expression 
for the equilibrium number of species on an island, which is the same as equation (7.2), though Levins used the inverse of extinction and colonization rates, the expected time to extinction D and the expected time to next colonization M (N is the number of species in the mainland pool). Ironically, MacArthur and Wilson did not give this simple equilibrium result in their 1963 paper, in which they first developed much of their theory, though it outlines many of the more advanced results subsequently discussed at length in 1967.
Turning to MacArthur and Wilson (1967), they discussed in their book “habitat islands on the mainland” (pp. 114–15), such as shown in figure 7.1, but rather than working in the direction of Levins’s description of a network of local populations, they emphasized how habitat islands are different from true islands in being surrounded by other habitats that might harbor competitors of the focal species present in the habitat islands. They went on to describe what we would now call source-sink dynamics (in the sense of Pulliam 1988), and they discussed the implications of such dynamics for interspecific competition. Apparently, MacArthur and Wilson were so focused on what happens in a particular island, whether a true or a habitat island, that they did not attempt to extend their model formally to networks of local populations in fragmented landscapes—in spite of the very first figure in their book (figure 7.1). They were interested in communities of species—how does species number vary with the area or isolation of an island—rather than in single species, which would have facilitated the development of models for habitat networks. Finally, MacArthur and Wilson did not construct a measure of isolation that would have been applicable to islands in a network, in which colonization does not occur from the mainland but from multiple other populations in the neighborhood of the focal island.
Subsequent research has attempted to merge the conceptual frameworks of the island theory and the classic metapopulation theory in two major ways: first, by developing single-species metapopulation models that take from the island model the explicit description of landscape structure in terms of the areas and isolations of habitat patches (I describe this line of research in the next section) and second, by developing multispecies models by making use of Levins’s description of habitat patch networks. This latter approach has led in the past few years to various models of metacommunity dynamics (reviewed in many chapters in Holyoak et al. 2005). There is a clear need for developing theory and models for metacommunities, but the task is difficult and the field is still searching for its basic concepts. Most of the current metacommunity models are not formally related to MacArthur and Wilson’s island model nor to Levins’s metapopulation model, for which reason they are not examined more closely in this chapter. One exception is the extension of Levins’s model to two or more competing species, which I comment on in the final section of this chapter.
Here I turn to models that mix assumptions from the island biogeographic model and the Levins metapopulation model. The qualifier “spatially realistic” indicates that the models take into account the actual spatial configuration of the habitat: how many patches are there in a network, how large are they, and how far apart are they located from each other? I show that the MacArthur-Wilson and Levins models are special cases of a spatially realistic metapopulation model.
The origin of these models is in Jared Diamond’s (1975) incidence functions, which are based on a straightforward idea. Consider the occurrence of a species on a set of islands with dissimilar areas. Diamond grouped the islands in classes of similar areas, for instance islands from 1 to 10 ha, from 11 to 100 ha, and so forth. He then calculated the proportion of islands in a particular area class on which a particular species had been detected during a survey. The incidence function describes how the proportion of occupied islands changes with area—usually the incidence increases with area. The islands could equally well be classified based on some other property, such as the number of species present, and the incidence function would be constructed in a similar manner. More generally, we may not group the islands at all but define the incidence function p(A) as the probability that the species is present on an island with area A.
In the case of mainland-island metapopulations, in which all migrants originate from the mainland, and assuming time-constant probabilities of extinction E and colonization C, the long-term probability of a species being present on an island is given by
as already noted by Levins and Heatwole (1963) in the island biogeographic context (this result is a property of the Markov chain defined by the model assumptions). The incidence function is now obtained by making assumptions about how the colonization and extinction probabilities C and E depend on the area or some other property of islands (in continuous-time models the probabilities become rates).
The incidence functions played some role, though not a very big one, in the vigorous debate that broke out in the 1980s about the factors that influence the assembly of island communities—or factors that do not influence community assembly, as many participants found that “null models,” which were presumed to involve no interspecific interactions, explained well the occurrences of species on islands. The volume edited by Strong et al. (1984) has many chapters on these issues (see Simberloff and Collins, this volume). At the same time, I was studying the dynamics of shrews and other small mammals on small islands. Stimulated by the work of Diamond and intrigued by the possibility of extracting some information about extinctions and colonizations from patterns of island occupancy, I constructed an incidence function by assuming that the annual extinction probability on island i is an inverse function of island area, 
and that the annual colonization probability declines exponentially with di, the isolation (distance) from the mainland, 
and β are model parameters (Hanski 1993). Assuming further that the colonization probability approaches 1 when isolation approaches zero, we have β = 1. The incidence function model is then given by
Using data on the occurrence of Sorex cinereus on a set of 40 islands studied by Crowell (1986) and Lomolino (1993) in North America, I estimated the values of the model parameters (Hanski 1993). The figure in box 7.1 depicts how the predicted probability of occurrence (the incidence) depends on island area and isolation. Naturally, one could make some other structural assumptions about how colonization and extinction probabilities depend on island area and isolation than what was made above. Some assumptions lead to incidence functions in which several parameters occur as a product and hence their values cannot be estimated independently without making extra assumptions.
BOX 7.1. Measurement of connectivity in metapopulations without a mainland
In island biogeographic models with all migrants originating from the mainland, isolation of an island is given by its distance to the mainland. In metapopulations without a mainland, migrants to a particular habitat patch i originate from existing local populations in the surrounding habitat patches. A measure of connectivity, which reflects lack of isolation, may be constructed by summing up the contributions from all possible source populations j. These contributions are weighted by three factors (see the illustration). First, the area of the source patch j, which reflects the numbers of potential emigrants from that patch. To gain further flexibility, the area may be raised to power 
, which reflects both the scaling of population size with patch area and the scaling of emigration with patch area. Second, the distance of the source patch j from the focal patch i, which influences the likelihood of individuals leaving patch j ever arriving at patch i. This likelihood is often assumed to be an exponential function of the distance dij, but some other distribution (“dispersal kernel”) could be used instead. Parameter α gives the rate of decline in the exponential distribution of migration distances from population j. Third, the contribution of patch j depends on the probability of patch j being occupied. In reality, only patches that are currently occupied may send out migrants, but in the mean-field model the contribution of a patch is weighted by its probability of occupancy (the mean-field concept is discussed below). Finally, connectivity of patch i may depend on its own area, possibly raised to the power 
to account for the scaling of immigration with patch area.
The two graphs illustrate how the pattern of patch occupancy depends in an analogous manner on isolation from the mainland in the case of islands off the mainland and on the above-described measure of connectivity in a metapopulation without a mainland. Black dots represent occupied, open circles unoccupied islands or habitat patches at the time of sampling. (B) Occurrence of the shrew Sorex cinereus on islands off the mainland. Isolation is here measured by distance to the mainland. The lines indicate the combinations of area and isolation for which the predicted probability of occupancy is greater than 0.1, 0.5, and 0.9, respectively (from Hanski 1993; data from Crowell 1986, Lomolino 1993). (C) Classic metapopulation of the silver-spotted skipper butterfly (Hesperia comma) on dry meadows in southern England. The line indicates the combinations of area and connectivity above which the predicted incidence of occupancy is greater than 0.5 (from Hanski 1994; data from Thomas and Jones 1993).
A brief digression is in place here. The incidence function exemplifies what is called the inverse approach in modeling (Tarantola 2005; for ecological applications see Wiegand et al. 2003, Ovaskainen and Crone 2009). Rather than estimating the parameters of ecological processes directly to predict patterns, here we use the pattern to estimate the parameters. The pattern consists of the probabilities of occupancy on a set of islands, the pi values, which in practice are often approximated by just a single snapshot of presence-absence data. It is better if data are available for several years (Etienne et al. 2004), but even a single snapshot has much information if presence-absence data are available for many islands. And for systems with low rate of population turnover, extinctions and recolonizations, it would not help much to have data for many years, because most islands would stay in the state (occupied or not) in which they were observed in the first year. It is exactly for such systems, for which the direct measurement of the processes of extinction and recolonization would be difficult or impossible because of low rates, that the “pattern-oriented” approach represented by the incidence functions is potentially most helpful.
Though difficult to estimate directly, the rates of extinction and recolonization are of self-evident importance to population ecologists and conservation biologists. In my own work on three species of Sorex shrews inhabiting small islands in lakes in Finland, I examined how differences in body size among the three species affect their foraging behavior and life histories, and how these effects might be reflected in population dynamics. One approach was based on incidence functions, with which I estimated for each species the scaling of extinction risk with island area and hence with the carrying capacity: parameter 
in equation (7.5) (Hanski 1992). I found that, while extinction risk decreased very rapidly with increasing island area for the largest species, the scaling was shallow for the smallest species, consistent with the hypothesis that environmental stochasticity plays a bigger role in the dynamics and hence also in the extinction of small-bodied than large-bodied vertebrates (Pimm 1991, Hanski 1998a). I shall return to this observation in the next section while discussing the species-area relationship. Here it remains to note an important caveat to all this modeling: equation (7.5) assumes that enough time has elapsed without any major environmental changes so that the focal species occurs on the islands in a stochastic quasi-equilibrium between recurrent extinctions and recolonizations. This assumption has to be considered case by case.
Let us then turn to metapopulations without a mainland. The essential difference from the mainland-island situation just discussed is that now isolation has to be measured in a different manner, as in metapopulations without a mainland recolonization is the result of migration from any one of several possible source populations in the neighborhood of the focal habitat patch. Box 7.1 describes a measure of connectivity that can be used in this context; connectivity is the reverse of isolation, measuring lack of isolation. The apparent complication that arises in comparison with the measure of isolation from the mainland is that the value of connectivity changes in time, with a changing pattern of occupancy and population sizes in the source populations. In stochastic models that keep track of which particular habitat patches are occupied this is not a problem, but such models are difficult to analyze (Ovaskainen 2001, Ovaskainen and Hanski 2004) or one is forced to rely on numerical simulations. An alternative is to use a trick called the mean-field approximation: connectivity of patch i depends not on which particular other patches happen to be occupied at a particular time but instead on the probabilities of occupancy of the other patches, the pi values. This may appear to be no solution at all, because surely the probability of occupancy is more difficult to determine than whether a patch is occupied or not. This is true for field studies, but for models the pi values are very convenient. Now our model consists of a set of equations like equation (7.5), in which pi for patch i depends on the corresponding p values for all the other patches in the network apart from i. This set of equations may be iterated until an equilibrium is reached, the set of pi values that satisfies all the equations simultaneously (Hanski 1994). This will not work if there is no equilibrium, but single-species patch occupancy metapopulation models typically converge to a unique equilibrium (Ovaskainen and Hanski 2001). Another issue is how good the mean-field approximation is. I return to this question in the discussion, but note already here that, as far as the prediction of the equilibrium state is concerned (quasi-equilibrium in stochastic models), the mean-field approximation works rather well for heterogeneous patch networks, in which the habitat patches have dissimilar areas and dissimilar connectivities (for transient dynamics, see Ovaskainen and Hanski 2002). Luckily for this line of modeling, the real networks are always heterogeneous.
Working together with Otso Ovaskainen, I have constructed and analyzed a range of spatially realistic metapopulation models, including both stochastic models and their deterministic approximations (for reviews, see Hanski 2001, 2005, Hanski and Ovaskainen 2003, Ovaskainen and Hanski 2004). Of particular relevance here is a general equation for the deterministic rate of change in the incidence of occupancy of patch i, because this model has the MacArthur-Wilson model and the Levins model as two special cases. The spatially realistic model is given by
where Ci depends on the connectivity of patch i (see box 7.1). Assuming a mainland pool of P identical and independent species and constant colonization and extinction rate parameters for island i, the equilibrium incidence is given by 
from which the basic MacArthur-Wilson model (equation [7.2]) follows by multiplying by P to obtain the equilibrium number of species. On the other hand, assuming a network of equally connected and equally large habitat patches, that colonization rate is proportional to the fraction of occupied patches (which is the same, at equilibrium, as the probability of any one patch being occupied, Ci = cpi), and further assuming constant colonization and extinction rate parameters, we arrive at the Levins model, equation (7.3), with the equilibrium 
An attractive feature of the spatially realistic metapopulation models is that they can be parameterized with empirical data, as I showed in the case of a mainland-island model for shrews (box 7.1). The same applies to models that do not have a mainland. Methods of parameter estimation have been reviewed by Etienne et al. (2004) and many applications to real metapopulations have been discussed by Hanski (2005). Box 7.2 gives an extended example on the Glanville fritillary butterfly.
BOX 7.2. The Glanville fritillary metapopulation in the Åland Islands in Finland and extinction threshold
The Glanville fritillary butterfly (Melitaea cinxia) has a classic metapopulation in a large network of about 4,000 habitat patches in the Åland Islands, southwest Finland, within an area of 50 by 70 km2 (map; Hanski 1999, Nieminen et al. 2004). The habitat patches are dry meadows with an average area of only 0.15 ha and never larger than a few ha (photograph). There is a high rate of population turnover, with around 100 local populations going extinct every year for various reasons (Hanski 1998b) and about the same number of new populations being established. The extinction rate declines with increasing patch area, and the colonization rate increases with connectivity (graphs on the left; data on annual extinction and colonization events have been binned in patch area and connectivity classes and only the average values are shown here; Ovaskainen and Hanski 2004). The graph on the right shows the size of the metapopulation as a function of the metapopulation capacity λM (Hanski and Ovaskainen 2000) in 25 habitat patch networks (these networks were delimited as clusters of patches in the entire large network shown in the map). The vertical axis shows the size of the metapopulation based on a survey of habitat patch occupancy in one year. The empirical data have been fitted by a spatially realistic model. The result provides a clear-cut example of the extinction threshold (from Hanski and Ovaskainen 2000).
MacArthur and Wilson (1963) originally developed their theory of island biogeography to explain a general pattern in the occurrence of species on islands: the species-area relationship. A couple of different functional forms had been suggested to describe the increasing number of species with increasing island area (e.g., Rosenzweig 1995), but the most common form is the one due to the Swedish ecologist Olof Arrhenius (1921) and used by MacArthur and Wilson, the power function species-area relationship, S = kAz, where S is the number of species on an island, or within an area delimited more arbitrarily, A is the area, and k and z are two parameters. This relationship can be linearized by taking logarithms, and the parameter z then gives the slope of the logarithm of S against the logarithm of island area.
At the level of single species, the incidence function describes how the probability of occurrence of a particular species changes (usually increases) with increasing island area. For instance, in the case of equation (7.5), the logit of 
increases linearly with the logarithm of island area, with the slope given by parameter 
Clearly, there must be some relation between the incidence functions for individual species and the species-area relationship for the community of species, especially if the species have independent dynamics on the islands as assumed in the basic island model, equation (7.1).
Starting from equation (7.4) and observing that 
Ovaskainen and Hanski (2003) calculated the slope of the power function species-area relationship as
where
Assuming further that extinction and recolonization rates scale with island area as 
is independent of island area A, and it is convenient to describe an incidence function with two quantities, the “critical” island area Ai* at which pi(A) = 0.5, and the slope of the incidence function at Ai*, which is proportional to xi (figure 7.2).
To actually predict the species-area relationship for a community of species based on their incidence functions, we need to know the distributions of the Ai* and xi values. There is no general theory from which these distributions could be inferred; hence we examined two large data sets for plants and birds (sources described in figure 7.2). In both cases, the exponential distribution fitted the −log A* values reasonably well, while the log x values were normally distributed (Ovaskainen and Hanski 2003). Furthermore, in both cases there was a negative correlation between the log Ai* and xi values, which is perhaps expected, because species that are vulnerable to environmental stochasticity have small xi and tend to require large areas to avoid extinction; hence they have large log Ai*. The negative correlation implies that species with large critical areas tend to respond more slowly to increasing island area than species with small A* (the examples in figures 7.2a and 7.2b are thus representative).
Figure 7.2. Two examples of incidence functions for the birds Troglodytes troglodytes (A) and Sphyrapicus ruber (B). Panel B also indicates the two parameters that are used to describe incidence functions, the critical island area A* and x, the slope of the incidence function at A*. The following panels show the estimated values of x and A* in a plant community (C; data from Moran 1983) and in a bird community (D; data from Thibault et al. 1990). The final panels show the species-area curves and their 95% confidence intervals for the plant (E) and the bird community (F) calculated on the basis of the single-species incidence functions as explained in the text (from Ovaskainen and Hanski 2003).
The species-area relationship can be calculated either by estimating the parameters for each species separately and by summing up the predicted incidences, or by first generating a hypothetical community of species with parameters drawn from the estimated distributions of Ai* and xi and then summing up their incidence functions (Ovaskainen and Hanski 2003). Figures 7.2e and 7.2f show the latter result for the plant and the bird communities. The species-area relationships thus derived correspond closely to the power function species-area relationships fitted to the same data. Though similar regression lines were obtained, arguably the result based on the incidence functions for individual species is more fundamental, because it is based on properties of individual species, and it may bring new insight into the community-level pattern. For instance, the decomposition of the species-area relationship into the constituent incidence functions helps explain why it has been so hard to arrive at a meaningful biological interpretation of the slope parameter z (Connor and McCoy 1979 and many subsequent papers). Consider a situation where z is small. The present model indicates that z is small either because the rate at which new species reach their critical areas A* with increasing island area is slow, which is a property characterizing the community of species, or because each species responds slowly to increasing island area (small x-value), which is a property characterizing individual species, or both.
Although the slope of the species-area relationship does not generally have a simple interpretation, in suitably circumscribed situations some progress can be made. As an example, assemblages of small-bodied birds and small mammals have a systematically smaller value of z than the corresponding assemblages of large-bodied species on the same set of islands (table 7.1). The explanation offered by Matter et al. (2002) relates to the greater impact of environmental stochasticity in the dynamics of small-bodied than large-bodied species, to which I referred above while discussing the scaling of extinction risk in shrews. Indeed, the respective explanations are the same: Matter et al. (2002) showed that the ranges of the critical island areas were about the same for both small-bodied and large-bodied species on the same set of islands, in which case the theory described above implies that the slope z of the species-area relationship directly reflects the average of the slopes 
of the species’ incidence functions.
TABLE 7.1
Estimated Slope Values (±SE) of the Power Function Species-Area Relationship for Five Assemblages of Birds and Small Mammals on Islands
The species-area relationship is one of the best-established generalizations in ecology. From the perspective of single-species incidence functions, the species-area relationship is obtained by summing up the rows of a matrix giving the occurrence (=1) of species (on columns) among a set of islands (on rows). The row sums give the numbers of species on islands; plotting these sums against the island areas gives the species-area relationship.
The island occurrences of species in the same matrix may be summed up along the columns to calculate on how many islands different species have occurred. The column sums then indicate the extent of species’ distributions among the islands. Analogous to the plot of species number per island against island area, the distribution of a species may be plotted against its carrying capacity (“species size”), which in practice is measured by the average abundance on islands where the species occurs. This is called the distribution-abundance relationship, and it is also very widely reported and analyzed in the ecological literature (Hanski 1982, Brown 1984, Hanski et al. 1993, Gaston 2003). And what is it like? Just as bigger islands tend to have more species, species with bigger “size’ (greater carrying capacity) tend to have more island occurrences (greater distribution) than species with smaller carrying capacity. There is no well-established functional form for the distribution-abundance relationship, but often the logistic function is used to model the fraction of islands or other sampling areas out of all islands or sampling areas that were occupied by a species as a function of its carrying capacity.
Given that the species-area relationship and the distribution-abundance relationship are obtained from the same matrix, by summing up the matrix elements either along the rows or along the columns, it is natural to ask how the two relationships might be related to each other in natural communities. Hanski and Gyllenberg (1997) derived both relationships from the spatially realistic metapopulation model given by equation (7.6). We assumed that the extinction rate is proportional to the inverse of the carrying capacity, and that different species have different population densities and hence different carrying capacities on the same set of islands. With these assumptions, small islands have fewer species than large islands because populations on small islands have smaller carrying capacities and hence greater risk of extinction. Likewise, species with lower density have narrower distributions than species with higher density, because the former have systematically smaller carrying capacities on the same set of islands and hence generally a greater risk of extinction. The model predicted several features of the observed species-area relationship, but the interesting point here is that species-area relationships with realistic slope values were predicted only when there were differences in the densities (abundances) among the species and when the more common species were more widely distributed than the less common ones, that is, when there was a realistic distribution-abundance relationship. In other words, the two relationships are so intimately related to each other that one does not occur without the other. I also note in passing that this model, with differences in species densities reflecting differences in their ecological requirements, effectively merges the two main hypotheses that have been proposed to explain the increasing number of species with increasing island area, namely, the extinction-colonization dynamics as in the MacArthur-Wilson model and habitat heterogeneity allowing more species with dissimilar ecological requirements to persist on larger islands (e.g., Rosenzweig 1995).
Reading of the ecological literature suggests that the days of simple models, such as the MacArthur-Wilson island biogeographic model and the Levins metapopulation model, have been passed. Current modeling efforts concerning the spatial occurrence and dynamics of species at the landscape level tend to be quite specific, often including much information about species’ life histories and information about the structure of the landscape. Two sorts of models of this type include statistical regression-type “habitat models” (e.g., Elith and Burgman 2003) and generic simulation-based models of population viability analysis (e.g., Akçakaya et al. 2004). Not surprisingly, in specific situations the predictive power of these models is much greater than that of simple general models. The cost is that the predictions are indeed specific, and hence these models are not that helpful in advancing our general understanding of the processes and phenomena at stake. Levins (1968) made the pertinent observation forty years ago: it is not possible to maximize simultaneously generality, realism, and precision, and therefore there is no single best-for-all-purposes model.
Concerning the more general theory in spatial ecology, one may discern a succession from the island model to metapopulation models to more general models of the spatial dynamics of species in any kind of environment, not just in patchy environments. Much of the general theory is concerned with the question of how spatiotemporal variation in population densities is generated and maintained by population processes (Durrett and Levin 1994, McGlade 1999, Dieckmann et al. 2000, Lande et al. 2003, Ovaskainen and Cornell 2006). This research has supported the early insight by Alan Turing (1952) that spatial dynamics may generate complex spatiotemporal patterns in species abundances in the absence of any environmental heterogeneity. Interest in such spatial pattern formation in ecology roughly parallels the previous excitement about nonlinear dynamics in single populations potentially generating complex temporal dynamics in the absence of any environmental stochasticity (May 1976a,b).
In this context, equation (7.6) and comparable deterministic models may appear overly simplistic, as these models make the mean-field approximation and thereby predict uniform density in a homogeneous environment. However, it should be remembered that, while most of the general theory about spatial pattern formation has been developed for homogeneous environments, real landscapes are always heterogeneous and include spatially fixed variation in habitat quality. In a patch network such as shown in figure 7.1d, spatial variation in patch areas, qualities, and connectivities greatly constrains population dynamics. In other words, the probability that a species is present in a particular part of the landscape may be influenced by its own dynamics and by interactions with other species, but it is also strongly influenced by the spatial structure of the landscape, which makes it less likely that spatial patterns due to population processes would dominate over patterns due to heterogeneous environment, especially in single-species models. This is the reason why the deterministic mean-field approximation often predicts surprisingly well the occurrence of metapopulations in fragmented landscapes.
The single-species metapopulation model introduced by Levins (1969) had been extended to competing species by the early 1970s (Levins and Culver 1971, Horn and MacArthur 1972, Slatkin 1974). This research soon demonstrated that the mean-field approximation led to a qualitatively wrong conclusion, namely, competitive exclusion of all but one of the competitors (Slatkin 1974). In contrast, in a model that properly accounts for the spatial correlation in the occurrences of competitors in a homogeneous patch network, two or more species may coexist in spite of strong competition, because strong competition effectively reduces the numbers of habitat patches in which the two species occur simultaneously (spatial pattern formation). Such spatial segregation enhances intraspecific competition in relation to interspecific competition and thereby facilitates regional coexistence. Indeed, the mean-field approximation fails badly for a homogeneous patch network and equal competitors, but if one or both of these assumptions are relaxed, and we examine the dynamics of at least somewhat dissimilar species in heterogeneous networks, the mean-field model predicts well the equilibrium distributions, including complementary distributions in the case of local migration (Hanski 2008). The message is that we should not be led astray by complex models examining interesting phenomena but in a context that is not relevant for populations in natural environments.
Turning from theory to one very practical issue, one legacy of the island biogeographic and metapopulation models is what is commonly called the habitat area and isolation paradigm in conservation biology. What is meant by this is that the spatial distribution of species is largely determined by the areas and isolations (more properly connectivities; see box 7.1) of habitat patches in a fragmented landscape. This prediction is often contrasted with the view that what really matters for the occurrence of species is not habitat area and isolation but habitat quality, and spatial variation in habitat quality from one patch to another. An extensive literature has grown around this issue (reviewed by Fahrig 1997, 2003, Hanski 2005). However, important as it is to know what really determines the occurrence of species in particular cases, not least for conservation and management, one should realize that there is no general answer beyond the observation that, of course, both the quality and the amount of habitat matter. How much they matter in particular situations must depend on the specific circumstances. Each empirical study is necessarily based on a limited number of habitat patches and variables that are measured, and exactly which patches are included makes a difference. Including more patches of very low quality will most likely increase the “significance” of habitat quality in explaining habitat occupancy; adding tiny patches (which an ecologist might be tempted to exclude because they do not often support a local population) would increase the “significance” of patch area; and including some very isolated patches might do the same for the “significance” of connectivity. The point is that there is no general answer, and one should not be misled into assuming that ten studies demonstrating the importance of habitat quality have somehow demonstrated the general unimportance of the spatial configuration of habitat for the dynamics of species living in fragmented landscapes. Incidentally, the literature on the species-area relationship contains a parallel debate about the importance of island area versus habitat heterogeneity on islands in explaining the increasing number of species on islands with increasing area (Williamson 1981, Rosenzweig 1995, Whittaker 1998).
I conclude by commenting on one striking difference between MacArthur and Wilson’s island biogeographic model and Levins’s metapopulation model—how were they received by the scientific community? The MacArthur-Wilson model quickly became very well known, it started to have great impact on basic research, and it was one of the building blocks upon which modern conservation biology was established in the mid-1970s (Simberloff 1988, Hanski and Simberloff 1997). In contrast, the Levins model remained little known and had very little impact on anything for the next ten years. Levins’s 1969 paper received fewer than ten citations per year until 1991 (ISI Web of Knowledge), by which time the MacArthur-Wilson volume had accumulated more than 2,200 citations, an incredible number for those years (any ecologist would be glad to have papers with the same citation record as the common misspellings of the MacArthur-Wilson classic). In the past fifteen years, the difference has become much smaller, and while 34% of the pooled citations to MacArthur and Wilson (1967) are in papers published since 2000, the corresponding figure for Levins (1969) is a whopping 65%. Amazingly, both publications have received their highest annual number of citations to date in . . . 2007. This is amazing for a paper and a book published in the 1960s, even allowing for the ever-expanding literature and hence increasing annual number of total citations.
So why were MacArthur and Wilson so successful early on, and why was Levins not? MacArthur-Wilson (1967) was published as the inaugural volume in a monograph series that was bound to succeed, whereas Levins (1969) was published as a short paper in a rather obscure journal (that is, obscure from the perspective of most ecologists). This difference surely mattered, but I suggest that another difference was even more important. From the very beginning, in fact from the introduction to the original description of the island biogeographic model in MacArthur and Wilson’s paper published in Evolution in 1963, the theory became associated with the species-area relationship. This is important, because the species-area relationship was something that scores of biologists had been working on previously, and something for which more data could be easily gathered. The MacArthur-Wilson model appeared to provide a ready recipe for empirical studies, and for studies that would be highly doable and seemingly highly relevant for a current high-profile theory in ecology. No wonder that ecologists seized the opportunity. And not only that, soon that theory appeared to make a major contribution to conservation as well! In contrast, the Levins model must have appeared a rather abstract exercise to the few ecologists who noticed it. The Levins model did not lead to instructions as to what ecologists should, or could, do in practice.
We now know that the expectations concerning the MacArthur-Wilson model were too high, that demonstrating the species-area relationship does not critically validate, or refute, the island model, and that the conservation applications were simplistic. One could even argue that the excessive emphasis on the species-area relationship may have distracted attention from single-species incidence functions, which would have provided a much richer material for research, and for no extra cost at all, because with exactly the same data that were collected to study species-area relationships one could have calculated the incidence functions for individual species. But this is all wisdom based on hindsight. The Levins model has experienced a renaissance partly because it deals with situations that are now prevalent in the terrestrial world everywhere, highly fragmented habitats without a mainland, but also because it is the basis for the spatially realistic models described in this chapter, which have provided the blueprint for empirical studies of metapopulation dynamics. The works of MacArthur and Wilson and of Levins have had lasting impact in ecology and conservation because they succeeded so beautifully in capturing great ideas in simple mathematical models.
I thank Elizabeth Crone, Otso Ovaskainen, Robert Ricklefs, and two anonymous reviewers for comments, and Sami Ojanen for technical help.
Akçakaya, H. R., M. A. Burgman, O. Kindvall, C. C. Wood, P. Sjögren-Gulve, J. S. Hatfield, and M. A. McCarthy, editors. 2004. Species Conservation and Management. Case Studies. Oxford: Oxford University Press.
Andrewartha, H. G., and L. C. Birch. 1954. The Distribution and Abundance of Animals. Chicago: University of Chicago Press.
Arrhenius, O. 1921. Species and area. Journal of Ecology 9:95–99.
Brown, J. H. 1984. On the relationship between abundance and distribution of species. American Naturalist 124:255–79.
Connor, E. F., and E. D. McCoy. 1979. The statistics and biology of the species-area relationship. American Naturalist 113:791–833.
Crowell, K. L. 1986. A comparison of relict versus equilibrium models for insular mammals of the Gulf of Maine. In Island Biogeography of Mammals, ed. L. R. Heaney and B. D. Patterson, 37–64. London: Academic Press.
Curtis, J. T. 1956. The modification of mid-latitude grasslands and forests by man. In Man’s Role in Changing the Face of the Earth, ed. W. L. Thomas, 721–36. Chicago: University of Chicago Press.
Diamond, J. M. 1975. Assembly of species communities. In Ecology and Evolution of Communities, ed. M. L. Cody and J. M. Diamond, 342–444. Cambridge, MA: Harvard University Press.
Dieckmann, U., R. Law, and J. A. J. Metz. 2000. The Geometry of Ecological Interaction: Simplifying Spatial Complexity. Cambridge: Cambridge University Press.
Durrett, R., and S. Levin. 1994. The importance of being discrete (and spatial). Theoretical Population Biology 46:363–94.
Elith, J., and M. A. Burgman. 2003. Habitat models for population viability analysis. In Population Viability In Plants, ed. C. A. Brigham and M. W. Schwartz, 203–35. Ecological Studies vol. 165. Berlin: Springer-Verlag.
Etienne, R. S., C.J.F. van ter Braak, and C. C. Vos. 2004. Application of stochastic patch occupancy models to real metapopulations. In Ecology, Genetics, and Evolution of Metapopulations, ed. I. Hanski and O. E. Gaggiotti, 105–32. Amsterdam: Elsevier Academic.
Fahrig, L. 1997. Relative effects of habitat loss and fragmentation on population extinction. Journal of Wildlife Management 61:603–10.
———. 2003. Effects of habitat fragmentation on biodiversity. Annual Review of Ecology, Evolution, and Systematics 34:487–515.
Gaston, K. J. 2003. The Structure and Dynamics of Geographical Ranges. Oxford: Oxford University Press.
Hanski, I. 1982. Dynamics of regional distribution: The core and satellite species hypothesis. Oikos 38:210–21.
———. 1992. Inferences from ecological incidence functions. American Naturalist 139:657–62.
———. 1993. Dynamics of small mammals on islands. Ecography 16:372–75.
———. 1994. A practical model of metapopulation dynamics. Journal of Animal Ecology 63:151–62.
———. 1998a. Connecting the parameters of local extinction and metapopulation dynamics. Oikos 83:390–96.
———. 1998b. Metapopulation dynamics. Nature 396:41–49.
———. 1999. Metapopulation Ecology. New York: Oxford University Press.
———. 2001. Spatially realistic models of metapopulation dynamics and their implications for ecological, genetic and evolutionary processes. In Integrating Ecology and Evolution in a Spatial Context, ed. J. Silvertown and J. Antonovics, 139–56. Oxford: Blackwell Science.
———. 2005. The Shrinking World: Ecological Consequences of Habitat Loss. Oldendorf/Luhe: International Ecology Institute.
———. 2008. Spatial patterns of coexistence of competing species in patchy habitats. Theoretical Ecology. DOI:10.1007/s12080-007-0004-y.
Hanski, I., and M. Gyllenberg. 1997. Uniting two general patterns in the distribution of species. Science 275:397–400.
Hanski, I., J. Kouki, and A. Halkka. 1993. Three explanations of the positive relationship between distribution and abundance of species. In Community Diversity: Historical and Geographical Perspectives, ed. R. E. Ricklefs and D. Schluter, 108–16. Chicago: University of Chicago Press.
Hanski, I., and O. Ovaskainen. 2000. The metapopulation capacity of a fragmented landscape. Nature 404:755–58.
———. 2003. Metapopulation theory for fragmented landscapes. Theoretical Population Biology 64:119–27.
Hanski, I., and D. Simberloff. 1997. The metapopulation approach, its history, conceptual domain and application to conservation. In Metapopulation Biology: Ecology, Genetics, and Evolution, ed. I. Hanski and M. E. Gilpin, 5–26. San Diego: Academic Press.
Holyoak, M., M. A. Leibold, and R. D. Holt, eds. 2005. Metacommunities: Spatial Dynamics and Ecological Communities. Chicago: University of Chicago Press.
Horn, H. S., and R. H. MacArthur. 1972. Competition among fugitive species in a harlequin environment. Ecology 53:749–52.
Huffaker, C. B. 1958. Experimental studies on predation: dispersion factors and predator-prey oscillations. Hilgardia 27:343–83.
Lande, R., S. Engen, and B.-E. Saether, editors. 2003. Stochastic Population Dynamics in Ecology and Conservation. Oxford: Oxford University Press.
Levins, R. 1968. Evolution in Changing Environments. Princeton, NJ: Princeton University Press.
———. 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bulletin of the Entomological Society of America 15:237–40.
———. 1970. Extinction. Lecture Notes in Mathematics 2:75–107.
Levins, R., and D. Culver. 1971. Regional coexistence of species and competition between rare species. Proceedings of the National Academy of Sciences U.S.A. 68:1246–48.
Levins, R., and H. Heatwole. 1963. On the distribution of organisms on islands. Caribbean Journal of Science 3:173–77.
Levins, R., and R. MacArthur. 1966. Maintenance of genetic polymorphism in a heterogeneous environment: Variations on a theme by Howard Levene. American Naturalist 100:585–90.
Lomolino, M. V. 1993. Winter filtering, immigrant selection and species composition of insular mammals of Lake Huron. Ecography 16:24–30.
MacArthur, R., and R. Levins. 1964. Competition, habitat selection and character displacement in a patchy environment. Proceedings of the National Academy of Sciences U.S.A. 51:1207–10.
MacArthur, R. H., and E. O. Wilson. 1963. An equilibrium theory of insular zoogeography. Evolution 17:373–87.
———. 1967. The Theory of Island Biogeography. Princeton, NJ: Princeton University Press.
Matter, S. F., I. Hanski, and M. Gyllenberg. 2002. A test of the metapopulation model of the species-area relationship. Journal of Biogeography 9:977–83.
May, R. M. 1976a. Models for single populations. In Theoretical Ecology: Principles and Applications, ed. R. M. May, 5–29. Oxford: Blackwell Scientific.
May, R.M. 1976b. Simple mathematical models with very complicated dynamics. Nature 261:459–67.
McGlade, J., editor. 1999. Advanced Ecological Theory. Oxford: Blackwell Scientific.
Moran, R. 1983. Vascular plants of the Gulf Islands. In Island Biogeography of the Sea of Cortez, ed. T. J. Case and M. L. Cody, 348–81 Berkeley: University of California Press.
Nicholson, A. J. 1933. The balance of animal populations. Journal of Animal Ecology 2:132–78.
Nieminen, M., M. Siljander, and I. Hanski. 2004. Structure and dynamics of Melitaea cinxia metapopulations. In On the Wings of Checkerspots: A Model System for Population Biology, ed. P. R. Ehrlich and I. Hanski, 63–91. New York: Oxford University Press.
Ovaskainen, O. 2001. The quasi-stationary distribution of the stochastic logistic model. Journal of Applied Probability 38:898–907.
Ovaskainen, O., and S. J. Cornell. 2006. Space and stochasticity in population dynamics. Proceedings of the National Academy of Sciences U.S.A. 103:12781–86.
Ovaskainen, O., and E. Crone. 2009. Modeling movement behavior with diffusion. In Spatial Ecology, ed. S. Cantrell, C. Cosner, and S. Ruan. Boca Raton, FL: CRC Press.
Ovaskainen, O., and I. Hanski. 2001. Spatially structured metapopulation models: global and local assessment of metapopulation capacity. Theoretical Population Biology 60:281–304.
———. 2002. Transient dynamics in metapopulation response to perturbation. Theoretical Population Biology 61:285–95.
———. 2003. The species-area relation derived from species-specific incidence functions. Ecology Letters 6:903–9.
———. 2004. Metapopulation dynamics in highly fragmented landscapes. In Ecology, Genetics, and Evolution in Metapopulations, ed. I. Hanski and O. E. Gaggiotti, 73–104. Amsterdam: Elsevier Academic.
Pimm, S. L. 1991. The Balance of Nature? Ecological Issues in the Conservation of Species and Communities. Chicago: University of Chicago Press.
Pulliam, H. R. 1988. Sources, sinks, and population regulation. American Naturalist 132:652–61.
Rosenzweig, M. L. 1995. Species Diversity in Space and Time. Cambridge: Cambridge University Press.
Simberloff, D. 1988. The contribution of population and community biology to conservation science. Annual Review of Ecology and Systematics 19:473–512.
Sinclair, A.R.E. 1989. Population regulation in animals. In Ecological Concepts, ed. J. M. Cherrett, 197–242. Oxford: Blackwell.
Slatkin, M. 1974. Competition and regional coexistence. Ecology 55:128–34.
Strong, D. R. J., D. Simberloff, L. G. Abele, and A. B. Thistle, eds. 1984. Ecological Communities: Conceptual Issues and the Evidence. Princeton, NJ: Princeton University Press.
Tarantola, A. 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. Philadelphia: SIAM.
Thibault, J.-C., J.-L. Martin, and I. Guyot. 1990. Les oiseaux terrestres nicheurs des iles mineures des Bouches-de-Bonifacio: Analyse du peuplement. Alauda 58:173–85.
Thomas, C. D., and T. M. Jones. 1993. Partial recovery of a skipper butterfly (Hesperia comma) from population refuges: Lessons for conservation in a fragmented landscape. Journal of Animal Ecology 62:472–81.
Turing, A. M. 1952. The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London, Series B 237:37–72.
Whittaker, R. J. 1998. Island Biogeography: Ecology, Evolution, and Conservation. Oxford: Oxford University Press.
Wiegand, T., F. Jeltsch, I. Hanski, and V. Grimm. 2003. Using pattern-oriented modeling for revealing hidden information: A key for reconciling ecological theory and application. Oikos 100:209–22.
Williamson, M. 1981. Island Populations. Oxford: Oxford University Press.
Wright, S. 1940. Breeding structure of populations in relation to speciation. American Naturalist 74:232–48.
