Jessica Gurevitch, Samuel M. Scheiner, and Gordon A. Fox - The Ecology of Plants - 2002 / The Ecology of Plants Jessica Gurevitch, Samuel M. Scheiner, and Gordon A. Fox; 2002 / Ch 17

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however, this local patchiness is greatly reduced (neighboring trees tend to experience similar burn damage), but a new spatial pattern emerges because intense fires tend to “spot”: flying embers ignite material at a distance from the initial fire. The spatial arrangements of individual trees, groups of trees, lakes and rivers, and topography all combine to determine fire spread and intensity. Thus, we can gain unique insights into fire ecology and the resulting responses of plant populations by studying how the elements in a landscape are arranged.

Landscape ecology is concerned with questions such as, How are individuals, species, and communities distributed within a landscape? How many types of patches are there in a landscape? What are the sizes, shapes, and distributions of those patches? How do these patterns affect the movement of individuals, materials, and energy among patches and across the landscape? What processes are responsible for which patterns? Do the processes occur at the level of the landscape (e.g., migration) or within communities (e.g., competition)? How do the patterns we find, and the interpretation of those patterns, differ as scale changes? The answers to these ques-

Figure 17.1

Diagram of a landscape showing a patchwork of meadows, forests, and shrublands. For sampling purposes we can lay down a 100 × 100 m grid that arbitrarily divides the area into smaller plots.

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scape elements, others are most interested in how ecological phenomena are affected by changes in scale, or in the effects of spatial patterns on ecological properties regardless of spatial scale.

Spatial Patterns

There are two general ways to study patterns on a landscape level. One, a spatially explicit approach, depends on determining the particular spatial arrangement of species and landscape elements (e.g., habitat patches, farms, roads). The other, often called the mean field approach, focuses on describing average parameter values.

To see this distinction, consider a species-area curve (see Chapter 12). Imagine that we do the following: On the landscape shown in Figure 17.1, we overlay a grid of 10 × 10 m squares, and we compile a list of every vascular plant species growing in each square. We can now build a species-area curve in two ways (Figure 17.2). Using the first, spatially explicit approach, we start in one spot. Imagine that this spot is the bottom left corner

Shrubland

Agricultural

field

Agricultural

field

Forest B

Forest A

Agricultural

field

Agricultural field

100 m

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Mean field approach (nonspatial curve)

Area

Figure 17.2

Two species-area curves. One curve is from a spatially explicit tally, which forms a stairstep pattern. Within a community, the number of species found as area increases will level off. When a community boundary is crossed, however, the number of species found will once again rise rapidly. The other curve, from a mass field analysis that is not spatially explicit, forms a smoothly rising pattern.

of the meadow. Our first data point is the number of species in the 10 × 10 m square in that corner. Next, we expand our square to 20 × 20 m and again count the total number of species. This number will include all of the species in the original square plus any new species in the larger square. We repeat this operation for a 30 × 30 m square. Initially, the number of species will rise rapidly. But as more and more of the meadow is captured in our ever-growing plot, the number of new species will drop off. At some point, it is likely that we will even stop finding new species, and the curve will level off. If we keep going, however, our plot will eventually run into the forest abutting the meadow. Now, suddenly, we will find a whole new suite of species, such as trees and for- est-floor herbs. The species-area curve will again rise rapidly with increasing area until, once again, it levels off as most of the forest gets sampled. This stairstep pattern will be repeated each time we cross a new community boundary.

In contrast, we can build our species-area curve using the mean field approach, which ignores the spatial arrangement of the squares. First, we calculate the average number of species in all of the 10 × 10 m squares. Next, we take all possible combinations of two squares, determine the total number of species in each pair of squares, and again calculate the average. (While we

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would not want to do this by hand, it can be done in a few seconds on a typical desktop computer.) We continue this process with all sets of three squares, four squares, and so forth. The result is a smoothly rising curve. Any discontinuities in the environment that would have produced a stairstep effect with the spatially explicit approach are, instead, averaged out.

Which of these approaches is preferable? The answer depends on the questions one wishes to address. The pattern produced by each of these species-area curves can provide important information about both landscape patterns and the processes that account for them. Next, we explore these patterns in more detail.

Defining Patches

The basic unit we use to examine spatial patterns is the patch. A patch is any specified area, either an area defined arbitrarily or, often, an area that is in some way relatively homogeneous or internally consistent. A patch can be defined by naturally occurring edges (e.g., the boundary between a meadow and a forest), but need not be. For example, each of our 10 × 10 m squares in the exercise above could be a patch. However, if we start with such artificially defined patches, we might want to aggregate them into “natural” units. Sometimes this is done based on compositional similarity—the extent to which adjacent patches share a similar set of species at similar frequencies (see Chapter 16). Of course, adjacent patches usually share at least some species, so the cutoff point for when patches should be joined or not is somewhat arbitrary. Often the decision is aimed at ending up with a manageable and interpretable number of patch types.

Remote sensing images have become a popular method for sampling large areas quickly. In this case, the minimum patch size is determined by the resolution of the remote sensing equipment. Typically this is 30 × 30 m, although newly available technology is pushing resolution down to 10 × 10 m, and even 1 × 1 m. Our aggregation of patches is again based on their similarity, but now similarity is determined by spectral quality rather than by shared species (see Box 16A). Because spectral quality is an indirect measure of community composition, it limits our view of landscape structure.

Aggregating patches and deciding on the number of patch types is, therefore, as much an art as a science. Often it depends on the ability of measurement instruments to distinguish between different patch types. With remote sensing, for example, the indirect measures we are using need to be related to the actual vegetation through a process called ground-truthing: one goes out to the field and determines empirically what the remote sensing devices are recording. Two communities with very different species compositions may be indistinguishable in spectral quality. Thus, if remote sensing

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were being used to classify a large region, one would be forced to lump those two communities into one patch type.

Because landscape ecology is built on the study of patches, it is, inevitably, built on a categorical view of the world. Categorical versus continuous views of nature have been a regular source of controversy in plant ecology. In Chapter 12 we explore the history of this controversy, beginning with the disagreements of Clements and Gleason. It is somewhat ironic that nearly 50 years after the Gleasonian continuum was declared triumphant, at least among English-speaking ecologists, a categorical framework has once more emerged to dominate landscape-level studies.

Quantifying Patch Characteristics

and Interrelationships

Once we have defined our patches, we can measure some of their attributes. For individual patches, we can measure sizes and shapes. A typical measurement of shape is the perimeter-to-area ratio, which indicates how much edge a patch has relative to its interior. Edges are important ecologically for many reasons. In particular, they are the focal areas for processes such as the movement of individuals or materials between the outside and inside of a patch of forest.

One approach to quantifying the shapes used in landscape ecology is to calculate the fractal dimension of the landscape elements. A fractal dimension is a fractional dimension—for example, something in between a one-dimensional line and a two-dimensional surface, or in between a two-dimensional surface and a threedimensional area. For example, as a line (which has one dimension) becomes increasingly convoluted, it becomes space-filling, finally turning into a surface with two dimensions. The fractal dimension is thus a measure of the complexity or degree of convolution of a line (such as the perimeter of a patch or the coastline of an island) or a surface. Fractal geometry can also be used to quantify patterns of branching and fragmentation.

Imagine that you have a very large map of a coastline showing very fine details. You have to measure the length of the coastline using a ruler with no markings. Thus, the smallest distance that you can measure is the length of the ruler. A small ruler will pick up more of the ins and outs of a coastline than a large ruler, so that using the smaller ruler will result in a longer measurement. As the coastline or the perimeter of a patch becomes increasingly convoluted, the effect of ruler size becomes greater. The magnitude of the ruler size effect is the fractal dimension. Pennycuick and Kline (1986) used just such an approach to quantifying fractal dimension to compare bald eagle nest density on two Alaskan islands that differed in the complexity of their coastlines. This approach might be useful for various problems in plant

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ecology as well.

Bruce Milne (1992) connected the concepts of fractal dimension and scaling explicitly by quantifying the degree of habitat fragmentation in a landscape. Milne quantified the fractal dimension of a New Mexico grassland and showed that the habitat was patchy. Beyond that simple result, he showed that the patchiness was scale-dependent. Scale dependence implies that species might perceive the landscape differently. A beetle that lived in grass clumps might perceive a very disconnected landscape, while a grazing pronghorn might perceive that same landscape as one large patch. Some areas of the landscape showed high connectivity at three different scales, so here the beetle and the pronghorn might perceive similar landscapes. Other areas consisted of isolated patches that might attract species that disperse readily, but then settle within a single patch (e.g., small rodents).

As researchers become concerned with going beyond the individual patch, they can examine the arrangement of patches in the landscape. Using a different approach than the one taken by Milne, researchers may ask, are patches mostly aggregated into like types or not? A checkerboard pattern, for example, represents the maximal interdispersion of two patch types in a landscape (Figure 17.3A). The maximal aggregation of two patch types would have all the patches of each type grouped into two large clumps (Figure 17.3B). A measure of this aggregation is connectedness (see below). If there are more than two patch types, then other measures of context come into play. For example, for a landscape with three patch types, a patch of type A could be surrounded entirely by patches of type B, or entirely by patches of type C, or by half of each. Clearly, such sec- ond-order patterns can get very complicated with a large number of patch types. We can quantify these patterns as the nearest neighbor probability, the chance that two patch types will be adjacent.

Scale

Definitions and Concepts

When we study an ecological pattern, we must establish the scale at which we are conducting our analysis. Before we examine the implications of this seemingly straightforward statement, however, we must point out a potentially confusing difference in how people in different fields discuss scale. Ecologists generally equate scale with the physical dimensions of the phenomenon being studied: a process that occurs over 10 km is at a larger scale than one that occurs over 10 cm. We follow this practice in this textbook. Some landscape ecologists refer, instead, to “coarse scale” and “fine scale” for larger and smaller physical areas, respectively (Forman and

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Figure 17.3

Three different patterns of patch aggregation in a landscape with two patch types. (A) Maximal dispersion of patches. (B) Minimal dispersion of patches. (C) Random dispersion of patches.

Godron 1986). However, cartographers and remote sensing researchers use exactly the opposite terminology, in which scale is the relative size at which an object of a standard size appears in a map or photo. Thus, a world map has a much smaller scale than a regional map (that is, a meter is much smaller on a world map). It is therefore important to be explicit as to how one is using the term “scale” (e.g., Turner and Gardner 1991), although some ecologists have advocated abandoning the term entirely, replacing it with its components (Csillag et al. 2000).

Scale is composed of several components, two of which are grain and extent. Grain is the size of the primary unit used in a study—for example, a 10 × 10 m square or a 10 km2 area in which species are counted. While the grain is usually the smallest distinguishable unit, or the finest level of spatial (or temporal) detail that can be resolved in a given data set, for some types of data it is possible to extrapolate to a smaller grain; in contrast, data aggregation will result in a larger grain. Extent is the total range over which a pattern is examined. The extent of a study might be across a local region of 100 km2 or across the entire globe. For example, a research project might have a grain of 1-ha squares and an extent of 10,000 ha in central Africa, which together define the scale of the analysis.

Some researchers recognize a third component of scale, focus or resolution. These terms refer to the sampling level or the area represented by each data point. Consider an analysis of diversity in a 1-km2 area that is divided into 1-ha squares. Those 1-ha squares become the focus of our sampling. In each square we randomly place 10 1-m2 quadrats and record the species contained in each quadrat. These data can be analyzed in two ways: using the average species richness of the 10 quadrats in each 1-ha square (in which case the grain equals 1 m2), or using the total number of species in the 10 quadrats in each 1-ha square (in which case the grain equals 10 m2). In both cases, though, the focus is 1 ha, because the 1-ha square is the basic unit of analysis. Changing the grain, extent, or focus of a study can result

in fundamentally different patterns (Figure 17.4; see also Figure 20.11).

Consideration of scale is central to understanding ecological processes. The same process may function differently at different scales. For example, the ecological process of herbivory is generally a very local phenomenon. However, the effects of an elephant herd might be manifested at the landscape level, and those of a locust swarm at the regional level. Population dynamics can be a local phenomenon, but if metapopulation dynamics is involved, then it becomes a regional phenomenon as well. The evolutionary process of extinction can occur at a local, regional, or global scale; its implications at these different scales can be very different.

Jonathan Levine (2000) found that at different scales, different processes explained the success of plant invasions. In northern California, Carex nudata (Cyperaceae), a tussock-forming sedge, forms clumps along river edges. A tussock (clump) creates a microhabitat that can shelter up to 20 species of angiosperms and bryophytes during winter flooding. Across a 7-km stretch of river, Levine found that the higher the species richness of a tussock, the more likely it was to be invaded by the nonnative species Agrostis stolonifera (creeping bent grass, Poaceae), Plantago major (common plantain, Plantaginaceae), and Cirsium arvense (Canada thistle, Asteraceae). This pattern was the result of processes working in opposite directions at smaller and larger scales. Very small-scale processes—those occurring within a single tussock—were investigated by an experimental manipulation of the number of native species on individual tussocks and by the introduction of seeds of the invaders into the experimental tussocks. At this small scale, as species richness increased, the germination, survival, and growth of the invaders declined. In contrast, at the larger scale of the 7-km stretch of the river, Levine found that seeds of all species (both natives and invaders) moved with the flow of the river, and so both native diversity and frequency of invasion increased downstream. Thus, although native diversity contributed to community resistance to invasion at very small scales,

Figure 17.4

An illustration of how changing the focus and extent of a study can alter the results—in this case, the relationship between species richness and productivity. Each diagram represents a region. In this region there are three landscapes, with three communities in each landscape and five plots in each community. In all cases, the grain of the study remains the plot. Species richness is measured as the total number of species in each plot. (A) Each point represents one plot, so the focus is the plot, and the extent is the entire region. An analysis of the relationship between species richness and productivity results in a slope of zero, or no relationship. (B) Each point represents one plot, so the focus is the plot. Each

at larger scales, other factors that changed with diversity (here, seed supply of invaders) were more important, creating a positive relationship between invader and native diversity.

Interestingly, there was even a suggestion of the potential importance of factors operating at much larger scales. At the beginning of the study in 1998, the river chosen for the experiment was one of the few whose banks were not flooded by high-water conditions, and thus had one of the few stretches of Carex tussocks available for study. In the following year, water levels dropped severely, threatening the survival of the Carex tussocks on this stretch of river, while they recovered elsewhere in the region. This variation in local flooding is related to a 3- to 7-year climatic cycle, the El Niño Southern Oscillation, that affects weather conditions

landscape is analyzed separately, so the extent is the landscape. Analyses of the relationship between species richness and productivity for each landscape result in a negative slope. (C) Each point represents the average of the plots in a community, so the focus is the community. Each landscape is analyzed separately, so the extent is the landscape. Analyses of the relationship between species richness and productivity for each landscape result in a negative slope. (D) Each point represents the average of the plots in a landscape, so the focus is the landscape. The extent is the entire region, and the relationship between species richness and productivity is positive. (From Scheiner et al. 2000.)

across the Pacific (see Chapter 18). How the temporal variation in stream flow affected invasions is not known. However, such landscape-scale variations in flooding, across temporal changes in weather happening at the scale of thousands of kilometers, could potentially influence regional invasion success by creating greater opportunities for invasion as tussocks become disturbed by flooding followed by drought. Thus, the process of invasion could be influenced in this system by processes occurring over a range of spatial and temporal scales, sometimes acting in opposition to one another.

Hypotheses that seek to explain large-scale patterns may be based on processes that operate at smaller scales than, or at the same scale as, the phenomena they seek to explain. If the underlying process on which the explanation rests operates at a smaller scale, then the hypoth-

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esis must either include an explanation for how local processes scale up to larger areas or assume that largescale patterns are simply the sum of local-scale patterns. Scaling up and down from one set of patterns to another is a major current focus of research in many different areas of plant ecology, from the modeling of carbon fluxes (leaf-canopy-ecosystem-global) to determining the causes of diversity patterns.

Spatial and Ecological Scale

How do we specify (and think about) scale? It is common for people to define scale in an exclusively spatial manner, but we can contrast this approach with a consideration of scale in an ecological sense. This latter meaning of scale is related to the concept of hierarchy in ecology. Using a strictly spatial definition, ecologists conventionally speak of local, landscape/regional, and continental/global scales, and they use area to distinguish these scales—for instance, local scales are often thought of as being up to 102 km2, landscape/regional scales as being between about 102 km2 to 108 km2, and continental/global scales as being 108 km2 and larger. It is usually important, however, to define scale in terms of the particular ecological processes involved. For example, 102 km2 may be about the scale at which local pollen dispersal occurs for pines or some other plants with winddispersed pollen, but it is much too large a scale for pollen dispersal by animals such as ants, beetles, or Australian possums.

Ecologically, we often recognize three basic scales: within a community, across communities, and across biomes. It is easiest to appreciate the distinctions among these scales by starting with biomes, the largest scale. Biomes are defined by distinct differences in physiognomy over large regions (e.g., grasslands, shrublands, forests; see Chapters 18 and 19). Patterns that cross biome boundaries involve variation in the physical sizes, morphologies, and ecological niches of the dominant species—for example, shrubs versus trees. Within biomes, moving across communities may result in more subtle, yet equally important changes—for example, from a forest dominated by fast-growing hickory (Carya spp.) trees to one dominated by slower-growing oaks (Quercus spp.). Even within a single community, such as the meadow in Figure 17.1, there may be variation in factors such as soil conditions, land use history, microhabitat, and herbivore activity.

These two different approaches to thinking of scale—spatial and ecological—are not entirely independent. The farther we move in space, the more likely it is that we will cross community or biome boundaries. But we may also cross community boundaries over very short distances if the environment changes rapidly. Moving away from the edge of a stream or lake, for example, can result in abrupt changes in the amount of water in

the soil or the amount of disturbance due to wave action (Wilson and Keddy 1986; see Figure 10.26). In arid montane regions, by moving a distance of just a few kilometers up the side of a mountain, one can travel from desert to temperate forest to boreal forest. Conversely, traveling across the taiga of Russia, one finds very similar forest communities stretching across many hundreds of kilometers.

Quantifying Aspects of Spatial Pattern and Scale

Given the complexity of the mosaic of environmental factors and biotic interactions that determine spatial patterns at different scales, how can we hope to describe or quantify these patterns? Ecologists have developed both spatially explicit and nonspatial methods of doing this. (By “spatially explicit,” we mean approaches that take the particular location of each data point into account.) Methods that are not spatially explicit—ordination and some statistical approaches to quantifying spatial asso- ciation—are older, and we discuss some of them in Chapter 16.

Recently, a spatially explicit graphical technique,

Geographic Information Systems (GIS), has become central to landscape ecology. GIS is a method for depicting the relationships between different kinds of information (“layers” in GIS terminology) at particular locations. For example, we might have several types of information that relate to each location in the landscape (Figure 17.5). One set of information might be the species composition of each patch. For each patch, we might also have information on soil type, nutrient levels, water table depth, slope, and the last time the patch was logged or burned. Our information could also include socioeconomic information such as land ownership, taxes on that piece of land, the number of people residing there, and the legal classification determining how the land can be used. GIS allows us to map these variables singly and in any combination to get a visual sense of potential patterns of correlation among them, so that we may pose hypotheses regarding causality. GIS by itself does not allow one to quantify these relationships or statistically test hypotheses about them.

Spatially based relationships can be quantified using spatial statistics. Many advances in the field of spatial statistics have been made since the late 1980s (Legendre and Fortin 1989). These advances include the use of geostatistics, taken from the field of mining geology, and techniques for studying spatial autocorrelation, taken from the field of geography (Liebhold et al. 1993; Liebhold and Gurevitch 2002). Older statistical techniques in ecology were developed to analyze and model spatial variation in ecological data (Pielou 1977; Greig-Smith 1983). These methods were useful in distinguishing certain kinds of spatial patterns, but did not take the actual spatial location of the data into account, and had

Trees

Topography

Streams

Sum

Figure 17.5

GIS is used to examine the relationships among multiple types of information, which are arranged in layers.

numerous shortcomings. The newer approaches have contributed substantially to our current understanding of the spatial aspects of organisms’ responses as individuals, populations, and metapopulations (summarized by Dale 2000; Legendre and Legendre 1998). Curiously, spatial statistics have rarely been used by ecologists in conjunction with GIS. Just as we combine graphical and statistical analyses with other types of data, together GIS and spatial statistics constitute a remarkably powerful set of tools.

Hierarchical patch dynamics is a new approach to examining the effects of scale (Wu and Loucks 1995). In

this approach, a patch is seen as existing in a hierarchy of successively larger patches. Each level of this hierarchy defines a scale. Ecological processes occur within patches, while links among patches are created by the movement of materials and energy. The pattern of linkages defines the next layer in the hierarchy and determines how processes propagate up the hierarchy. For example, gas exchange by a single leaf can be combined with that of the other leaves on a plant to determine the dynamics of gases around the entire canopy of a tree, which in turn can be used to model an entire forest stand.

We have discussed a number of the issues involved in describing and studying patterns in nature. As you have doubtless concluded, there is no single way to study or even describe such patterns (Levin 1992). Viewing similar questions from different vantage points (such as the spatial versus ecological scales discussed above) or at different scales can lead to very different answers. This may be frustrating if you are expecting simple truths, but as we have seen, considering information at more than one scale, while difficult, can also lead to new insights and understanding of the underlying biology. We now turn to some approaches to studying the processes responsible for some of the types of patterns we have just examined.

Toward a Theoretical Basis for Landscape Patterns: Island Biogeography Theory

Our exploration of landscape processes starts with archipelagos, chains of islands. Robert MacArthur and Edward O. Wilson’s theory of

island biogeography (1967) was developed to explain patterns of species presence and absence on islands. That theory was important in the development of much of today’s landscape ecology.

Although the initial description of the theory applied it to oceanic islands, many types of communities have island-like properties. An inclusive definition of an island is any area that is suitable for the survival and

reproduction of a species and is surrounded by unsuitable habitat. A landscape made up of isolated woodlots consists of a set of islands from the point of view of an herb that lives only in woodland understories. Mountaintops are islands for alpine species. Plants can be specialized for life on particular types of soil or terrain, such as serpentine barrens (see Box 15A) or rock outcrops, which also function as islands. The theory of island biogeography therefore has applications far wider than oceanic islands.

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The basic idea of the theory is quite simple: The number of species on an island is determined by the rate of immigration of new species to the island and the rate of local extinction on the island (Figure 17.6). The rate of arrival of new individuals from the mainland determines the immigration rate—the rate at which new species arrive. (This immigration rate differs from that used in other contexts where it refers to the rate at which individuals arrive.) As the number of species on the island increases, the immigration rate declines, because an arriving individual is increasingly likely to be a member of a species already on the island. The relationship between species number and extinction rate is a bit more complicated. Assume that the island can hold a fixed total number of individuals. As more species take up residence on the island, the potential population size of each gets smaller. The chance that a species will go locally extinct—disappear from the island—increases as its population size gets smaller. Thus, the rate of extinction increases as the number of species rises. If we make the further assumption (as MacArthur and Wilson did) that these processes are roughly constant over long periods, then the number of species on the island eventually reaches a steady state, given by the value of the horizontal axis where the immigration and extinction curves intersect.

A key insight of MacArthur and Wilson was that larger islands can hold more individuals and, consequently, extinction rates on larger islands are lower.

(A)

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Moreover, islands closer to the mainland have higher immigration rates because it is more likely that individuals will arrive there from the mainland. Using their assumption that the islands had reached an equilibrium number of species, MacArthur and Wilson predicted that the greatest number of species would be found on large islands near mainlands (the source of immigrant species), and that the smallest number would be found on small islands far from mainlands (Figure 17.6C,D).

Real archipelagos, of course, are more complex: among other things, immigrants can arrive not just from the mainland, but also from other islands. A more important limitation of island biogeography theory is that it treats all species as if they were interchangeable. Clearly, this is not the case. Even very similar species have distinctive ecologies. MacArthur and Wilson acknowledged this limitation, although many ecologists subsequently trying to apply their theory did not. The equilibrium assumptions needed to predict species number are unlikely to hold, and empirical studies have provided little support for the theory’s predictions. Despite these limitations, the theory is useful. It provides a starting point for examining patterns and gives us a way of thinking about the movement of individuals in a landscape.

Metapopulation Theory

The theory of metapopulations, originally developed by Richard Levins (1969) and subsequently expanded

(B)

Immigration rate ( )

A2

A3

Mainland

A1

(C)

A2

A1

s1

Figure 17.6

(A) Mainland and archipelago. Islands A1 and A2 are the same distance from the mainland, while A1 and A3 are the same size. (B) The equilibrium number of species on island A1 is determined by the balance between the immigration and extinction rates. (C) The larger island (A2) will have a higher immigration rate and a lower extinction rate, resulting in a greater number of species at equilibrium. (D) The nearer island (A3) will have a higher immigration rate but the same extinction rate, resulting in a greater number of species at equilibrium.

Number of species (s)

BOX 17A

Metapopulation Models

Metapopulation models have the general form dp/dt = immigration rate – extinction rate where p is the percentage of patches occupied by a given species, 0 ≤ p ≤ 1, and dp/dt indicates the rate of change in occupation over time. Typically we are interested in solving this equation for p, which predicts the frequency of occupied patches in a landscape. Obviously, a species’ biology sets its immigration and extinction rates.

It is important to realize that in these models, immigration and extinction rates may not be independent of each other, and either or both may depend on the frequency of occupied patches, p. It is easy to see how the immigration rate might depend on p: the number of seeds arriving in a patch could depend on the number of occupied patches.

The extinction rate might also depend on p for the same reason: as the number of seeds being dispersed increases (with increasing p), there could be a reduced chance of local extinction. This phenomenon is called the rescue effect (Brown and Kodric-Brown 1977) because new immigrants “rescue” the local population from extinction.

If both the immigration rate and the extinction rate are functions of p, the model takes the form

dp/dt = ip(1 – p) – ep(1 – p)

That is, the immigration rate is a function of the probability of immigration, i, times the fraction of available sources of immigrants, p, times the fraction of available patches, 1 – p. The extinction rate is a function of the probability of extinction, e, times the fraction

of patches with populations that can go extinct, p, times one minus the fraction of sites that can provide immigrants, thereby reducing the probability of local extinction, 1 – p.

If both extinction and immigration rates are independent of the fraction of occupied patches, the model reduces to

dp/dt = i(1 – p) – ep

Gotelli (1991) refers to occupancyindependent immigration as the propagule rain because immigration provides a steady “rain” of new individuals to each population. All four possible combinations of occupancydependent and occupancy-independ- ent immigration and extinction rates are conceivable and represent the extremes of a more general metapopulation model.

upon by Illka Hanski (1982, 1999), was another important stimulus for landscape ecology. In some respects, metapopulation theory is similar to the theory of island biogeography, but applied to populations rather than sets of species. Begin by thinking of an archipelago, but one far from a continent. The population size of a species on any given island is a result of local population dynamics plus immigration minus emigration (this is the same as Equation 7.1). A population on a given island may go extinct, but new immigrants from other islands may repopulate that island. The equilibrium between immigration and extinction determines the total size of the metapopulation—the number of surviving populations in the archipelago and their average size. Like the theory of island biogeography, metapopulation theory applies, of course, to many situations other than oceanic islands. The ideas of Levins were undoubtedly influenced by those of the population geneticist Sewall Wright, who envisaged a very similar process as part of his shifting balance theory of evolution (Wright 1931, 1968). We discuss the details of metapopulation models in Box 17A.

Metapopulation theory can be used to look at both population and community patterns. The theory predicts the distribution of population sizes for a species across a set of islands or patches, and how it changes over time. If one is willing to assume that species have

equivalent migration and extinction probabilities, the theory also predicts the distribution of population sizes among species within a community and the frequency of occurrence of species among communities. It is possible to create more complex versions of the theory by relaxing the assumptions of species and patch equality. Susan Harrison recently coined the term “metacommunity dynamics” for the extension of metapopulation theory to explain community patterns (Harrison 1999).

One important conclusion that comes from extending population dynamics to a landscape is that not all local populations need be self-sustaining. A sink population may be continually on the road to extinction, but may be maintained by constant immigration from a source population. The idea of source and sink populations is important because it emphasizes that local populations may often not be at equilibrium. Thus, testing theories based on equilibrium assumptions may be misleading. Island biogeography theory, for example, makes predictions based on equilibrium assumptions. During the 1960s, especially in response to island biogeography theory, equilibrium was often the dominant assumption among ecologists. Since the 1980s, the pendulum has swung in the opposite direction, and nonequilibrium is generally assumed, at least at a local level. A metapopulation perspective moves the equilibrium assumption up one level to the entire landscape. While individual