THE DOMAIN OF THIS CHAPTER is the development and testing of the MacArthur-Wilson Species Equilibrium Model. Naturally, most testing (as well as theoretical extension) followed rather closely the initial presentation (MacArthur and Wilson 1963, 1967) of this exciting, innovative conceptualization. My objective in this chapter is to focus mainly on this earlier research. As I discuss at the end of this chapter, papers citing the MacArthur-Wilson book have become very numerous in recent years. For this reason, an exhaustive review of current work is beyond the scope of my chapter. Rather, I focus on how the main aspects of the model, as presented by MacArthur and Wilson, have been evaluated in what I consider to be the most notable papers, many of which come from the older literature. As certain other chapters in this volume attest, the MacArthur-Wilson Species Equilibrium Model continues to inspire new research ideas, some far removed from the original kernels planted in the 1960s; because of my historical emphasis, I leave it to these other chapters to chart such future directions.
The MacArthur-Wilson Species Equilibrium Model was first presented as a graph of gross extinction and immigration rates against the number of species present on an island (MacArthur and Wilson 1963, 1967). In its most general form it makes two assumptions (figure 3.1):
1. The rate of immigration of new species (those not yet on the island) decreases monotonically with increasing number of species already present. It reaches zero when all species in the source area (there are P of them) are on the island.
Figure 3.1. The graphical version of the MacArthur-Wilson Species Equilibrium Model. The model is for a particular island. Ŝ is the number of species at equilibrium (when gross immigration equals gross extinction), and P is the number of species in the source pool. Rate curves are monotonic but nonlinear. The intercept of the dashed line on the ordinate is the turnover rate at equilibrium.
2. The rate of extinction of species increases monotonically as the number of species increases (the more species there are, the more to go extinct).
These two assumptions imply that an equilibrium between immigration and extinction will eventually occur, at which time the immigration and extinction rates will have the same value, called the turnover rate at equilibrium.
Both of these model results, equilibrium and turnover, were predictions bold for their time, and as such drew substantial controversy. An equilibrium in numbers of species runs counter to a previous view that far islands would have fewer species than near islands because of lower dispersal rates, but that, given enough time, they would approach the number of species on near islands (both ultimately limited by the number of “available” species in source areas and perhaps by opportunities for in situ speciation). Species turnover was even more controversial: many lists and manuals giving the species of some taxon found on a particular island had been and were continuing to be published; how could the species on islands be dynamic, such that the very identities of catalogued species change from one survey to the next? The degree to which equilibrium and turnover in fact have been found by investigators will concern us shortly, but first I note a few niceties for the MacArthur-Wilson Species Equilibrium Model.
The graphical model was first presented with nonlinear species immigration and extinction curves. MacArthur and Wilson(1967) argued that the immigration curve should be concave, declining more rapidly at first because the better dispersers would be the first to arrive, leaving poorer and poorer dispersers as the only species not on the island and thereby reducing the absolute rate of decline of the species immigration curve. Their major argument for the concavity of the species extinction curves, as elaborated a bit later by Wilson (1969) in a “Brookhaven Symposia in Biology” volume, was completely different: the more species, the greater the likelihood of deleterious, i.e., extinction-producing, species interactions (as a first approximation this extinction rate would be proportional to the square of the number of species on the island). A fair bit later, Gilpin and Armstrong (1981) presented their species-by-species theory, showing that the same argument MacArthur and Wilson used to justify the concavity of immigration curves applied to the concavity of extinction curves—if all species possible (P of them) are present and one loses species, one will lose the most extinction-prone species first. Put another way, Gilpin and Armstrong showed that it is simply the variation in the individual-species extinction and immigration probabilities (rates) that can give concavity.
Despite the greater realism of the nonlinear model, a linear version, first presented in detail in Wilson’s (1969) Brookhaven paper, gives us a feel for some of the important properties of this more limited version of an equilibrium model. He wrote the linear model as
where S is the number of species on the island at time t, λA is the per species immigration rate, μA is the per species extinction rate, P is the number of species in the source pool, and Ŝ is the number of species at equilibrium. Figure 3.2 graphs this model.
The differential equation (3.1) can be solved for the colonization curve, or the curve relating number of species on the island to time since the colonization process began; it is a convex exponential (figure 3.3), i.e.,
The convex form of the colonization curve is also a prediction that can be tested (the model above would not lead uniquely to this form, however). Inspection of equation (3.2) allows two additional insights. First, the rate of approach to equilibrium varies positively with both the immigration and extinction parameters (even though extinction diminishes the number of species). Second, equilibrium is approached at a decreasing rate (the slope of the colonization curve diminishes with time). This implies that islands not at equilibrium yet not too far from equilibrium are going to be strongly influenced by the same factors (the immigration and extinction rates, and whatever affects those quantities) as are islands effectively at equilibrium.
Figure 3.2. A. The linear version of the MacArthur-Wilson Equilibrium Model. Conventions as in figure 3.1. B. The colonization curve (species on an island versus time since beginning of the immigration/extinction process).
I now discuss the degree to which empirical tests supported the idea that islands are in a state of species equilibrium (turnover is considered in the next section). Although the discussion is grouped by system, it is arranged roughly in order of decreasing correspondence to the equilibrium prediction.
Figure 3.3. A. Birds of the Channel Islands, California (USA). Number of breeding species S for each island plotted against survey year. The number written over the line connecting each pair of points is the percent turnover between those surveys (Jones and Diamond 1976). B. Colonization curves of four mangrove islets, Florida (USA). E-2 is the nearest island and E-1 is the farthest island (Simberloff and Wilson 1970). C. Oscillating equilibrium in marine epifaunal invertebrates. Gross immigration and extinction rate curves (top) and colonization curves (bottom) (Osman 1978). The number of species changes seasonally as a result of corresponding changes in the immigration-rate curves.
1. Arthropods of red mangrove islands. Shortly after publication of MacArthur and Wilson’s book, Wilson and his student Simberloff performed a major test of the MacArthur-Wilson Species Equilibrium Model, reported in a set of papers entitled “Experimental zoogeography of islands.” (Wilson and Simberloff 1969, Simberloff and Wilson 1969; the history of these experiments is recounted in Wilson’s chapter in this volume). The mangrove Rhizophora mangle grows as isolated units in shallow marine waters; the areas of such “islands” can range from a few cm2 (a single recently rooted propagule) to groups of many clustered trees. Wilson and Simberloff hired a pest-extermination company to place sheeting over a number of moderately sized such islands and gas the arthropods within; this “defaunation” killed nearly all of the arthropods inhabiting the islands, and then Simberloff and Wilson monitored the recolonization of the islands. The islands typically recovered to their predefaunation species numbers in something less than a year, although the most distant island had a slower approach, not fully achieving its previous value even after two years (Simberloff and Wilson 1970). This is expected from equation (3.3). The form of the colonization curve (species versus time) was convex (figure 3.3b), also in accordance with the theory (see also next section).
2. Birds of the Channel Islands. Breeding birds of nine islands off the California coast were first surveyed in 1917. Jones and Diamond (1976) performed a number of surveys on each island, beginning approximately fifty years later; these repeated surveys extended over a period of four years (figure 3.3a). The data showed a great deal of constancy in the number of species, although species identities were quite different (see next section).
3. Birds of the islands in the Aegean Sea. During 1988–92, Foufopoulos and Mayer (2007) resurveyed five islands that were first surveyed by Watson (1964) in 1954–61. No species count changed by more than one species, thereby providing strong evidence for an equilibrium species number.
4. Birds and plants of Krakatau. In 1883 a huge volcanic eruption destroyed two-thirds of the Indonesian island of Krakatau and buried its remnants and two neighboring islands under 30–60 meters of ash, with no apparent plants or animals surviving. Unlike for the mangrove islands discussed above, prior records of the bird or plant inhabitants of the islands were unavailable for this unanticipated “natural experiment.” For birds, however, the recolonization (as documented by Dammermann [1948]) appeared to MacArthur and Wilson (1967) to be leveling off after only 25–36 years, a pattern that they interpreted as major support for their equilibrium theory. Subsequent studies, however, showed that the conclusion was premature, equilibrium perhaps not being quite attained even a century after the eruption (Bush and Whittaker 1991, Thornton et al. 1993). Plants, in contrast, showed a much slower rate of recovery, and in MacArthur and Wilson’s (1967, figure 22A) illustration there was little if any indication of convexity in the colonization curve. The most recent censuses, about 100 years later (Bush and Whittaker 1993), show otherwise: equilibrium appears perhaps nearly attained for seed plants (figure 3.4, top left) and ferns (compiled by Thornton et al. 1993). For plants, MacArthur and Wilson (1967) predicted that extinction rates might actually decline at first during the period when initially arriving species facilitate the establishment and persistence of subsequent species; figure 3.4 (top right) reproduces the relevant figure from their book. In fact, nonmonotonic curves have been reported (Thornton et al. 1993), but for immigration rather than extinction (figure 3.4, bottom left). However, as Lomolino et al. (2005) pointed out, this discrepancy most likely reflects how immigration is defined in the two treatments (initial immigration [so sensu stricto], versus recent immigration plus establishment, the quantity available from censuses widely spaced in time). Interestingly, a similar nonmonotonicity is apparent, perhaps to a slightly lesser extent, for birds (figure 3.4, bottom right). In any event, recent species-by-species analysis of extinction among Krakatau plants (Whittaker et al. 2000) concludes that successional loss of habitat (as well as to a lesser extent other habitat disturbance or loss) largely accounts for the extinction of well-established species.
5. Marine epifaunal invertebrates on rocks. To simulate colonization of rocks, Osman (1978) set out artificial panels in the marine subtidal of Massachusetts. This experiment produced an oscillating equilibrium (figure 3.3C), in which the number of species increased toward a regular cyclical rise and fall of species. The low point of the cycle occurred in the winter months and the high point in the late summer, as would be expected in this highly seasonal environment.
Figure 3.4. Top left. Species number versus time for seed plants of Krakatau (Rakata; Thornton et al. 1993). Top right. Immigration and extinction curves vs. time for a biota showing succession (MacArthur and Wilson 1967, fig. 23). Bottom left. Immigration and extinction curves versus number of species present for seed plants of Krakatau (Thornton et al. 1993). Bottom right. Immigration and extinction curves versus number of species present for resident land birds of Krakatau (Rakata; Thornton et. al. 1993).
6. Birds on islands off Australia and New Zealand. Censuses conducted over periods of 50–124 years for fifteen islands in the Australasian region showed that their number of passerine species did not fluctuate around an apparent equilibrium; rather, in fourteen of fifteen cases the species counts increased, up to 900% of the original values. Abbott and Grant (1976), who compiled these data, argued that direct human changes were insufficient to account for these systematic increases. Rather, they suggested (somewhat presciently) that climatic warming might have been responsible. Abbott and Grant (1976) entitled their paper “Nonequilibrial bird faunas on islands,” and these islands certainly stand in marked contrast to the Channel Islands discussed above.
7. Plants and ants on islands of the Bahamas. Surveys of both plants and ants spanning nearly two decades on approximately 200 Bahamian islands by Morrison (2002, 2003, in prep.) showed a similar “nonequilibrial” situation as for the birds of the previous example. However, the direction of change was the opposite: islands lost plant and ant species during the second decade of the study rather than gained them. Foliar cover of plant populations whose species did not completely go extinct showed a steady decline over that time. The relative abundance of ant populations that did not go extinct declined in the second decade of the study as well. Although several hurricanes struck the region (see no. 9 below), the direct impact of hurricanes did not appear to be the main cause. Morrison suggested that decreasing precipitation and increasing temperatures in the region, along with potential increased herbivory of plants due to hurricane and drought stress, could be contributing factors.
8. Birds on Skokholm Island. Abbott and Grant (1976) compiled data on numbers of bird species for a small island off the British mainland, recorded 1928–67, with time off for the war years. The species number fluctuated between 5 and 13, with substantial temporal autocorrelation (figure 3.5). These are large percentage changes, so they might be interpreted, as Abbott and Grant did, as evidence against equilibrium. However, MacArthur and Wilson’s (1963, 1967) original theory went well beyond the simple deterministic, graphical or algebraic model presented above, including a stochastic version with per-unit-time probabilities of immigration and extinction rather than fixed rates. The implication for present purposes is that the “equilibrium” number of species is expected to vary around some mean, rather than be constant once an average equilibrium is attained. Box 3.1 reproduces and extends somewhat the MacArthur-Wilson mathematics to show that the variance/mean number of species will fall between 0 and 1. For the Skokholm data, the mean is 6.59 and the variance is 4.37, so that variance/mean ~ 2/3. This relatively high value is to be expected (box 3.1) from the high extinction rate that should characterize the low populations on this very small island (see also comments in the next section under no. 5).
9. Hurricane effects on Bahamian lizards and spiders. In 1996 the massive Hurricane Lili swept east to west across the Bahamas, bringing with it a storm surge of up to ca. 5 meters (Spiller et al. 1998). Such an inundation was devastating for many small, very low islands of the region, including the eleven islands to the west of the main island of Great Exuma, directly in the path of Lili. As part of an introduction experiment, Losos and Spiller had been collecting faunal data for these eleven islands, as well as eight protected islands to the east of Great Exuma, up to the very moment the hurricane struck. After its passage, they retrieved their boat from a tree (so the story goes, anyway) and recensused the islands. On the exposed islands, every lizard and web-spider individual originally inhabiting the islands was gone. However, spiders on the devastated islands were not entirely absent: webs of a few individuals of a species never before found on the islands (Metazygia bahama) were found clinging to bare rock (subsequently this species completely disappeared). It is notable in this regard that the first colonist of Krakatau was also a spider (Thornton 1996)! On the protected islands, no lizard population became extinct as a result of the hurricane, and the likelihood of extinction for spider populations on those islands was negatively related to their population size.
Figure 3.5. Number of breeding passerine bird species for Skokholm Island through time (figure drawn from data in Abbott and Grant 1974). Note the substantial variability in number of species over the time period studied.
BOX 3.1. Derivation of the limits of
the variance/mean for number of species
around equilibrium.
Note: λ and μ are MacArthur and Wilson’s (1967) notation and refer to gross rates in this derivation.
MacArthur and Wilson show that
We can graph the gross immigration and extinction rate as follows (this corresponds to our figure 3.1)
Then
at equilibrium. Letting λŝ = μŝ = X (the common value of the two rates at equilibrium), it follows that (see figure)
Substituting into the previous equation, we get
If 
var/mean ≤ 1/2, the MacArthur-Wilson result, but in general var/mean falls between 0 and 1.
When was the original species equilibrium recovered for this natural defaunation? The answer depends on the organism. For spiders, recolonization was rapid and in one year, the number of species on average was the same as before the hurricane struck (figure 3.6; the islands were all less than 15 km from the main island of Great Exuma). In complete contrast, the number of species of lizards was still at zero on the exposed islands, and at the last survey date (2001) only two of the islands had been colonized by lizards (in protected areas, three of five islands naturally having lizards were colonized, but none of the eight introduction islands was). Thus it appears that equilibrium depends on the organism: for highly vagile organisms like spiders, which disperse mainly by ballooning through the air, equilibrium can be recovered quickly, just as it was for Simberloff and Wilson’s mangrove arthropods discussed above. For lizards, which have to disperse by rafting or floating (Schoener and Schoener 1984), attainment of equilibrium may take a very long time, indeed a longer time than the next devastating catastrophe, implying that lizards may never be at equilibrium. Certainly, spiders and lizards on the same set of islands differ in the likelihood that they will be at equilibrium at a randomly chosen point in time.
Figure 3.6. Mean number of individuals and of species (± one standard error) for web spiders on islands immediately before, immediately after, and one year after devastating hurricanes struck two regions of the Bahamas (Schoener and Spiller 2006). The patterns for two different hurricanes in two different regions are nearly identical.
Two footnotes to these results are interesting. First, even though the number of species of spiders attained equilibrium after one year, the total number of spider individuals fell short of the value before Lili struck (figure 3.6). Second, the pattern in both number of species and number of individuals was repeated with nearly the same relative values after the storm surge of Hurricane Floyd in 1999 wiped out the spiders on a more northerly group of islands, those off the main island of Great Abaco (figure 3.5; Schoener and Spiller 2006). That two hurricanes would recently occur, for both of which predefaunation data were available, seems serendipitous, although the likelihood of further confirmation of these patterns is perhaps not small, given the increase in hurricane frequency presently characterizing the Caribbean.
10. Arthropods in soybean fields. An even more extreme example of draconian extinction being frequent relative to how quickly equilibrium is attained was described by Price (1976). Croplands are highly temporary habitats for which “defaunation” is a scheduled human activity; combined with seasonal variation, this results in catastrophic extinction followed by a period of little to no recolonization. Once the crop has been replanted and is growing again, arthropods begin to colonize it, but they do not have time to reach an equilibrium before the next catastrophic harvest.
These examples allow us to make the following conclusions concerning the existence of equilibrium:
First, equilibrium can be steady (a constant number of species), cyclical (a regular fluctuation in number of species), or moving directionally (a slow, undirectional change in numbers of species brought on by a systematic change in immigration and/or extinction rates, e.g., due to climate change). Many examples of the first possibility were discussed above, and Osman’s (1978) work on marine epifaunal communities illustrates the second. The third is not clearly demonstrated by any of the examples above, except possibly the birds of Australian and New Zealand islands and the plants and ants of Bahamian islands; however, one could certainly argue that those two examples are nonequilibrial, and indeed that is what their investigators have done.
Second, even for a steady average equilibrium, there is expected to be a variance according to the stochastic version of the MacArthur-Wilson equilibrium model (as well as for any other such type of model). In the case of Skokholm Island, the variance was large but within theoretical expectations.
Third, a system approaching equilibrium can have similar properties to one at equilibrium, e.g., with respect to area and distance effects (see below). Because number of species is expected to approach equilibrium at a decreasing rate (second derivative negative), as in equation (3.2), communities are expected to show qualitatively the same effects of factors affecting immigration and extinction rates, even if those communities are moderately far from the equilibrium number, and perhaps even over the majority of the colonization period. Hence it would not be fair to argue that, because an island community is not at equilibrium, a species-area effect as predicted by the equilibrium model (if island area is related to extinction rate; see below) will not occur.
Fourth, the more frequent the disturbance rate, the less likely equilibrium is to be attained. Disturbance, as in lizards of the Bahamas and arthropods of soybean fields, can wipe out a biota before, and sometimes well before, there is time to attain equilibrium.
Fifth, for a given rate of disturbance, equilibrium is more likely to be attained by organisms that are good dispersers (giving a higher immigration rate and thus a faster approach to equilibrium as in equation [3.2]). An example is lizards and spiders on the same Bahamian islands; the latter attain species equilibrium quickly after catastrophic hurricanes, whereas the former may never do so.
The second prediction of the MacArthur-Wilson Species Equilibrium Model, that species lists will vary in composition even after equilibrium is attained (as well as on the way to equilibrium) is even less intuitive than the equilibrium prediction itself; we now review evidence for species turnover. The most commonly used measure of this quantity is relative turnover, given as
(This equation can actually be viewed as having two averages, one in the numerator and the other in the denominator, so the 2’s in each of these averages cancel out. A second kind of measure, absolute turnover, does not normalize by species counts but simply computes the average of the absolute numbers of species immigrating and becoming extinct over the time period.)
1. Arthropods of red mangrove islands. The Simberloff-Wilson colonization curves show a roughly monotonic approach to equilibrium (figure 3.3, top), and this is accompanied by a patchy record of individual-species presences and absences, with particular species immigrating and then going extinct, some repeatedly, during the colonization process. Moreover, once the old equilibrial number is regained, the composition of arthropod species is substantially different from that determined shortly before the artificial defaunation.
2. Birds of the Channel Islands. Using a formula similar to equation (3.3), Diamond (1969) found very high turnover for birds of the Channel Islands separated by censuses 51 years apart. The conclusion was challenged by Lynch and Johnson (1974), who argued that, among other problems, species were missed during one or the other census, thereby artificially inflating the reported turnover rate—a phenomenon labeled “pseudoturnover” by Simberloff (1974). However, subsequent censuses by Jones and Diamond (1976) annually over a period of several years showed that in fact turnover was substantial, primarily because of entire missed sequences of immigration followed by extinction for particular species—“cryptoturnover” (Simberloff 1974). In fact, their year-by-year data showed turnover at 0.5–4.9%, whereas the two censuses in Diamond’s (1969) original study gave 0.3–1.2%, if anything too small. Hence, if the original two censuses missed species, they were more than compensated for by entirely missed immigration/extinction sequences for particular species during the long interval between the censuses.
Shortly after the data were published, Diamond and May (1977) presented an elegant mathematical treatment of how measured (“apparent”) turnover is expected to decline with increasing time between censuses (box 3.2). For the “island” treated by Diamond and May—the Farne archipelago (near Skokholm; see above)—predictions match data rather well (figure 3.7). The turnover rate per year T(1) equals 0.13 or 13%. For intervals exceeding about ten years (T ≥ 10), turnover is underestimated by about an order of magnitude. Note that the possible variety of species for this high-latitude site is limited by a rather low diversity of immigrants, so to some extent the same species wink in and out. Also note that most of the species are migrants and present in very small numbers, further contributing to a high turnover rate.
BOX 3.2. Derivation of the relation of apparent turnover T after time t (the period between two successive censuses) to t.
Let Ii(t) and Ei(t) for Species i be the probability of, respectively, being present at t yrs if initially absent and of of being absent at t yrs if initially present. The incidence (which gives the fraction of time periods occupied by a given species [or the fraction of islands at any time occupied by a given species]) is given by
Equilibrium species number sums over the incidences:
The apparent rates (those quantities measured by the investigator over a census period), Λ(t) and M(t), sum up the Ii’s and Ei’s for all species:
Note that Λ(t) = gross immigration and M(t) = gross extinction. They are equal at equilibrium.
The apparent turnover T(t) is calculated as
which is our equation (3.1) divided by the length of the time interval. At equilibrium, S1 = S2 = S*, so substituting from the above equations, we get the apparent turnover after time t as
3. Birds of the Aegean islands. Despite a very strong tendency toward equilibrium in species numbers, the five islands studied by Foufopoulos and Mayer (2007) showed a great deal of turnover over the same period, comparable to values for other temperate islands as reviewed above.
4. Birds and plants of Krakatau. MacArthur and Wilson’s (1967) original estimates for extinction of birds in this archipelago are now known to be much too high, perhaps by a factor of about 3 (Thornton et al. 1988): Their estimates are 0.5–1.6% per year, whereas recent estimates are 0.25–0.42% per year. Similarly, previous extinction rates for the plants of Krakatau are “significantly overestimated” (Whittaker et al. 2000): New data reduced the pseudoturnover contribution, and the extinctions that are thought to have occurred involved human introductions as well as rare or ephemeral species. As stated above, losses were mainly due to successional loss of habitat and to a lesser extent, other habitat disturbance or loss.
Figure 3.7. Apparent turnover rate (T) of breeding land bird species on the Farne Islands, expressed as the fraction of breeding species immigrating or becoming extinct per year and calculated from differences in the species list for pairs of censuses t years apart. Circles, mean observed T; vertical bars, observed mean ± 1 standard deviation. Solid curve, mean predicted T; dotted curves, predicted mean ± 1 standard deviation (Diamond and May 1977).
5. Birds of tropical islands other than Krakatau. Values of turnover for the Channel and Aegean Islands, which are temperate, are large. In contrast, certain tropical islands (those not subjected to recent disturbance) have much lower turnover. Abbott and Grant (1976) noted that, over a 72-year interval, the Tres Marías Islands off western Mexico had only two immigrations. Even more extreme, Slud’s (1976) data show that the Neotropical Cocos Island had no turnover in 72 years, and One Tree Island in the Great Barrier Reef region had no turnover during six continuous years of observation (Heatwole et al. 1981). Most sensationally, a seven-year survey (1984–90) by Mayer and Chipley (1992; see this paper for additional references), with additional censuses in 1954 and 1976, found no immigrations and only one extinction for Guana Island in the Caribbean. This stability is also in contrast to the Australasian islands discussed in the equilibrium section (no. 6).
However, some tropical islands show higher turnover. In 1986 an extensive hydroelectric project flooded a huge area in the Caroni Valley of Venezuela, creating islands in Lago Guri that had formerly all been part of a single land mass. Surveys by Terborgh and colleagues (1997; see Terborgh’s chapter in this volume) found that a new equilibrium was achieved in just seven years on the smaller islands, while the larger islands are still declining. Similar phenomena occurred in relation to the massive changes when the Panama Canal was constructed (review in Lomolino et al. 2005). Here, as in Lago Guri, turnover was somewhat lower the larger the island; it was also lower for far than near islands (except for the nearest three islands; Wright 1985). The general patterns are consistent with the MacArthur-Wilson Species Equilibrium Model (see next section) or a modified such model (see Wright [1985] for details). However, while turnover is substantial for these tropical islands, they are perhaps not comparable examples to those of the preceding paragraph, as the islands were in a recently very much disturbed state, being essentially young landbridge islands relaxing to a new equilibrium. Further, the islands studied were very close to the mainland, having indeed been recently a part of it. Finally, as pointed out in the previous section, the extinction component of turnover for the Krakatau archipelago is now known to be much smaller than was originally thought, despite the recently disturbed nature of that region.
Thus there may well be a difference for birds in turnover between the average temperate versus tropical island. Why might this occur? Mayer and Chipley (1992) suggested it is because tropical birds have lower immigration rates (they are locally more sedentary), lower mortality, and are nonmigratory.
6. Spiders on Bahamian Islands. What kinds of species show turnover? The question is easiest to answer for extinction, which shows a strong relation to population size when looked at empirically or theoretically (e.g., the above studies for Bahamian spiders, Channel Island birds; see the theoretical review in Schoener et al. 2003). This brings up the issue: How important, in terms of total population numbers of all individuals combined, are species showing turnover? Indeed, something of a contradiction runs through the various theoretical papers written by MacArthur: some papers assume a community of competitors that is commonly at population-size equilibrium (MacArthur 1968); other papers postulate that turnover, which involves the entire disappearance of species (to say nothing of changes in abundance) is commonplace.
An attempt to answer this question precisely was made for Bahamian spiders by Schoener and Spiller (1987), who calculated the percentage of all individuals combined belonging to populations becoming extinct over particular intervals, ranging from one to five years. Using one-year intervals, 2.8% belonged to populations becoming extinct. Using five-year intervals, still only 4.8% did so. Turnover, while quite large in terms of species number (about 35% per year), does not involve the most abundant species, those that should often have the greatest food-web effects and in any event are of most interest to ecosystem, as opposed to biodiversity, ecologists. In this system, often the same species become extinct and reimmigrate, much as portrayed in Hanski’s (1982) core-satellite scheme. Population-persistence curves, which give the fraction of species populations remaining n years after a particular census, show this more precisely (figure 3.8). The curve for all web-spider species combined levels off quite sharply (even on a semilogarithmic scale). Interestingly, the individual species vary in the degree to which a leveling off occurs: Gasteracantha cancriformis has a practically exponential decline, i.e., a straight line on a semilog plot (produced by a constant per time probability of a population becoming extinct). In contrast, Eustala cazieri and Metapeira datona show a marked curvature even on a semilog scale, implying that many of their populations persist for long periods of time. The mostly ephemeral nature of the populations going extinct is similar to the situation for Krakatau plants (Whittaker et al. (2000).
These examples allow us to make the following conclusions about species turnover:
First, complete turnover events (immigration followed by extinction of a particular species) are often missed in surveys, which typically are separated by substantial intervals. While it is possible that the opposite type of error will occur (designating a species absent that was in fact present because of an incomplete survey, thereby inflating the turnover estimate), for intermittent censuses, missed complete sequences are expected to be common enough so that turnover will typically be underestimated.
Figure 3.8. Population-persistence curves for web spiders on 108 islands of the Bahamas. Top. All species combined. Bottom. Individual species curves (Schoener and Spiller 1987). Note that, overall, while some species become extinct rather quickly, about the same percentage persist throughout the study period. The bottom panels show the four commonest species, which differ considerably among themselves, and sometimes in comparison to the overall pattern.
Second, turnover tends to be greater for small islands and for far islands, in accordance with the MacArthur-Wilson Species Equilibrium Model (see next section).
Third, turnover can be very low for tropical islands, but for those recently disturbed or created, this is not necessarily the case.
Fourth, species turning over may comprise a low fraction of the total number of individuals in the biota—this results from the very strong relation between extinction rate (one of the two components of turnover) and population size. Such species can be important for species-diversity studies but would seem epiphenomenal for ecosystem studies.
The MacArthur-Wilson Species Equilibrium Model makes predictions about the effects of an island’s distance from the source of immigrants and about an island’s area, as follows. Assume
1. Near islands have higher immigration rates than far islands, for the same number of species present; and
2. Small islands have higher extinction rates higher than large islands, for the same number of species present. This is because average population size is smaller for the smaller islands, hence the per species extinction likelihood is greater—note that a decreasing relation between extinction likelihood and population size has been repeatedly demonstrated, e.g., Jones and Diamond (1976), Terborgh and Winter (1980), Schoener and Schoener (1983b), Schoener and Spiller (1987), Pimm et al. (1988), Laurance (1990).
These assumptions imply two results (figure 3.9). First, near islands (of the same area as far islands) have more species. Second, large islands (at the same distance as small islands) have more species. Both predictions are consistent with numerous examples from the literature (reviewed in Lomolino et al. 2005). Note that the graphs of figure 3.9 also imply that absolute turnover (intercept on the ordinate) is greater for near than far islands and greater for small than large islands (for relative turnover, equation [3.3], which can be different, see Williamson [1978]).
Figure 3.9. Left. The distance effect for the MacArthur-Wilson Equlibrium Model. Far islands have lower immigration rates than near islands, resulting in a smaller number of species present at equilibrium. Right. The area effect for the MacArthur-Wilson Equilibrium. Model. Large islands have lower extinction rates than near islands, resulting in a larger number of species present at equilibrium. Axes as in figure 3.1.
Species-distance relations have had a variety of explanations, only one of which follows from the original MacArthur-Wilson (1963, 1967) model.
First, far islands are less likely to be at species equilibrium than near islands because of their lower immigration rates, but given enough time will eventually achieve the same number of species as otherwise similar near islands. This is a nonequilibrium explanation for the common observation of biotic poverty on isolated islands.
Second, far islands have a less diverse range of habitats, thereby supporting fewer species that depend on those habitats. This explanation says that far and near islands are not “otherwise similar,” but differ in the key feature of habitat complexity. Lack (1976) used this idea to explain the lower species diversity of birds on far islands. The explanation is somewhat circular for the entire biota, of course, as a lower habitat diversity for birds would probably imply a lower plant-species diversity, and that would in turn beg explanation.
Third, because of a lower immigration rate far islands may reach equilibrium at a smaller number of species than do near islands. This is the MacArthur-Wilson (1967) explanation, and it is a bit difficult to express without mathematics; the graphical model (figure 3.9 left) is more transparent: an island having a lower immigration rate will balance with its extinction-rate curve at a smaller number of species. It of course differs from the first explanation in that this predicted state of affairs is expected to last forever (at least until the immigration rates change).
Tests distinguishing the first from the third explanations are rare except for short-term experiments such as that of Simberloff and Wilson on mangrove arthropods discussed above (see Schoener [1988] for other examples). Schoener and Schoener (1983a) were able to distinguish the second from the third explanation for Bahamian resident birds and lizards, which showed distance (and area) effects. The fraction of vegetation in different height categories was used to construct habitat-diversity indices, and Lack was correct that far islands had a lower habitat diversity than near islands. However, accounting for that relation in partial correlation still resulted in significantly negative distance relations. This last result is certainly consistent with the MacArthur-Wilson Species Equilibrium Model, although some of the islands, as least for lizards, may not be at species equilibrium (see above).
The list of explanations for the species-area effect is even longer than that for the species-distance effect (Spiller and Schoener, in press):
First, some kind of random sampling could produce the effect, independently of a well-defined mechanistic process. For example, imagine only that large islands have more individuals of a given kind of organism than do small: draw (or allow to colonize; see “third” below) more individuals from the source’s species-abundance distribution for large than small islands, and more species will result on large islands.
Second, populations are larger on larger islands, implying lower extinction rates there. This is the MacArthur-Wilson assumption, and like the distance effect is somewhat difficult to express without mathematics; note from the graph (figure 3.9, right) that an island having a lower extinction rate will balance with the immigration-rate curve at a greater number of species. The assumption relating population size to extinction likelihood is very well supported by data, as discussed above.
Third, interception area (or shoreline) is larger for larger islands, implying a greater immigration rate for larger islands (not just a smaller extinction rate). This so-called “target effect” has been shown for a variety of organisms (reviewed in Lomolino et al. 2005). These include the striking result of Buckley and Knedlhans (1986) in which species diversity of seaborne plant propagules is linearly related to shoreline length for islands off Australia, and the demonstration of Lomolino (1990) that immigration rates of mammals to islands in the St. Lawrence River were positively correlated with island area (see also Rey 1981, Schoener and Schoener 1981, Hanski and Peltonen 1988). It is possible, of course, to add this effect to MacArthur and Wilson’s original model, giving a more complicated set of curves. The corresponding effect for area is the “rescue effect” of Brown and Kodric-Brown (1977), in which extinction rate varies with distance: the nearer the island, the more likely populations on that island will be “rescued” from extinction by numerical reinforcements from the mainland; the greater flow from the mainland also could enhance genetic diversity on the island and prevent inbreeding depression, again reducing the chance of extinction. Oddly, few demonstrations of this rescue effect seem to exist additional to the arthropods-on-thistle-head example in Brown and Kodric-Brown’s seminal paper. Smith (1980) showed that talus-inhabiting pikas (Ochotonia princeps) had lower extinction rates on “islands” near to a source of immigrants (see also Wright 1985, Lawrance 1990). One system, Bahamian web spiders, shows all four possible relations—the traditional area and distance relations of MacArthur and Wilson, as well as the relations of immigration to island area and extinction to island distance (Toft and Schoener 1983).
Fourth, habitat diversity is higher on larger islands, leading to the ability to support a greater diversity of ecologically distinct species there. Perhaps even more than for distance, the relation of species number to habitat diversity to area is likely to hold; the altitudinally zoned, diverse vegetation characterizing higher islands, which tend to be larger, constitutes a good example. Indeed, sometimes the relation of species number to habitat diversity is stronger than that for area, e.g., the study by Watson (1964) of birds on the Aegean Islands; an overview is presented by Ricklefs and Lovette (1999).
Fifth, abiotic disturbance is larger on smaller islands, implying a greater extinction rate there. Evidence for this idea comes again from Bahamian lizards: larger islands, which tend to be higher, were less likely to lose their lizards as a result of the storm surge that accompanied Hurricane Floyd (Schoener et al. 2001): lizards could survive the inundation if on high enough ground. This example also illustrates a consequence of the correlation between two island traits—maximum altitude and area. Altitude was in fact more important than area in forestalling extinction (Schoener et al. 2001); however, when altitude was not taken into account in the statistical analysis, area was significant.
Sixth, within-island multiplication of species is greater for larger islands. This idea was demonstrated conclusively by Losos and Schluter (2000) for Caribbean Anolis lizards, and it is discussed elsewhere in this volume (Losos and Parent). It has also been measured and modeled for endemic land mammals by Heaney and colleagues (summary in Heaney 2004).
Plots of species-area relations are commonplace in the literature, and they fall into two general categories, a linear relation on a semilogarithmic scale (as implied by an exponential function)
and a linear relation on a log-log scale (as implied by a power function)
Figure 3.10. Species-area plots showing the semilog and log-log relation, top and bottom respectively. Top. Shetland land birds. Bottom. Malaysian faunal region land birds. (Schoener 1976)
where S is number of species, A is area, and c1, c2, c, and z are constants typically to be fitted to the data. Which description is better, equation (3.4) or (3.5)? Connor and McCoy (1979) interpret their review of 100 data sets to say that the two fit about equally. Clear examples of each of the two are given in figure 3.10, in which arithmetic bird species number increases linearly with log(area) for the Shetland islands, whereas logarithmic bird species number increases linearly with log(area) for the Malaysian region (note, incidentally, that the plot for Malaysian islands on a semilogarithmic scale is especially accelerating for the largest islands, perhaps due to within-island species multiplication).
What is the form of the species-area relation implied by the MacArthur-Wilson model? Using equation (3.1) above as a starting point, Schoener (1976) has shown that where abundances at equilibrium are complementary (defined as abundances summed over all species equaling ρA, where ρ is the density of all individuals combined and A is island area),
where λA, μA, and P are as in equation (3.1) and μN is proportional to μA. Equation (3.6) results from assuming (1) μA = μN/
, where is the average population size and (2) = ρA/S (other possible assumptions are in Schoener [1976]). Substituting these into equation (3.1) and solving the resulting quadratic in S gives equation (3.6). For this expression, unlike the descriptive power or exponential functions, the number of species asymptotes at P, the number in the source (note that within-island diversification by in situ speciation is not in the model). In other words, no matter what the area, there can be no more species on the island than that number available for colonization, a property that must be true for any MacArthur-Wilson-like model. Note also that the slope on a log-log plot (z = dlogS/dlogA) is not constant but goes from 0 to 0.5 in this model (a model in which individual species abundances are additive, not complementary, extends the range of z to 1.0; Schoener [1976]).
In the equilibrium species-area model (equation [3.6]), the greater the λA (per species immigration constant) the smaller dlogS/dlogA. Indeed z is smaller for less remote islands within a single archipelago (also z for an archipelago of habitat islands, where immigration is presumably very high, can be very small, e.g., Watling and Donnelly [2006]; see Holt, this volume, for review). But far archipelagoes have smaller z’s than near archipelagos (figure 3.11; also see Connor and McCoy 1979). This is probably because of a differentially high λ (per species immigration rate) among birds that have been able to colonize such archipelagoes. To elaborate, for far archipelagos most immigration is from other islands within the archipelago; for this component, both P and the immigration rate (which varies with P) are relatively small. Immigration from outside the archipelago (say from some large continental source) is minimal despite the large number of species in the pool, P, because of the much lower λ. For near archipelagos, most of the colonization is from sources external to the archipelago, and this gives an immigration curve with a large intercept on the rate axis as well as a large P. Figure 3.12 illustrates this argument. Various evidence additional to that just cited suggests that this model is on the right track. For example, the species-area slope for birds on islands of Burtside Lake, Minnesota (United States) is unusually high, but P is very large and the islands are very small (Rusterholz and Howe 1979).
Figure 3.11. The species-area slope (log-log) or z versus distance to the nearest source (as measured from the edge of the archipelago to the nearest large land mass). Numbers give area of the largest island; clear circles are archipelagoes with only islands less than 710 mi2; shaded circles are archipelagoes with largest island greater than 1500 mi2 (details in Schoener 1976).
The species-area representation of the MacArthur-Wilson model (equation [3.6]) also suggests a relation between the per species extinction rate μA and the steepness of the species area slope z: the greater the extinction rate, the greater the slope. Assuming that an increase in predation intensity can be represented by an increase in per species extinction likelihood, this implies that a biota subjected to predation should have a larger log species/log area slope (see also Holt 1996, Holt et al. 1999). However, Ryland and Chase (2007) used a different extension of the MacArthur-Wilson Species Equilibrium Model to get the opposite result: the greater the predation intensity on a biota, the smaller the species-area slope. In the Ryland-Chase extension, the contribution that predators make to the per species extinction rate is assumed additive, not multiplicative as in equation (3.6), and this seemingly minor change in functional form reverses the direction of the prediction. In Holt’s chapter in this volume, this analysis is generalized to allow for the extinction factor or addend to itself be a function of area, and in that case results can be more complicated. In neither approach, however, is there a mechanistic or biological justification for the functional form of the respective assumption about how predation affects extinction. Moreover, using a completely different approach, a continuous-time Markov model, Holt (1996; Holt et al. 1999) predicted that the higher the trophic rank, the larger the species-area slope. This result follows from the colonizing properties of predators and prey: higher trophic ranks cannot colonize unless a member of the next lowest rank is present. However, they won’t always colonize even when that is true—this is a necessary but not sufficient condition. This leads to a larger (or at best equal) species-area slope, the higher the trophic rank in a given community. Finally, equation (3.6) suggests another way that predators might have larger species-area slopes than prey: the lower the population density ρ of the group in question, the larger the slope.
Figure 3.12. Equilibrium for near and far archipelagoes. See text for explanation and definition of symbols (Schoener 1976).
The preponderance of data collected so far (Hoyle 2004, Ryland and Chase 2007), including ten-year averages for web-spider data from 64 islands from the central Bahamas (near Staniel Cay; Spiller and Schoener [in press]), supports the prediction that predators should have steeper species-area slopes (z’s) than prey. There is even a rough correspondence between (surmised) low population density and greater z among birds (Schoener 1976); raptors are in the low-density group. However, more northerly Bahamian spider data (from the Abaco region; Schoener and Spiller 2006) if anything suggest the opposite, supporting the prediction from equation (3.6).
A final form for the species-area relation has been suggested by Lomolino and Weisen (2001; see precursor ideas in Lomolino [2000] and Lomolino et al., this volume), one having essentially an S-shaped segment, i.e., a greater rate of increase for intermediate-sized islands than either for small or large islands; note that the low slope for the smallest islands (where variation in species number is expected to be greatest because of stochastic factors) is the feature of this concept that makes it very different from any of the species-area curves proposed so far, descriptive or mechanistic. Some evidence for such a slope was indeed given in MacArthur and Wilson’s (1967) book for a particular case—plants on the Micronesian atoll of Kapingamarangi (Niering 1963). However, their explanation was quite different: freshwater lenses are absent on islets below a certain area, giving a very low and constant species number there. The upper, leveling-off portion of the “S” has the same explanation as that for equation (3.6) above: any system with an upper limit to the number of species available to immigrate to an island (P in this case) will have a species-area curve that will tend to level off in its upper portion. In their analysis of 102 insular data sets, Lomolino and Weiser (2001) showed that an increasing portion of the species-area curve is quite general: the initial flat portion of the species-area curve typically included a substantial portion of an archipelago’s islands. The authors also point out that the final portion of the species-area curve, should there exist within-island species multiplication, may again accelerate.
I would like to close by reminding the reader of the word “chronicle” in the title of this chapter. This word attempts to bolster the legitimacy of my approach of dealing with the mostly older papers (see the introduction). At an early stage of preparing my presentation, I was concerned about the following question: Were most papers that dealt specifically with the MacArthur-Wilson Species Equilibrium Model in fact older? This would necessarily be true were most papers that cited the MacArthur-Wilson book and paper older. Optimistically, I went to the Science Citation Index to see what the more recent papers had to say, hoping to lace my presentation with a few appropriate citations. MacArthur and Wilson’s 1963 paper has a reasonable number of new citations, showing a modest if mildly erratic rise to about twenty-five citing references per two-year period (figure 3.13). However, I was shocked to find that MacArthur and Wilson’s 1967 book in recent years (2000–2007 inclusive) had over 2,000 citations, dashing any hope for an easy resolution of my question.
The pattern of citations itself is very interesting (figure 3.13). The number of citing references for the book increases sharply from 1967 until about 1985, at which point it levels off, showing an apparent “citation equilibrium.” However, in 2000 the number of citations begins another steep climb that continues unabated to the present time. Does the recent pattern of increasing citations imply that the influence of the MacArthur-Wilson theory, at least sensu lato, is again on the rise? Or is it simply a by-product of a recent increase in the overall numbers of citations, no matter what the significance of the work?
To analyze further, one would like some measure of the increase in citations that might be expected simply from the increase in number of citing papers, perhaps a comparison to a work that could serve as a “citation standard.” I was hard pressed to think of any such ecological work, given the ups and downs that so many ideas have received in this field. Then I had an inspiration: surely Darwin’s On the Origin of Species is a work that has not waned in influence and has had many years to achieve a constant citation rate per citing reference. All one had to do was normalize the MacArthur-Wilson numbers by dividing the latter by the number of citations of Darwin’s enshrined work. Strikingly, using this measure we find a completely different result than just using the raw number of citations: the MacArthur-Wilson book reached the apogee of its influence in 1975, after which it underwent an almost linear decline. One has the nagging feeling, however, that something has gone wrong with the analysis, and this is reinforced by looking at the citation curve for On the Origin of Species alone. It steadily increases, exceeding the MacArthur-Wilson book at about 1986 (where the lines cross in figure 3.13) and then continuing upward, even at a slightly increasing rate.
Figure 3.13. Absolute number of citations (left vertical axis) or normalized (by On the Origin of Species) number of citations (right vertical axis) per two-year period, against two-year interval. Triangles: Absolute number for MacArthur and Wilson’s (1963) paper. Circles: Absolute number for MacArthur and Wilson’s (1967) book. Squares: Absolute number for On the Origin of Species (all editions listed). Crosses: Normalized MacArthur-Wilson (1967) citations, i.e., circles divided by squares.
So where does this leave us? Is the true phenomenon of importance the relentless rise of Darwin, rather than anything to do with the MacArthur-Wilson statistics? If so, how can we explain the increasing popularity of Darwin—is that just due to the increasing number of citing references, or is something more going on? No doubt this topic will be discussed at length in 2009 during the 150th anniversary of On the Origin of Species.
I thank R. Holt, M. Lomolino, J. Losos, R. Ricklefs, and an anonymous reviewer for insightful comments on previous versions of this chapter and NSF Grant No. DEB-0444763 for support.
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