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CAN RAPOPORT'S RULE BE RESCUED? MODELING CAUSES OF THE LATITUDINAL GRADIENT IN SPECIES RICHNESS.

STEVEN D. GAINES [1]

Department of Ecology, Evolution and Marine Biology, University of California, Santa Barbara, California 93106 USA

Abstract. The latitudinal gradient in species richness, wherein species richness peaks near the equator and declines toward the poles, is a widely recognized phenomenon that holds true for many taxa in all habitat types. Understanding the causative mechanism or mechanisms that generate the latitudinal gradient in species richness (LGSR) has been a major challenge, and the gradient remains unexplained. A different latitudinal trend (named "Rapoport's rule"), in which the mean size of species geographical ranges tends to decline toward the equator, has been hypothesized by G. C. Stevens to play a key role in generating the LGSR when coupled with a version of the "rescue effect," in which local populations toward the fringes of geographical ranges are sustained by immigration. The Stevens hypothesis is now commonly cited as a potential explanation for the LGSR and has provoked numerous empirical studies in macroecology and biogeography. However, important aspects of the hypothesis are not obvious in Stevens's v erbal model and may go unrecognized, despite their major implications for empirical work related to large-scale ecological and evolutionary processes. Here we present mathematical simulation models that test the logical structure of the Stevens hypothesis, examine effects on global patterns of species richness produced by the mechanisms (Rapoport's rule and the rescue effect) explicitly identified by Stevens, and investigate the additional effect of competition.

We find that Rapoport's rule on its own generates an LGSR opposite that of the real world, with species richness peaking at the poles rather than at the equator. The same qualitative result (a "reverse" LGSR) appears when rescue-effect regions, as described by Stevens, are added to the model. Building upon Stevens's verbal model, we then develop an explicit version of competition and show that competition alone tends to equalize species richness across all latitudes. However, when both Rapoport's rule and competition are included in the model, we find that a qualitatively correct LGSR is produced. Unlike previous hypotheses regarding the LGSR, this version of the model does not rely on a latitudinal gradient in the intensity of competition to produce an LGSR. However, detection of this LGSR depends on the spatial scale at which species richness is sampled, with the LGSR appearing only with regional, not local, sampling. In contrast, when competition is explicitly added to the model with both Rapoport's rule and the rescue effect, an LGSR that is qualitatively consistent with that of the real world does appear in both local and regional samples. This expanded version of the Stevens hypothesis potentially could explain the real-world LGSR, but all three elements (Rapoport's rule, the rescue effect, and competition) are crucial and must operate sufficiently strongly and in specific ways. The LGSR becomes apparent in the model only when parameter values for Rapoport's rule and the rescue effect are large, possibly unrealistically so, and when all points on Earth are filled to the competitively defined "community species saturation level." These findings highlight the complexity of the hypothesis and the need to consider all three of its components during empirical tests.

Key words: biodiversity; biogeography; competition; latitudinal diversity gradient; macroecology; Rapoport's rule; rescue effect; species diversity; species geographical range; Stevens hypothesis.

INTRODUCTION

Species richness tends to peak near the equator and decline toward the poles, forming a latitudinal gradient recognized by biologists for well over a century (Wallace 1878). This pattern holds true for many taxa (see Fischer 1960, Simpson 1964, Cook 1969, Arnold 1972, Currie and Paquin 1987, Clarke 1992, Rex et al. 1993) and is among the most prominent features of the natural world. Accordingly, substantial theoretical and empirical attention has been devoted to understanding the gradient's underlying cause (see Pianka 1966, Rohde 1992, Huston 1994, Krebs 1994). Yet despite numerous hypotheses the latitudinal gradient in species richness (LGSR) remains "the major, unexplained pattern in natural history" (R. E. Ricklefs, quoted in Lewin 1989: 527).

Recently interest in large-scale ecological patterns such as the LGSR has resurged. Among the most influential and provocative of this work is that of Stevens (e.g., 1989, 1992). Stevens's newly-formulated hypothesis to explain the LGSR is generating substantial interest among ecologists, evolutionary biologists, biogeographers, conservation biologists and others concerned with species richness. The hypothesis often is discussed as a significant advance in thinking about large-scale biological processes and rapidly has found its way into textbooks and popular books as a potential general explanation for the LGSR (e.g., Eldredge 1992, Lewin 1989, Wilson 1992).

Stevens's (1989, 1992) hypothesis suggests a link between the LGSR and another global-scale pattern: the tendency for species geographical ranges to decrease in size toward the equator. Based on analysis of empirical datasets, Stevens (1989) suggests that this trend exists for a wide range of species groups. He refers to the latitudinal trend in range size as "Rapoport's rule," after E. H. Rapoport, who documents it for a more limited set of species and subspecies (Rapoport 1982). Noting that some exceptions to Rapoport's rule are also exceptions to the LGSR, Stevens (1989, 1992) hypothesizes that the LGSR is generated by an interaction between Rapoport's rule and the phenomenon known as the "rescue effect" (see Brown and Kodric-Brown 1977). Stevens speculates that at the edge of every species geographical range individuals disperse into areas where they survive but cannot reproduce because of insufficient local conditions or resources. Stevens calls these individuals "accidentals." Populations of accidental s are nonviable in and of themselves; they are maintained entirely by the rescue effect: dispersal from viable populations located closer to the center of the range. If both Rapoport's rule and rescue effect regions exist, the Stevens (1989, 1992) hypothesis proposes, a latitudinal gradient in species richness would appear as a byproduct: Toward the equator, where ranges are smaller due to Rapoport's rule, the ratio of rescue effect area to geographical range area increases, creating more opportunity for species richness to be inflated, thereby generating the LGSR.

Despite widespread interest in explaining the LGSR and the fact that much recent empirical research on the subject has been influenced by the Stevens (1989, 1992) hypothesis, one central question remains untested: Could the Stevens hypothesis suffice to explain the observed latitudinal gradient in species richness? An immediate need exists to evaluate the hypothesis from a theoretical perspective and to examine the issues it raises for empirical work regarding both Rapoport's rule and the LGSR. Here we report on mathematical simulation models created to examine the Stevens (1989, 1992) hypothesis. We first explore impacts on species richness of Rapoport's rule and the rescue effect individually and in combination. We then proceed to add an additional factor to the models: competition. Although competition is frequently discussed in connection with the LGSR, no explicit description of it is included in Stevens's papers (Stevens 1989, 1992).

Specifically, we ask the following questions: (1) Does Rapoport's rule alone generate a species richness gradient like that of the real world? (2) Does inclusion of rescue effect regions around geographical ranges lead to a species richness gradient like that found in nature, either in the presence or absence of Rapoport's rule? (3) Does inclusion of competition in the model generate a species richness gradient like that of the real world, either in the presence or absence of Rapoport's rule? (4) Can Rapoport's rule, the rescue effect, and competition interact to generate a species richness gradient like that of the real world? (5) How are these results qualitatively altered solely by the spatial scale of samples of species richness?

BASIC STRUCTURE OF THE MODEL

The model discussed in this paper has a structure conceptually similar to that used by Colwell and Hurtt (1994), and therefore we use similar terminology in describing it. However, because we are interested in complex interactions between several phenomena operating on a global scale, we model the mechanisms as occurring on the surface of a sphere, rather than in a one-dimensional approximation, as is appropriate in models such as those of Colwell and Hurtt (1994).

Our model consists of a sphere (representing the earth) inhabited by artificial species. Each species has only three biological/ecological traits: the size, shape, and location of its geographical range. In any given run of the model, all species ranges have identical shapes, such as circles on the surface of the sphere (i.e., "skullcaps"). This simplified model allows us to test the effects of specific mechanisms of interest (e.g., Rapoport's rule or the rescue effect) in isolation from additional factors--biotic interactions, historical events, physical features of the globe, climate, and variation in shapes of individual geographical ranges--that are not central to the issues raised by Stevens.

The basic model can be thought of as follows. Picture a solid sphere and imagine a tiny dot at some randomly chosen site on the sphere. The dot represents the location of the centerpoint of a species' geographical range. For a given run in which all species have circular ranges, randomly choose a radius, defined in terms of degrees along a great circle segment on the surface of the sphere, from a uniform distribution between zero and some specified maximum value. (We used maximum radius values of 5[degrees], 10[degrees], 20[degrees], and 40[degrees].) Draw a circle around the centerpoint, using the chosen radius. The area within the circle represents the geographical range of the species. Our models are intended to provide static distributions on a globe, and thus species geographical ranges in the models are static. This process is repeated for some number n species, so that the sphere becomes covered with many species ranges. The latitude and longitude of each centerpoint are each chosen randomly, with rep lacement, from distributions that give every location on the sphere equal probability of being selected. The basic model can be altered to make geographical range size a function of latitude (therein forcing Rapoport's rule), to allow noncircular ranges, to add rescue effect regions onto edges of geographical ranges, or to include competition.

Rapoport's rule, as discussed by Stevens (1989, 1992), is primarily concerned not with the overall area of species geographical ranges, but only with the north-south extent of each species geographical range. This is what Stevens (1989, 1992) referred to as the species latitudinal range. For example, if a species in our model has a circular range that does not overlap a pole, its latitudinal range is equal to the diameter of the circle. If, however, a species has a circular range that does overlap a pole, its latitudinal range is equal to the latitude of the pole (90[degrees]) minus the latitude of whichever point on the circle is furthest from the pole. Note that in this case, latitudinal range will always be smaller than the diameter of the circle. We refer to this as the "polar truncation effect."

To create the model we used MATLAB software (MATLAB 4.2c, 1984-1994, The MathWorks, Inc., Natick, Massachusetts). In most cases we used 20 000 species per run and 10 runs of each version of the model, because this produced results with sufficiently low variance to show clearly the resulting patterns. Exceptions will be explained as they arise.

METHODS FOR SAMPLING SPECIES RICHNESS

Choice of sampling method can strongly affect how species richness patterns are described and, depending on underlying causative mechanisms, whether or not they are detected (Colwell and Hurtt 1994). We used two separate approaches, corresponding to "regional" and "local" diversity. Various methods of sampling regional diversity have been used, including quadrats and transects (e.g., Simpson 1964, Colwell and Hurtt 1994). We used a line transect extending from north pole to south pole as a meridian of longitude. Species richness was quantified by counting the exact number of species geographical ranges that overlapped each 10[degrees] latitudinal section of the transect. The second method uses point samples of species richness (e.g., Rex et al. 1993), quantifying local, rather than regional, species richness. In our model, "points" for the point samples were taken as extremely short fragments of the transect. The points were 1 X [10.sup.-5] degrees long (roughly equivalent to 1 m on the earth's surface), effe ctively 0[degrees] wide, and spaced every 2[degrees] from 89[degrees] S to 89[degrees] N. Species richness was obtained by exact counts of how many species geographical ranges overlapped a given point. We analyze model output using both sampling methods.

EFFECT OF RAPOPORT'S RULE ON SPECIES RICHNESS

To evaluate the effect of Rapoport's rule, we made the maximum possible radius of each species geographical range a function of the latitude of that species' range centerpoint. If a species centerpoint was placed, by chance, at a high latitude, the radius of its geographical range was drawn from a uniform distribution with a higher maximum possible value than if its centerpoint were at a lower latitude. The function we used to generate this relationship between maximum possible radius and latitude was linear:

[r.sub.max] = [r.sub.e] + s([l.sub.c]) (1)

where [r.sub.max] is the maximum possible radius for the geographical range of a species, [l.sub.c] is the latitude of its centerpoint, [r.sub.e] is the maximum possible radius at the equator, and s controls the strength of Rapoport's rule. For all trials, we set [r.sub.e] equal to 5[degrees] in order to minimize the impact of the polar truncation effect on generation of the desired increase in species latitudinal ranges with increasing latitude. We varied the strength of Rapoport's rule by setting s equal to 0.1 (weak Rapoport's rule) or 0.5 (strong Rapoport's rule).

Imposition of Rapoport's rule strongly affected the latitudinal species richness pattern in the model, but the resultant pattern was opposite the real-world species richness gradient: Species richness was greatest at high latitudes and lowest at the equator (Fig. 1). Therefore, the effect of Rapoport's rule, acting in the absence of other factors, is to create a "reverse" species richness gradient. Larger latitudinal ranges at high latitudes caused more species to co-occur on the transect at high latitudes than at low latitudes. The strength of the reverse species richness gradient depended on s, the strength of Rapoport's rule (Eq. 1). These results hold true for both regional and point samples of species richness. These findings demonstrate the manner in which Rapoport's rule can complicate the question of the species richness gradient. For species groups in which Rapoport's rule is a true phenomenon, the mechanism or mechanisms that produce the species richness gradient must not only create a gradient in s pecies richness from tropics to poles. It or they must also counteract the reverse species richness gradient introduced by Rapoport's rule. In the context of this version of our model, counteracting the effect of Rapoport's rule could be produced only by placing many more species centerpoints onto the sphere at low latitudes, inflating species richness there but also making the density of centerpoints nonuniform over the surface of the sphere.

According to Stevens (1989), empirical studies show a positive correlation between overall geographical range area and latitude, making this characteristic of the model not overly troublesome. However, this correlation is apparently somewhat weaker than that for latitudinal ranges vs. latitude, and Stevens's hypothesis was concerned only with the latter. What would happen to the species richness gradient if geographical range widths were independent of latitude, but latitudinal ranges tended to increase with latitude? Overall area of ranges would still increase, of course, as a consequence of increased latitudinal ranges, but would patterns of species richness be altered qualitatively from what was observed when circles were used? We addressed these questions by separating the effect of larger range widths from the effect of larger latitudinal ranges. We used diamond-shaped ranges instead of circles, so that range width could be kept independent from latitudinal range. Results were qualitatively identical to those obtained using circular ranges: Rapoport's rule, acting in the absence of other factors, caused species richness to increase from the equator to the poles.

IMPACT OF THE RESCUE EFFECT

Stevens's (1989, 1992) hypothesized mechanism for generation of the LGSR includes a role for the rescue effect. He suggests that beyond the edges of a species geographical range individuals disperse into unfavorable habitats, where their populations persist solely through dispersal. We included the rescue effect in the next two versions of the model, first with only the rescue effect and next with both the rescue effect and Rapoport's rule. The latter is an explicit test of Stevens's hypothesis as described verbally in his publications (Stevens 1989, 1992).

We began by assuming, like Stevens (1989, 1992), that dispersal ability was independent of geographical range size and that all species had equal dispersal distances. The rescue effect was modeled as follows:

[r.sub.total] = [r.sub.range] + [r.sub.rescue] (2)

where [r.sub.total] was the total radius of the area occupied by the species, [r.sub.range] was the radius of the geographical range of the species, and [r.sub.rescue] was the radius of the rescue effect region surrounding the geographical range. All radii were defined in terms of degrees along a great circle, with three sizes of rescue area (0[degrees], 1[degrees], or 3[degrees]).

Next we tested Stevens's full hypothesis by modeling the interaction between Rapoport's rule and the rescue effect. Six parameter combinations were tried, using three [r.sub.rescue] values (0[degrees, 1[degrees], or 3[degrees]) and two strengths of Rapoport's rule (s = 0.1 or 0.5, Eq. 1). Results from this version of the model showed that inclusion of the rescue effect did not qualitatively alter the observed pattern in species richness, either with or without Rapoport's rule (Fig. 2). The impact of the rescue effect was only quantitative: Species richness increased across all latitudes as [r.sub.rescue] increased. These conclusions did not depend on sampling method (regional vs. point). The overall increase in species richness generated here by the rescue effect is intuitively apparent. It is generated by the across-the-board increase in area occupied by each species, making it possible for more species to overlap the sampling transect. The rescue effect simply acts to inflate mean range size at all latit udes. We also found that the assumption of a proportional rescue effect (i.e., that species with larger ranges have proportionately larger rescue effect regions) had no qualitative effect on these findings.

In sum, Stevens's verbal model (Stevens 1989, 1992), modeled mathematically here, falls short of generating a LGSR qualitatively consistent with that of the real world.

POTENTIAL ADDITIONAL FACTORS

If the combination of Rapoport's rule and the rescue effect generates a reverse gradient in species richness, we are left with the question of how to generate a "correct" LGSR. To counteract the effects of range size, something must generate a nonrandom distribution of species range centerpoints on the surface of the globe. Various possibilities exist, but one mechanism often invoked as potentially responsible for this process is competition. With regard to the LGSR, two major and somewhat contradictory hypotheses have been proposed. The first suggests that competition increases toward the equator, generating increased species richness via increased adaptive specialization over evolutionary time scales (Dobzhansky 1950, Williams 1964, and see Pianka 1966). The second hypothesis proposes that competition decreases toward the equator because of increased predation or other forms of disturbance, allowing coexistence of many potentially competing species (Paine 1966, and see Pianka 1966). In the context of our m odels, both hypothesized mechanisms would have the desired effect: nonrandom placement of species ranges, with range centerpoints being more likely to occur near the equator.

Stevens discusses competition briefly with reference to the rescue effect (Stevens 1989:251), and it can be argued that implicit in his verbal model is the notion that communities are often saturated with species and that accidentals would inflate local species richness by oversaturating communities. The underlying cause of the saturation may be assumed to be competition: Only a limited number of species are able to maintain viable populations because of competition for local resources. Rapoport (1982:188, for example) also discusses the potential effect on diversity patterns of limited overlap of species ranges. Limits in overlap presumably could be driven by interspecific competition. We made this notion explicit and added it to our model of Stevens's hypothesis. Although the hypotheses described above posit latitudinal gradients in the intensity of competition, compelling evidence to support either of the hypotheses as a general solution has not been found in the decades since they were first proposed. Co nsequently we start with the most basic possibility: no latitudinal gradient in strength of competition. Does inclusion of competition in the model, even without a latitudinal gradient in competitive intensity, alter the impact of Rapoport's rule and/or the rescue effect?

EFFECT OF COMPETITION

Competition was incorporated by setting a community species saturation level (i.e., the maximum number of coexisting species that can persist in viable populations, given competition for local resources). Saturation level was set equal at all locations on the sphere. This assumption simplifies the natural world, where saturation levels, if they exist, may vary spatially and temporally, even if no latitudinal gradient exists. Species geographical ranges were placed onto the earth sequentially in the same random fashion used previously. For each attempt to place a new species, the model determined the density of "competitors" by counting species already occurring at three points within the new range. All other species are assumed to be competitors. The new species either remained ("survived") or was removed from the model permanently, depending on the number of species at the most species rich of the three points. Probability of survival declined linearly from 1 (if no other species were present) to 0 (if the number of previously existing species equaled or exceeded the predefined saturation level). At intermediate levels, stochasticity was allowed. The new species might either survive or be removed, with the probability of surviving defined by a linear function. Strength of competition was varied for different model runs by setting community species saturation level as either 40 (weak competition) or 5 (strong competition).

Important differences in results can occur depending solely on how many attempts are made to place species during each run of the model. Three options exist: (1) a set number of attempts but not enough to allow all points on the earth to reach saturation level, (2) attempts made continuously until a set number of species is successfully placed on the earth, and (3) attempts made repeatedly until the earth is "full" of species, meaning all locations have reached saturation level. "Attempts" are perhaps best conceived of as speciation events, making option I represent a scenario in which the earth is initially devoid of species, and species richness patterns are then determined after a set number of speciation events. Option 3 represents a time scale long enough for speciation events to fill the world. Option 2 merely produces a variation on the results of options 1 and 3, and we will not consider it in detail.

We first modeled option 1. Ten runs were performed for each of 4 parameter combinations, using weak or strong competition (saturation level = 40 or 5) and presence or absence of Rapoport's rule (s = 0 or 0.5, Eq. 1). Each run consisted of 1000 attempts, a number that did not allow the world to fill to saturation. Maximum range radius at the equator was 5[degrees] Resulting patterns of species richness (Fig. 3A, B) did not depend on sampling method (regional vs. point). In the absence of Rapoport's rule, species richness was constant across all latitudes, and the strength of competition did not affect the pattern quantitatively or qualitatively. In the presence of Rapoport's rule, however, species richness was at a minimum at the equator and increased with latitude, as in previous versions of the model.

Under these conditions, strength of competition influenced the latitudinal pattern of species richness both quantitatively and qualitatively. Weak competition produced a consistent increase in species richness with latitude. Strong competition, however, dampened the gradient in species richness: Species richness reached a maximum at 30[degrees] N or S latitude and remained at that maximum all the way to the poles. This discrepancy arises from the fact that, given strong competition and the larger ranges possible with Rapoport's rule, high-latitude locations could reach saturation level quickly. At low latitudes, ranges tended to be smaller, because of Rapoport's rule, and thus fewer species overlapped the transect. Consequently, points there did not reach saturation level. With Rapoport's rule and strong competition, point samples (Fig. 3B) plateaued at 5 species per point sample; correspondingly, saturation level was 5. With weaker competition (saturation level = 40), not even high latitude points reached saturation level, and no plateau was produced. Point samples with weak competition (Fig. 3B) all fall below the saturation level of 40.

We then switched to option 3 and determined that, with strong competition, the world was full of species after 100 000 attempts. On average, only 1 of the last 1000 attempts successfully placed a species on the earth. Five runs of 100 000 attempts were conducted. All incorporated strong competition (saturation level = 5) and Rapoport's rule (s = 0.5, Eq. 1). Results are strikingly different from option 1. Regional sampling shows that a peak in species richness appears at the equator, mimicking the real world LGSR (Fig. 3C). Still, no gradient appears with point sampling of species richness (Fig. 3D). This discrepancy between regional and point sampling is an important outcome of the underlying mechanisms in the model. The interaction between competition and Rapoport's rule leads to nonrandom placement of species ranges onto the sphere: Filling the world with small ranges at low latitudes and large ranges at high latitudes permits more species to be packed into a given region at low latitudes. Regional sampl ing detects this species packing, but point sampling shows no gradient because all points on the earth are at saturation level. The same issue is discussed by Rapoport (1982:188). Detection and perceived existence of this type of LGSR therefore depends entirely on whether data are collected using point or regional samples. The same result would occur with weaker competition (e.g., saturation level = 40) if sufficiently more attempts were made, enough to fill all points on the world to saturation level. The influence of spatial scale of samples on detection of the LGSR is therefore an important consideration in the design of empirical studies.

This version of the model highlights three points. First, a latitudinal

gradient in regional species richness may result in part from competition, even if there is no latitudinal gradient in strength of competition. Second, it is possible for a gradient in regional species richness to be generated solely by an interaction between Rapoport's rule and competition, without invoking the rescue effect. Third, a gradient in regional species richness can exist in the absence of any gradient in local species richness; if a gradient in local species richness also exists, it could be produced by a separate mechanism. In any case, LGSRs in the natural world have been demonstrated for both local and regional species richness, so the Rapoport's rule--competition interaction cannot be the complete story.

INTERACTION OF RAPOPORT'S RULE, RESCUE EFFECT, AND COMPETITION

Five runs of the model were made incorporating competition (saturation level = 5), the rescue effect ([r.sub.rescue] = 3, Eq. 2), and Rapoport's rule (s 0.5, Eq. 1). A total of 100 000 attempts were made in each run, allowing the earth to become saturated with species. The model removed a new species, or allowed it to remain, based solely on the number of other species geographical ranges already present; rescue effect regions were not a factor in this decision, corresponding with Stevens's (1989, 1992) assumption that accidentals are rare and do not have an impact on local resources. With these parameter values, the interaction between Rapoport's rule, the rescue effect, and competition produced results that were qualitatively different from those seen in the previous versions of the model and consistent with the empirically observed LGSR (Fig. 3E, F). Both point sampling and regional sampling showed an equatorial peak in species richness and a consistent decline in species richness toward the poles. Local sampling did reveal a latitudinal gradient in local species richness, while regional sampling showed a higher equatorial peak in species richness than in the preceding model version.

Although it was shown earlier that the rescue effect alone did not produce a qualitative change in latitudinal patterns of species richness, in the presence of Rapoport's rule and competition the rescue effect played a crucial role by elevating species richness on a local level. As Stevens (1989, 1992) hypothesized, dispersal from the core of the geographical range may create a ring-shaped rescue effect region around the edge of the range. Because the species escapes from competition in the rescue effect region, species richness is inflated, or oversaturated. Where ranges on average are smaller, the ratio of rescue effect region to geographical range area is higher, so the potential for inflation of species richness increases. The equatorial peak present in point samples of species richness (Fig. 3F) in this version of the model was a dramatic consequence of this phenomenon. Elevated species richness occurred in regional samples (Fig. 3E), but it is important to note that components of this equatorial peak w ere generated by two separate processes: species packing (produced by the interaction between Rapoport's rule and competition seen previously) and inflation by the rescue effect. Results from this version of the model did not change qualitatively if width of rescue effect regions was assumed to be proportional to geographical range size.

CONCLUSION

By employing three distinct elements (Rapoport's rule, the rescue effect, and competition), this version of the Stevens hypothesis is complex, despite the straightforward LGSR produced by the model (Fig. 3E, F). All three elements are crucial, as none on its own produces the real-world pattern. Rapoport's rule generates a reverse gradient, the rescue effect generates no qualitative change in species richness patterns, and competition tends to equalize species richness across latitude. Nor do pairwise interactions of the three factors produce an equatorial peak in species richness, except for the peak in regional species richness observed with Rapoport's rule and competition. (This clearly illustrates the complexity of understanding the real-world gradient, as different species groups actually may have gradients in regional, but not local, diversity, and they may be produced by different mechanisms or different combinations of the same mechanisms.) Therefore, if for a particular species group there exists a l atitudinal gradient in both regional and species richness, it is possible for the expanded Stevens hypothesis correctly to explain the underlying mechanism only if all three elements are present.

Moreover, each factor must operate sufficiently strongly and in a highly specific manner. If the strength of Rapoport's rule is diminished or the size of rescue effect regions reduced, the latitudinal gradient may become less prominent and effectively disappear. Furthermore, the "correct" LGSR that is produced in this model is relatively weak, with richness decreasing from equator to pole by at most a factor of only 2 or 3; in nature, species richness in some taxa decreases by 1 to 2 orders of magnitude between the tropics and polar regions (Latham and Ricklefs 1993).

The parameter values used to generate the data in Figs. 3E and 3F are probably far larger than observed in real world systems. For example, empirical data presented by Stevens (1989) show that for all species groups analyzed mean range size increases 4[degrees] over a span of 40[degrees] latitude, much weaker than the Rapoport's rule included in our model. (We chose extreme parameter values so that any effects would be obvious.) In addition, few data exist, to our knowledge, to support Stevens's speculation that geographical ranges are fringed by rescue effect regions. Moreover, we consider the 3[degrees] width of rescue effect regions in the model to be extremely large, perhaps unrealistically so, especially given that many ranges have radii much smaller than 3[degrees]. In light of these observations, it is unlikely that the LGSR is a simple consequence of the interplay between these three processes.

Finally, although we noted that empirical evidence for a latitudinal gradient in competition is lacking and therefore included no such gradient in the model, it can be argued that the model's combination of Rapoport's rule and the rescue effect in fact creates an effect on coexistence that is indistinguishable from that of a latitudinal gradient in competition (M. Huston, personal communication). Increasing the size of a species' geographical range by adding a region where the species is not involved in competitive interactions elevates the number of species that can coexist at a given location. Because this effect is proportionately greater toward the equator in the model when Rapoport's rule is included, a latitudinal gradient in competition is effectively created. The limit on the maximum number of species at a location increases toward the equator just as it would if competitive interactions decreased at lower latitudes (Huston 1979).

ACKNOWLEDGMENTS

We thank J. Brown, S. Navarrete, R. Nisbet, G. Stevens, and participants in the UCSB Population Biology seminar (595P) for helpful discussions, and C. Blanchette, J. Byers, J. Caley, R. Colwell, D. Currie, L. Goldwasser, M. Huston, M. Moritz, S. Navarrete, R. Sagarin, E. Solomon, G. Stevens, and D. Zacherl for comments on drafts of the manuscript. This material is based upon work supported by three fellowships to P. Taylor--a National Science Foundation Graduate Research Fellowship, a National Science Foundation Research Training Grant Fellowship in Spatial Ecology (NSF grant GER93-54870), and a University of California Departmental Regents Fellowship--and three research grants to S. Gaines, from the Andrew W. Mellon Foundation, the United States Department of Energy, and the National Science Foundation (0CE94-02690).

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Manuscript received 20 November 1997; revised 14 October 1998; accepted 14 October 1998.

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Author:TAYLOR, PETER H.; GAINES, STEVEN D.
Publication:Ecology
Geographic Code:1USA
Date:Dec 1, 1999
Words:6130
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