Random processes and geographic species richness patterns: why so few species in the north?

ECOGRAPHY 24: 43 49. Copenhagen 2001 Random processes and geographic species richness patterns: why so few species in the north? Folmer Bokma, Jurjen Bokma and Mikko Mönkkönen Bokma, F., Bokma, J. and Mönkkönen, M. 2001. Random processes and geographic species richness patterns: why so few species in the north? Ecography 24: 43 49. In response to the suggestion that the latitudinal gradient in species richness is the result of stochastic processes of species distributions, we created a computer simulation program that enabled us to study random species distributions over irregularly shaped areas. Our model could not explain latitudinal variation in species richness of New World mammals and created species richness patterns that differed dramatically from previous stochastic models. The interplay of speciation and species migration in our simulation generated the highest species richness in the middle of a landmass, not in the middle of its latitudinal stretch as was found in previous one-dimensional models. The discrepancy between the results of this study and previous empirical studies suggests that the effect of randomness in species-richness distribution is on a continental scale restricted by other, dominant determinants that limit the effect of chance. F. Bokma, Zoological Lab., Uni. of Groningen, The Netherlands (present address: Dept of Biology, Uni. of Oulu, Box 3000, FIN-90014 Oulu, Finland). J. Bokma, Snelliusstraat 15, NL-9727 JK Groningen, The Netherlands. M. Mönkkönen (correspondence, mikko.monkkonen@oulu.fi), Dept of Biology, Uni. of Oulu, Box 3000, FIN-90014 Oulu, Finland. Ever since Darwin (1839) and Wallace, biologists have been fascinated by the richness of taxa in the tropics. Diversity of many groups of organisms increases from high to low latitudes largely irrespective of taxonomic level and spatial scale of measurement (Gaston and Williams 1996). Various mechanisms have been proposed for the latitudinal diversity gradients. These mechanisms include gradients in the effects of competition, mutualism, predation, spatial patchiness, environmental stability, environmental predictability, productivity, area, habitat diversity, ecological time, evolutionary time and solar energy (for reviews see Pianka 1966, Rohde 1992, 1999, Rosenzweig 1995, Gaston and Williams 1996). No consensus view on the primary cause of the pattern seems to be emerging. It is possible that at different taxonomic levels, at different spatial scales, and in different sets of environmental conditions the factors determining species richness differ (Böhning-Gaese 1997, see also He et al. 1996, Davidowitz and Rosenzweig 1998). Still, the latitudinal diversity gradient remains present in almost all cases. This could be an indication that the basic pattern of species distribution is the not the result of a single evolutionary or ecological process but rather of a complex of several factors. It could also indicate that the bounded nature of global environments alone in the absence of any environmental gradients produces equatorward increasing species richness patterns (Pielou 1977, Colwell and Hurtt 1994, Willig and Lyons 1998, review by Colwell and Lees 2000). In other words, the latitudinal gradient in species richness might be partly a result of ecological and evolutionary processes, and partly a conuence of a combination of probability and the boundaries of landmasses in nature. The first logical step in an attempt to understand the cause of species distributions is to consider the effect of stochastic processes, i.e. to ask what patterns emerge if no geographical gradients in ecological or evolutionary Accepted 17 July 2000 Copyright ECOGRAPHY 2001 ISSN 0906-7590 Printed in Ireland all rights reserved ECOGRAPHY 24:1 (2001) 43

processes are assumed. A basic analytical model assessing the degree to which random determination of the limits of species ranges produces latitudinal gradients in species richness was proposed by Willig and Lyons (1998). This model accounted for a significant proportion of latitudinal variation in marsupial and bat species diversity in the New World. Willig and Lyons concluded that the ubiquity and similarity of latitudinal gradients in species richness for different taxa could be a conuence of pervasive stochastic mechanisms rather than a product of a dominant underlying environmental gradient to which all species respond. Upon the findings of Willig and Lyons study and other, similar studies (e.g. Colwell and Hurtt 1994) it has very recently been claimed that stochastic mechanisms can account for a large share of the equatorward increase in species richness (Colwell and Lees 2000). Willig and Lyon s (1998) and earlier null-models to account for the distribution of species (Colwell and Hurtt 1994) and of species range sizes (Lyons and Willig 1997) have so far been one-dimensional. These models have focused only on the latitudinal stretch of a landmass without taking into account two-dimensionality in species ranges and in the shapes of real-world areas. Given that the distributions of species on a landmass occur in two dimensions, and given that the longitudinal dimension of landmasses varies across latitudes, any model of random species distribution should be (at least) two-dimensional (see also Lyons and Willig 1999). Here we use a stochastic modeling approach to assess the role of random processes in determining species diversity patterns for two-dimensional areas of various shapes. In the simulations, we first define the size and shape of an area and distribute a certain number of species over that area. We then let the pattern of species distribution change assuming certain probabilities of migration, extinction and speciation but without any geographical gradients in the probabilities. Because diversity gradients are particularly well documented for the New World (e.g. McCoy and Connor 1980, Willig and Selcer 1989, Kaufman 1995, Kaufman and Willig Table 1. Decision table of the model, that shows the effect of speciation S, migration M and extinction E on the number of species x in adjacent quadrates Q1 and Q2 before (t=1) and after (t=2) change of generation. v indicates that speciation, migration or extinction occurs at the change of generation. x t=1 (Q1) x t=1 (Q2) S M E x t=2 (Q1) x t=2 (Q2) 1 0 1 0 1 0 v 0 0 1 0 v 1 1 1 0 v v 0 1 1 0 v 2 0 1 0 v v 1 0 1 0 v v 2 1 1 0 v v v 1 1 1998, Willig and Lyons 1998), we provide a specific prediction from our model for species diversity patterns in a landmass resembling the Americas. Thus we investigate whether stochastic processes can really explain a large extent of observed species richness patterns, as was suggested on the basis of previous, less realistic models (Colwell and Hurtt 1994, Willig and Lyons 1998, Colwell and Lees 2000). Model description Parameters We used a computerized dynamic model of simplified biological evolution implemented as a stochastic state machine. The model consists of a world divided into quadrates, in some of which species existence is possible. Each quadrate is identified by a unique set of two integers (r1, r2). The neighboring four quadrates are denoted with integers with increment or decrement of one of those of the focal quadrate. The shape of a simulation area can be adjusted by declaring quadrates uninhabitable. Thus, an area of any two-dimensional size and shape can be entered. Species are characterized by a uence of bits and are located in one or several, adjacent or non-adjacent quadrates. In simulations, all initial species ranges have for simplicity been set to a single quadrate. The initial distribution of species is achieved by placing species in quadrates of randomly selected coordinates (r1, r2). Time goes by in steps of one generation, and three evolutionary events may occur at the change of each generation: 1) migration: a species that existed in quadrate Q1 exists after migration also in quadrate Q2, that is adjacent to quadrate Q1. 2) Speciation: one new species, C2, enters a quadrate previously occupied by species C1, resulting in an increase of the species number by one in this quadrate, and 3) extinction: a species that existed in a quadrate during the previous generation ceases to exist. Probabilities of migration (P m ), speciation (P s )and extinction (P e ) for each quadrate are expressed as per species and per generation, and are equal for all quadrates and species, i.e. there is no geographical patterning of these processes. Likewise, there is no spatial dependence between the quadrates in speciation and extinction. For example, extinction of species C1 in quadrate Q1 does not affect species C1 in quadrate Q2. The simulation program allows one to vary these probabilities. All combinations of events and their effect on species richness are given in Table 1. Because our model distinguishes individual species, speciation, migration and extinction must take place uentially. The uence in which these events take place at every change of generation in the model is 44 ECOGRAPHY 24:1 (2001)

speciation migration extinction. However, the separation of speciation and extinction has an effect on total species richness: a uential model and a model in which speciation and extinction occur simultaneously would yield different results. To correct for this effect, we applied a correction to P s (see Appendix). Speciation, extinction and species numbers Considering only speciation, the number of species x in any particular quadrate at time t depends on the number of species in that quadrate in the previous generation as: x t =x t 1 +rx t 1 in which r is the net rate of speciation; r=p s P e. Thus, the number of species in any particular quadrate after n generations relates to the initial number of species x 0 as: x n =(1+r) n x 0 The species number x increases for (1+r) 1, that is for P s P e and decreases for (1+r) 1, that is for P s P e. Net speciation results in exponential increase of species numbers in a quadrate over time, and net extinction results in exponential decrease. Conuently, absolute differences in species numbers between quadrates change exponentially over time as well. Effects of migration Considering migration only, the number of species x in any particular quadrate at time t depends on the number of species in the adjacent quadrates in the previous generation. The number of species that migrate from a quadrate is determined as m x t 1 where m=p m. From the quadrate Q2, 1/k mx t 1 (k is the number of adjacencies) species migrate to the adjacent quadrate Q1. The total migration into Q1 is 1/k m x t 1, with indicating quadrates adjacent to Q1. This expression equals m times the average number of species in all k adjacent quadrates. The number of species in quadrate Q1 after any number n of generations cannot be derived from this equation, since the average number of species in the adjacent cells depends on the initial conditions of other quadrates in the simulation. The effect of migration on species distribution can however be derived, considering two quadrates, Q1 and Q2 that are adjacent. As long as x Q1 x Q2 quadrate Q2 receives more species from Q1 than the other way around. Thus, migration tends to decrease the differences in species numbers between quadrates, ultimately until species numbers are equal in all quadrates. Speciation-migration interaction The combined effect of speciation and migration can also be seen by comparing adjacent quadrates Q1 and Q2. If it is assumed the quadrates are adjacent only to each other, the number of species in each of both quadrates depends on the situation in the previous generation as follows: Q2 Q1: x t =x t 1 +rx t 1 +mx t 1 Q1 Q2: x t =x t 1 +rx t 1 +mx t 1 Subtraction of these equations yields the difference in species numbers between interacting quadrates Q1 and Q2: Q1 Q2 Q1 Q2 x t x t =(1+r m)(x t 1 x t 1 ) which implies that the difference in species numbers between the quadrates remains constant in case r=m. The difference increases exponentially in case r m and decreases exponentially in case r m. The difference always changes at a lower rate than the number of species in either of the quadrates, so the ratio x Q1 /x Q2 approaches 1. In other words, the differences relative to the number of species decrease over time. Simulations We first compared the results of our two-dimensional simulation model with those from the analytical one-dimensional model of Willig and Lyons (1998). For this comparison, we extended Willig and Lyons model to two-dimensional space. Willig and Lyons (1998) stochastic model of species distribution supposes that the probability that the distribution of a species on a landmass includes a point P depends on the position of P on the latitudinal stretch of the landmass. Point P along this stretch can be expressed in terms of relative distance from this point to the northern (p) and southern (q) end. The probability P that a species distribution area includes point P is P P =1 p 2 q 2 =2pq. The number of species expected to occur at point P then becomes 2pqS, where S is the number of species in the species pool. Expected species richness attains a maximum at the midpoint of latitudinal bounds (for more detailed description, see original paper by Willig and Lyons 1998). A two-dimensional extension can be easily obtained. Consider the point P on the longitudinal stretch of the landmass, with distance s to the western end and distance t to the eastern end of the landmass. The probability of P being included by the east-west extension of a species range will be P P =1 s 2 t 2 =2st. The probability that a species range in two dimensions includes point P is the product of these two ECOGRAPHY 24:1 (2001) 45

one-dimensional probabilities, which is 4pqst. Finally, the number of species expected to occur in a point on a two-dimensional space becomes 4pqstS. For comparisons with Willig and Lyons model (1998) we chose the parameters in the simulations so that the results would be of the same magnitude as those provided by the analytical model. We started with 1418 species (see below) and set P s =P e and P m =0.1. Species were let to evolve on a square world of 20 20=400 quadrates. We compared the results of our simulation with the predictions of this two-dimensional extension of Willig and Lyons (1998) model by correlation analysis. (Note that the correlation of two variables can be calculated not only for two rows of variables, as in conventional correlation analysis, but also for two matrices as in our case.) In trial runs it was found that 50 generations were enough to create a relatively stable geographic pattern of species richness. Because diversity gradients are particularly well documented for the New World taxa (e.g. McCoy and Connor 1980, Willig and Selcer 1989, Kaufman 1995, Willig and Lyons 1998), we provide a specific prediction from our model for species diversity patterns on a landmass resembling the Americas. The latitudinal gradient of species richness has been well documented for New World mammals (Kaufman and Willig 1998). In our simulations the initial species number was set at 1418, which is the known number of extant New World land mammals (i.e. excluding pinnipeds, cetaceans and sirenians, and including 255 bat species). Distribution of New World land mammals was analysed by Kaufman and Willig (1998) using 2.5 latitudinal bands. Therefore, to allow comparison we adjusted the size of quadrates in our simulation. A grid of 280 280 km quadrates was superimposed on an equal-area (Peter s) projection of the New World. Since the latitudinal stretch of the New World is ca 14000 km (126 ) this latitudinal stretch was divided into 50 quadrates (2.5 ). Quadrates consisting of 50% water (estimated by eye) were declared uninhabitable, which resulted in 420 inhabitable quadrates covering the landmass. We then let the simulation run for 50 generations with P s =P e =P m =0.1. It is important to note that absolute species numbers produced by the stochastic simulation are not as important as the spatial patterns emerging from it. Almost any combination of parameter values produces the same overall spatial patterns but absolute species numbers may differ greatly. For comparisons, we transformed the results of our two-dimensional simulations into one-dimensional gradients calculating average species numbers of the quadrates in each 2.5-degree latitudinal band. Simulation results are compared with Willig and Lyons (1998) stochastic model and with real-world data presented by Kaufman and Willig (1998). Results If speciation, extinction and migration occur randomly in space, our model clearly implies that in an unbounded model world no spatial patterning of species richness appears: any differences between quadrates tend to level off with time. Spatial gradients in species richness in our random model stem from the bounded nature of the world, e.g., the distribution of terrestrial taxa is limited by the edges of terrestrial areas. Analysis of the effects of migration showed that its net effect depends on the number of adjacent quadrates. Because the chance of migration P m is independent of the position of a quadrat on a landmass, quadrates adjacent to uninhabitable quadrates receive fewer species from adjacent quadrates than they lose by migration. (The probability of migration is equal for all quadrates, but the number of adjacent quadrates from which species can migrate is lower for quadrates adjacent to uninhabitable places than for other quadrates.) As a result, species adjacent to uninhabitable places ( edge quadrates ) foster less species. Conuently, quadrates adjacent to these edge quadrates receive less species from the edge quadrates and inhabit less species themselves. Thus, in a bounded world chance produces a pattern with highest species richness in the middle of an area. It is therefore not surprising that our two-dimensional simulation model yields results qualitatively similar to Willig and Lyons (1998) model when applied to a square shaped world. The simulation results were in accordance with a two-dimensional extension of Willig and Lyons model with r=0.972. When comparing one-dimensional representation of the results, it seems that our simulation model provides a smoother pattern, where the decrease in species density away from the midpoint of a geographical stretch is more gently sloping than that predicted by Willig and Lyons model (Fig. 1). This apparent difference depends on the parameter values of the simulation. With lower relative probability of migration a steeper gradient is produced. Also in one-dimensional solutions correlation between the two models is high (r=0.989 in latitudinal direction and r=0.978 in longitudinal direction). Using the New World map with 2.5 quadrates and starting with the number of extant terrestrial New World mammal species, the model yields a pattern of species richness with two peaks in the centers of North and South America (Fig. 2a). This pattern is expected by the simple boundedness of an area with the geographic configuration of the New World. The one-dimensional representation of the model results shows that the peaks in species richness are predicted to be just south of the equator in South America and at mid-latitudes in North America (Fig. 2b). A comparison of our simulated pattern with Willig and Lyons (1998) results for the Americas shows little 46 ECOGRAPHY 24:1 (2001)

Fig. 1. Species richness on a 20 20 quadrate square geometric world, respresented as a one-dimensional gradient. 1418 species were distributed with P s =P e and P m =0.1. Presented distribution was obtained after 50 generations (solid line). The dotted line is the gradient predicted by a stochastic model (Willig and Lyons 1998). One-dimensional correlation of obtained and predicted gradients: r=0.989. against real world data (data from nature). When the configuration of existing landmasses is considered, results of our two-dimensional simulations are dramatically different from earlier stochastic models. Our model predicts the highest species richness in the middle of larger areas rather than in the middle of the latitudinal stretch of a landmass, as predicted by earlier one-dimensional models (Willig and Lyons 1998). Our results illustrate that stochastic mechanisms cannot account for latitudinal gradients in species richness when models are extended from one dimension to two dimensions. The simulation results differ so dramatically from real world data that we believe incorporation of historical events like the great American interchange in our model would not have qualitatively altered this mismatch. In the case of the New World, our model predicts a two-peaked latitudinal pattern. The largest discrepancy with the real-world latitudinal diversity pattern is the resemblance (r=0.292; Fig. 2b). Predictions of Willig and Lyons model applied separately to North- and South-America agrees well with simulated species-richness. Correlation between the two models is r=0.849 for North America and 0.871 for South America. The real-world mammal species richness pattern shows a decrease of species richness away from the equator. This decrease is not monotonic, however, but smaller peaks in species richness can be seen at ca 20 N and 35 N (Fig. 3). There is no overall correlation between our simulated data and New World mammal data (r= 0.006). It is striking, however, that the simulated species richness pattern correlates with the real world data for South America (r=0.810). Just as in our simulated pattern species richness in the New World peaks at ca 10 15 S. For northern latitudes the match between our model and New World mammal data is poor (r= 0.812). Discussion Our simulation model suggested that random species distribution processes result in high species richness in the middle of a landmass. Applied to a regularly shaped area, the results of our model match the results of earlier stochastic models by Colwell and Hurtt (1994) and Willig and Lyons (1998). Species diversity increases toward the middle of a latitudinal domain in the absence of any geographical gradients in ecological and evolutionary processes. The novelty in our model is that it can be applied to areas of any size and shape, and therefore can be tested Fig. 2. a) Species richness on a 420-quadrates world resembling the New World. 1418 species were distributed with P s =P e and P m =0.1. Presented distribution was obtained after 50 generations. Species numbers per quadrate are presented as a percentage of the sum of species numbers of all quadrates. Species richness increases from the edge towards the middle of landmasses. b) Same as above but represented as a one-dimensional gradient. Solid line is the model prediction. The dotted lines respresent the gradient predicted by a stochastic model (Willig and Lyons 1998): for the entire landmass and for both subcontinents separately. ECOGRAPHY 24:1 (2001) 47

Fig. 3. A comparison of a gradient of species richness predicted by the model with American mammal data (Kaufman and Willig 1998). Open dots represent the results of a 50 generations simulation with P s =P e =P m =0.1 started with 1418 species. Closed dots represent Kaufman and Willig s data from nature. low predicted diversity at mid-latitudes and the high predicted diversity at high latitudes. Therefore, two patterns emerge that need an ecological and/or evolutionary explanation: 1) the high extant diversity in Central America and 2) the low extant diversity in southern and central parts of North America. Traditionally, studies have addressed the question why there are so many species at low latitudes, but our model suggests that random processes predict high extant diversity in northern South America. The variation in species richness, which is not accounted for by random processes, is open to evolutionary and ecological explanations. Strong latitudinal correlation between an environmental factor (e.g., environmental stability or the amount of captured energy) and a species richness gradient does not imply a causal link. Correlative studies that examine possible determinants of species richness patterns (e.g. Willig and Selcer 1989, Cotgreave and Harvey 1994, Kaufman and Willig 1998, Willig and Lyons 1998) lead to the best descriptor, not necessarily to the primary cause of species richness. It would be valuable to try to falsify potential determinants that correlate with latitudinal gradients in species richness by investigating their correlation with species richness in two dimensions, or at least longitudinally. If a stochastic model yields species distributions that match well with observed species distributions, this can principally be explained in two ways: either the species distributions are random, or the species richness pattern is caused by some factor(s) so that it appears to be a random distribution. Many interacting processes may produce the same result as chance. The principle of parsimony implies that of alternative explanations for a pattern the one that requires least assumptions should be selected (pluralitas non est ponenda sine necessitate, Ockham s razor). Therefore, if random processes are able to predict a pattern then randomness enjoys priority over other explanations. It is most likely that the actual species distribution results from both stochastic processes and ecological processes, of which we do not a priori know the relative contributions. A logical next step would be to start incorporating evolutionary (historical) and ecological factors into the model to see what is the lowest possible number of assumptions needed to yield the observed species diversity gradients. Since the world is rather spherical instead of flat (Pythagoras -530 but see Newton 1687) a technical problem occurs, as our method involves a two-dimensional representation of a curved surface. Because equal-area maps are not equal-distance maps, migration from one quadrate to another is regardless of any ecological or evolutionary assumptions at higher latitudes relatively more likely in the real world than in our model. As a conuence the model yields a somewhat distorted pattern, distortion increasing with increasing latitude. Distortion is unavoidable when dealing with originally spherical surfaces on two-dimensional planes. We believe that using an equal-distance map (Mercator projection), would have created even larger distortions as it exaggerates land areas at high latitudes (see e.g. Rosenzweig 1992), and, as shown by our results, land area per se seems to have a large impact on model species richness predictions. A large proportion of studies dealing with species diversity patterns has focussed on the New World. Further studies on the distribution of Old World species richness would be of great interest. In Eurasia, for example, major landmasses are unlike in the Americas situated well north of the equator. This area therefore provides a good background to further evaluate the role of random processes to species diversity. This would improve our insight in the latitudinal diversity gradient if more attention was paid to biodiversity gradients other than latitude (e.g. Turner et al. 1987, Mönkkönen and Viro 1997). After the traditional focus on the high biodiversity of the tropics, the question why there are so few species in the north might improve our understanding of global biodiversity. Acknowledgements The program we developed has more possibilities than employed for this article and attempts are made to improve the present version. The program is available by request. Funding from the Academy of Finland to MM is greatly acknowledged. We thank Gerald J. Niemi and Paul R. Martin for helpful comments on an earlier draft of the manuscript. References Böhning-Gaese, K. 1997. Determinants of avian species richness at different spatial scales. J. Biogeogr. 24: 49 60. Colwell, R. K. and Hurtt, G. C. 1994. Nonbiological gradients in species richness and a spurious Rapoport effect. Am. Nat. 144: 570 595. 48 ECOGRAPHY 24:1 (2001)

Colwell, R. K. and Lees, D. C. 2000. The mid-domain effect: geometric constraints on the geography of species richness. Trends Ecol. Evol. 15: 70 76. Cotgreave, P. and Harvey, P. H. 1994. Associations among biogeography, phylogeny and bird species diversity. Biodiv. Lett. 2: 46 55. Darwin, C. R. 1839. The zoology of the voyage of H.M. Ship Beagle, under the command of Captain Fitzroy, R.N., during the years 1832 to 1836. Smith, Elder and Co., London. Davidowitz, G. and Rosenzweig, M. L. 1998. The latitudinal gradient of species diversity among North American grasshoppers (Acrididae) within a single habitat: a test of the spatial heterogeneity hypothesis. J. Biogeogr. 25: 553 560. Gaston, K. J. and Williams, P. H. 1996. Spatial patterns in taxonomic diversity. In: Gaston, K. J. (ed.), Biodiversity: a biology of numbers and difference. Cambridge Univ. Press, pp. 202 229. He, F., Legendre, P. and Lafrankie, J. V. 1996. Spatial pattern of diversity in a tropical rain forest in Malaysia. J. Biogeogr. 23: 57 74. Kaufman, D. M. 1995. Diversity of New World mammals: universality of the latitudinal gradients of species and bauplans. J. Mammal. 76: 322 344. Kaufman, D. M. and Willig, M. R. 1998. Latitudinal patterns of mammalian species richness in the New World: the effects of sampling method and faunal group. J. Biogeogr. 25: 795 805. Lyons, S. K. and Willig, M. R. 1997. Latitudinal patterns of range size: methodological concerns and empirical evaluations for New World bats and marsupials. Oikos 79: 568 580. Lyons, S. K. and Willig, M. R. 1999. A hemispheric assesment of scale dependence in latitudinal gradients of species richness. Ecology 80: 2483 2491. McCoy, E. D. and Connor, E. F. 1980. Latitudinal gradients in the species diversity of North American mammals. Evolution 34: 193 203. Mönkkönen, M. and Viro, P. 1997. Taxonomic diversity of the terrestrial bird and mammal fauna in temperate and boreal biomes of the northern hemisphere. J. Biogeogr. 24: 603 612. Newton, I. 1687. Philosophiae naturalis principia mathematica. Pianka, E. R. 1966. Latitudinal gradients in species diversity: a review of concepts. Am. Nat. 100: 33 46. Pielou, E. C. 1977. The latitudinal spans of seaweed species and their patterns of overlap. J. Biogeogr. 4: 299 311. Rohde, K. 1992. Latitudinal gradients in species diversity: the search for the primary cause. Oikos 65: 514 527. Rohde, K. 1999. Latitudinal gradients in species diversity and Rapoport s rule revisited: a review of recent work and what can parasites teach us about the causes of the gradients. Ecography 22: 593 613. Rosenzweig, M. L. 1992. Species diversity gradients: we know more and less than we thought. J. Mammal. 73: 715 730. Rosenzweig, M. L. 1995. Species diversity in space and time. Cambridge Univ. Press. Turner, J. R. G., Gatehouse, C. M. and Corey, C. A. 1987. Does solar energy control organic diversity? Butterflies, moths and the British climate. Oikos 48: 195 205. Willig, M. R. and Selcer, K. W. 1989. Bat species gradients in the New World: a statistical assesment. J. Biogeogr. 16: 189 195. Willig, M. R. and Lyons, S. K. 1998. An analytical model of latitudinal gradients of species richness with an empirical test for marsupials and bats in the New World. Oikos 81: 93 98. Appendix If speciation and extinction take place simultaneously in a population of species of size x in generation t, the number of species in the next generation t+1 is: sim sim x t+1 =x t +P s x t P e x t In a speciation-first uential model x t+1 becomes: x t+1 =x t +P s x t P e (x t +P s x t ) In order to keep x t+1 in a uential model equal to x t+1 of a simultaneous model, the chance of speciation should be increased, so that sim x t+1 =x t+1 Thus, the above equations can be rewritten: [x t +P s sim x t P e x t ]=[x t +P s x t P e (x t +P s x t )] Simplification of this equation yields the probability of speciation P s required to keep species numbers in a speciation-first uential model equal to numbers in a simultaneous model, expressed in terms of P s sim : P s = P s sim 1 P e We used this equation in correcting the probability of speciation P s in simulations. As we regard this conversion a technical artifact, for convenience, probabilities referred to in the text are uncorrected values of P s. ECOGRAPHY 24:1 (2001) 49