Tree dispersion, abundance, and diversity in a tropical dry forest
Stephen P. Hubbell
Science, New Series, Vol. 203, No. 4387. (Mar. 30, 1979), pp. 1299-1309.
That tropical trees are clumped, not spaced, alters conceptions of the organization and dynamics. Stephen P. Hubbell
Summary.
Patterns of tree abundance and dispersion in a tropical deciduous (dry) forest are summarized. The generalization that tropical trees have spaced adults did not hold. All species were either clumped or randomly dispersed, with rare species more clumped than common species. Breeding system was unrelated to species abundance or dispersion, but clumping was related to mode of seed dispersal. Juvenile densities decreased approximately exponentially away from adults. Rare species gave evidence of poor reproductive performance compared with their performance when common in nearby forests. Patterns of relative species abundance in the dry forest are compared with patterns in other forests, and are explained by a simple stochastic model based on random-walk immigration and extinction set in motion by periodic community disturbance.
A widely held generalization about tropical tree species is that most occur at very low adult densities and are of relatively uniform dispersion, such that adult individuals of the tree species are thinly and evenly distributed in space. If true, this generalization has potentially profound consequences for the reproductive biology, population structure, and evolution of tropical tree species (I). In this article the adequacy of this generalization is judged with respect to a particular tropical forest, a large tract of which has been mapped in detail (2)
The origins of this generalization can be traced back at least to Wallace (3), who stated the following concerning his impressions of species densities in Malaysian forests:
If the traveller notices a particular species and wishes to find more like it, he may often turn his eyes in vain in every direction. Trees of varied forms, dimensions, and colours are around him, but he rarely sees any one of them repeated. Time after time he goes toward a tree which looks like the one he seeks, but a closer examination proves it to be distinct. He may at length, perhaps, meet with a second specimen half a mile off, or may fail altogether, ti1 on another occasion he stumbles on one by accident.
Dobzhansky and co-workers (4) enumerated the species in several 1-hectare stands of Amazonian rain forest, and concluded that "the population density of a half or more of the tree species in Amazonian forests is likely to be les than one individual per hectare." One or both parts of this generalization (low density, uniform dispersion) now appear in most ecology texts (5), and theories have been proposed to explain both the causes and consequences of low density and uniform dispersion of adult tropical trees. Janzen (6) and Connell (7) independently proposed theories to explain low density and spacing between adults. Janzen focused attention on the effects of host-specific herbivores that attack ber of seeds arriving at that point" (6, p. 501).
Connell, in his earlier rain-forest studies, focused more attention on the dispersion and survival of young tree seedlings. In experimental studies of the fate of seedlings in an Australian rain forest, Connell, Tracy, and Webb (8) showed that survival was better in seedlings planted under adults of different species than under adults of the same species. They did not identify the causes of the differential mortality, but did suggest that herbivores attracted by adjacent adults would more often tend to defoliate and kill nearby seedlings rather than distant seedlings. - Janzen and Connell both argued that such host-specific attack by herbivores would reduce the local density of any given species, open up habitat to invasion by additional species, and thereby maintain high species diversity (9)
Many explanations have been offered for the high species diversity in tropical forests, and these have been classified into equilibrium and nonequilibrium hypotheses by Connell (lo), who now believes that high diversity is only maintained because of frequent disturbance.
seeds. He noted that a high proportion of seeds falling under the parent tree are killed by such seed "predators," so called because the death of the seed is virtually assured; and he suggested that only those viable seeds transported some distance away from the parent would escape discovery and manage to germinate. The predicted result: "most adults of a given tree species appear to be more regularly distributed than if the probability of a new adult appearing at a point in the forest were proportional to the numPotential consequences of a low-density uniform dispersion of adult trees in tropical species might include lower outcrossing success, reduction in deme size, and requirements for long-distance pollination. Thus, the generalization that adults of tropical tree species are widely spaced has also spawned a number of hypotheses about unusual breeding systems in tropical trees (!I), or special pollinator movements over long distances (12). It now appears that the majority of tropical tree species is facultatively or obligately outcrossed; and the frequency of dioecy in tropical trees is very high by temperate zone standards (13, 14). Animals rather than wind, in most cases, are the agents of cross-pollination.
References and Notes
1. P. S. Ashton, Biol. J. Linn. Soc. 1, 155 (1969).
2. A companion paper (S. P. Hubbell, J. E. Klahn, G. Stevens, R. Ferguson, in preparation) describes the forest site in detail, and discusses the patterns for each tree species.
3. A. R. Wallace, Tropical Nature and Other Essays (Macmillan, London, 1878), p. 65.
4. G. A. Black, Th. Dobzhansky, C. Pavan, Bot. Gaz. 111, 413 (1950); J. M. Pires, Th. Dobzhansky, G. A. Black, ibid. 114, 467 (1953).
5. P. A. Colinvaux, It1troduction to Ecology (Wiley, New York, 1973), p. 477ff.; J. M. Emlen, Ecology: An Evol~rtiotlaiy Approach (Addison-Wesley, New York, 1973), p. 400; C. J. Krebs, Ecology: The Experimet1tal Analysis of Distribrttiorl and Abutdance (Harper & Row, New York, 1972), p. 520; R. H. MacArthur and J. H. Connell, The Biology of Populutiot1s (Wiley, New York, 1967), p. 37; R. H. MacArthur, Geographical Ecology (Harper & Row, New York, 1972), p. 191; E. R. Pianka, Evolutiotlary Ecology (Harper & Row, New York, ed. 2, 1978), p. 296; P. Price, Insect Ecology (Wiley Interscience, New York, 1975), p. 49; J. L. Richardson, Dimetlsions of Ecology (Williams & Wilkins, Baltimore, 1977), p. 219; R. E. Ricklefs, Ecology (Chiron, Portland, 1973), p. 721. Most texts present several of the competing theories to explain the low density of tropical tree species.
6. D. H. Janzen, Am. Nut. 104, 501 (1970).
7. J. H. Connell, in Dynamics ofPopulations, P. J. den Boer and G. R. Gradwell, Eds. (PUDOC, Wageningen, Netherlands, 1970), pp. 298-312.
8. J. H. Connell, J. G. Tracy, 0. 0. Webb, unpublished result cited in (7).
9. D. H. Janzen, in Taxotlorny and Ecology, V. H. Heywood, Ed. (Systematics Association, special volume No. 5), (Academic Press, New York, 1973), chap. 10; Antlu. Rev. Ecol. Syst. 2, 465 (1971).
10. J. H. Connell, Scietlce 199, 1302 (1978).
11. H. G. Baker, ColdSprit~g Harbor Symp. Quant. Biol. 24, 177 (1959); A. A. Fedorov, J. Ecol. 54, 1 (1966); A. Kaur, C. 0. Ha, K. Jong, V. E. Sands, H. T. Chan, E. Soepadmo, P. S. Ashton, Nature (Lotldon) 271, 440 (1978).
12. D. H. Janzen, Science 171, 203 (1971); G. W. Frankie, in Tropical Trees: Variation, Breeding, and Conservatiot1, J. Burley and B. T. Styles, Eds. (Linnean Societv .,. Svmvosium Series No. 2. 1976); pp. 151-159.
13. K. S. Bawa, Evolution 28, 85 (1974).
14. -and P. A. Ovler. ibid. 29. 167 (1975)
15. I do not mean td suggest that clumping is any sort of evolved adaptation to rarity.
16. For example, if a rare species occurs only as widely scattered, very large adults, we may sus- pect that the population is a relict of an earlier successional episode when the species was once more abundant. Alternatively, if a rare species is locally abundant with a high proportion of juveniles, it may be self-replacing and may be rare only because its microhabitat is local and rare. This sort of evidence is only circumstantial, at best, since it is always possible to construct a life table for any observed age distribution con- sistent with population growth, constancy, or decline. However, the reasonableness of some life tables necessary to balance some age distributions may be questionable. For example, to balance a population consisting only of extreme- ly old individuals requires (i) highly episodic reproduction so that there is very little overlap of generations, (ii) extremely delayed age at matu- rity, (iii) high survival from seed to adult, or else extremely high adult fecundities, and (iv) synchronous fruiting, at long intervals, corresponding to (i). [Also see (53).]
17. K. S. Bawa and P. A. Opler (14),and M. N. Melampy and H. F. Howe [Evolution 32, 867 (1978)l have reported that, although sex ratios may 6 e skewed-from 1: 1, the stamhate and pistillate plants are distributed at random with re- spect to one another in the species they studied. Determining what "sex ratio" means in plants can be difficult at times, particularly if there is temporal variation in the number of staminate and pistillate flowers offered by individual plants. See also R. W. Cruden and S. W. Hermann-Parker libid. 31, 863 (1977)l and K. S. Bawa (ibid. p. 52).
18. G. W. Frankie. H. G. Baker. P. A. Ovler. . . J Ecol. 62, 881 (1'974).
19. Information on the effects of root and crown competition and thinning on the growth of plantation stands is available in references on tropical trees [T. C. Whitmore, Tropical Rait1forests of the Far East (Clarendon, Oxford, 19731.
20. Potentially positive effects of density on reproductive success are also possible, within certain density ranges. For example, if pollinators are limiting seed set, a group of adult trees might exhibit greater per capita seed set than isolated adults if per capita rates of pollinator visitation were greater in the group.
21. The mathematical model is similar in aspects to verbal models developed by J. H. Connell (10) and J. Terborgh [Am. Nat. 107, 956 (1973)l.
22. The site is a nearly level plain at 100 m elevation, 1092'N, 85"18'W. Soils are uniform, pale orange-brown silty clays derived from low bluffs of rhyolitic tuff, and exposed basaltic flows 200 to 600 m north of the site. Soils are 1.4 to 2.2 m deep, underlain by basaltic flows. Because the forest is seasonally dry for 6 months, a shallow layer of leaf litter (A-0) and humus (A-1) persists all year. A small seasonal creek flows through the site in a channel cut to the basaltic bedrock.
23. E. R. Heithaus, Ann. Mo. Bot. Gard. 61, 675 (1974); L. K. Johnson and S. P. Hubbell, Ecology 56, 1398 (1975); S. P. Hubbell and L. K. Johnson, ibid. 58, 949 (1977); ibid, in press.
24. D. H. Janzen, Evolutior~21, 620 (1967); R. Daubenmire, J. Ecol. 60, 147 (1972).
25. L. R. Holdridge, Life Zone Ec.ology (Tropical Science Center, San Jose, Costa Rica, 1967); J. 0.Sawyer and A. A. Lindsey, Vegetation of the Life Zones of Costa Rica (Indiana Academy of Sciences, Indianapolis, 1971); C. E. Schnell, Ed., 0.T. S. Handbook (Organization for Tropical Studies, San Jose, Costa Rica, 1971).
26. This total represents approximately a third of all woody species reaching a size of 2 cm dbh or larger in the Dry Forest Life Zone of Costa Rica.
27. Cattle enter the forest on occasion and cause some damage to the shrub understory by tram- pling and browsing. This disturbance is discussed in relation to the dispersion patterns found (2); cattle disturbance cannot have produced the clumped dispersion patterns of adult trees, nor the concentrations ofjuveniles around adults found in nearly all species (2).
28. Potential mammalian seed dispersers common in the forest include deer, howler monkeys, var- iegated squirrels, and agoutis. Visits by whitefaced monkeys, spider monkeys, tapirs, peccaries, and pacas have also been recorded. Oth- er mammals include armadillos, coatimundis, and kinkajous. Frugivorous birds are also com- mon.
29. It is difficult to find intact seeds of species such as Hyrnenaea courbaril, Enterolobirtm cyclocurprtm, Cassia spp., and many others that have not been attacked by weevils.
30. Many of the unknown plants were identified in the field by P. A. Opler and confirmed by vouch- er specimens sent to W. Burger (Field Museum, Chicago) or R. Leisner (Missouri Botanical Gardens, St. Louis). A few specimens could not be identified to species because they lacked reproductive structures.
31. Individual maps for each quadrat were drawn on graph paper (scale 2 mlcm) in the,field. Plants within 5 m of the quadrat perimeter were mapped first; plants inside the inner square (10 by 10 m) were added afterward, the penmeter plants being the reference points. Independent checks of the accuracy of mapping showed that two people could locate most plants to within 1 m of each other, and of the plant's true position. Mapping took two people 3 months. Gross for- est structure, by dbh class, is as follows: 2.0 to 4.9 cm, 9788 stems; 5.0 to 9.9 cm, 2606 stems; 10.0to 14.9 cm, 1359 stems; 15.0 to 19.9 cm, 650 stems; 20.0 to 29.9 cm, 768 stems: 30.0 to 39.9 cm, 41 1 stems; 40.0 to 49.9 cm, 117 stems; 50.0 to 99.9 cm, 163 stems; > 100 cm, 27 stems.
32. It was not possible to check the reproductive ca- pacity of every tree or species. Information was obtained on 61 tree species. The somewhat arbitrary rule of setting the lower limit for adult size at the smallest-sized individual of the species in flower or fruit was used. In general, this rule means that we probably have virtually all of the adults in the "adult" class. By the same token, however, some nonreproductive subadults are probably included with the adults. For purposes of this study, it was better to risk overestimating the number of adults since, in general, there were many more juveniles than adults.
33. Maps were drawn to a scale of 1: 450. Locations of individual plants were marked by a letter. The letter A represented plants < 2.5 cm dbh; B represented plants between 2.5 and 5.0 cm dbh; and thereafter. letters revresented 5 cm dbh in- crements.
34. M. Morisita, Mern. Fuc. Sci. Kyush~ Utliv. Ser. E 2. 215 (1959). Morisita's disoersion index is a ratio of the observed probabiliiy of drawing two individuals randomly without replacement from the same quadrat (over q quadrats), to the ex- pected probability of the same event for individuals randomly dispersed over the quadrats. The index is unity when individuals are randomly dispersed, regardless of quadrat size or the mean density of individuals per quadrat. Values greater than unity indicate clumping, and values between 0 and 1 indicate uniformity. An Fstatistic can be computed to test for significant depar- ture of the index (symbolized by I,) from unity (randomness).
35. Circular, clear plastic overlays with concentric rings drawn to scale every 5 m were made by Thermofax copier. These overlays were cen- tered in turn over each adult tree on the map, and the numbers of adults and juveniles were counted in each successive 5-m annulus, out to 100 m. Density was computed by dividing the number of counts by the area of the annulus in question.
36. Small-seeded species with wind, bird, or bat dispersal should exhibit greater dispersal distance than large-seeded species dispersed by ground mammals. The greater dispersal distance ex- pected in small-seeded species should tend (i) to obscure the local difference in seed production by both crowded and scattered adults and (ii) to increase the apparent per capita reproduction of scattered versus crowded adults. Consider, for example, the limiting case in which regional av- eraging of juvenile production is so complete that all quadrats have an equal density of juveniles. Then quadrats that have a small number of adults will show a high ratio of juveniles to adults, whereas quadrats that have many adults will show a low ratio, regardless of any density dependence.
37. The expected nearest neighbor distance, d, is given by d = lI(Z\Tp), where p is the mean density in number of adults per square meter [P. Clark and F. C. Evans, Ecology 35, 445 (1954)l.
38. Of the 21 remaining species of trees, shrubs, or vines represented by one individual, eight species could not be identified from our vegetative samples.
39. Cochlospermum r.itifolilrm Spreng. (Cochlospermaceae) is a deciduous species found in large light gaps. It grows rapidly and has wood of low density. Its large dry-season flowers (see cover) are pollinated by big anthophorid bees, and its seeds are wind-dispersed. Hymetluea courbaril L. (Leguminosae) is a slow-growing, evergreen species commonly found in riparian areas, and it has dense wood. It has fairly large flowers which are bat-pollinated. Its seeds are large and encased in a thick, woody pod, are at- tacked on the tree before dispersal by bn~chid weevils, and are food for a number of mammals ID. H. Janzen, Science 189, 145 (1975)l. Licania arborea Seem. (Rosaceae) is a slow-growing, evergreen, high-canopy species of the ma- ture dry forest, with very dense wood. It has small, bee-pollinated flowers, and its seeds are dispersed by bats, birds, and monkeys. Tabebuia roseu DC (Bignoniaceae) is a tree of intermediate stature characteristic of forest edge and large light gaps. It is relatively fast growing, with wood of intermediate density. It is decid- uous and produces a showy display of large flowers in the dry season. Pollination is by large anthophorid bees. Its pods are straplike, con- taining numerous small, flat, winged, wind-dispersed seeds. Thouirlidium decarldrurn Radlk. (Sapindaceae) is a medium-sized, deciduous tree of mature forest, with a moderate growth rate and medium to dense wood; its seeds are winddispersed.
40. P. S. Ashton (I) and M. E. D. Poore [J.Ecol. 56, 143 (1968)l also report clumping in rain-forest tree species in Malaysian dipterocarp forests. Their large rain forest maps cannot be used very effectively to test the Janzen-Connell hypothesis since only trees with a circumference 30 cm were mapped, and they did not distinguish be- tween adult and juvenile trees.
41. Because the demographic neighborhoods of adjacent adults are not completely independent (some juveniles are counted in the neighborhoods of more than one adult), compensation for this partial nonindependence should be made by a downward adjustment of the degrees of freedom of the regression analysis of variance. Accordingly, the average redundancy of juvenile counts was determined by the ratio of apparent number of juveniles in the neighborhoods of all adults to the actual number of juveniles in the union of all adult neighborhoods. The number of degrees of freedom was reduced by dividing the number of adults by the mean juvenile redundancy. This procedure results in a conservative estimate of the true degrees of freedom (R. Lenth, Department of Statistics, Univ. of Iowa, personal communication), and corresponds to the case of completely dependent demographic neighborhoods. For example, suppose that there are only three coincident adults in the population, which consequently have completely identical demographic neighborhoods in which every juvenile is counted three times. Therefore, mean juvenile redundancy is 3, and the number of independent data sets is found by dividing the number of adults by 3, giving one degree of freedom. A similar procedure was followed in testing the slopes of the adult density curves, with mean adult redundancy being used to adjust the degrees of freedom. However, even without the downward adjustment of degrees of freedom, the significance tests are con- servative since individual trees counting in the neighborhoods of more than one adult should always bias the slopes in the positive, not in the negative, direction. The slopes found were all negative or zero.
42. Fifteen meters away from the parent tree might be enough to permit the seeds to escape from discovery by seed predators. However, D. E. Wilson and D. H. Janzen [Ecology 53, 954 (1972)l found that inSc.heeleu palm there was no reduction in the percentage of seeds attacked under the palm and at a distance of 8 m, when seed density was held constant.
43. This analysis, of course, is analogous to con- structing a vertical life table, which assumes that the population size has been approximately sta- tionary for some time. While the conclusions are not necessarily invalid if the population is increasing, their validity cannot be confirmed.
44. In the 30 most common tree species, immediately adjacent adults with touching crowns are not infrequent.
45. Th. Dobzhansky and S. Wright, Generics 28, 304 (1943).
46. J. A. Endler, Geographic Vuriation, Speciution, u/ld Clines (Princeton Univ. Press, Princeton, N.J., 1977).
47. N. G. Hairston, Ecology 40, 404 (1959).
48. The self-compatible species may do little if any selfing (R. Cmden, personal communication). K. S. Bawa (personal communication) has shown that most of the hermaphroditic species in the dry forest he has studied are self-incompatible. Therefore, in Fig. 6, hermaphroditic species for which self-compatibility data are lacking are pooled with the obligately out- crossed hermaphrodites.
49. I chose the midpoint of the dbh range as the pivotal size in order to be conservative in my esti- mate of the number of species showing positive skewness. This means that, when positive skewing is detected, it is actually very pronounced.
50. Similar patterns of skewing have been reported in rare species by Connell (10) and by D. H. Knight [in Tropicul Ecologicul Systerns (Ecology Series No. ll), F. B. Golley and E. Medina, Eds. (Springer-Verlag, New York, 1975), pp. 53-59]. Knight studied late second-growth stands of semi-evergreen forest on Barro Colorado Island, Panama.
51. In 7 years of work at the study site, I have seen no juveniles produced by these species in spite of repeated flowering (seed set was not observed).
52. Of the 59 species in the forest studied, which oc- cur with an average density (ignoring dispersion) of greater than one individual per hectare, I know of at least 42 species that occur locally in much higher densities elsewhere in Guanacaste; and I expect the same is true for most of the remaining - species.
53. The most parsimonious explanation for the data on rare species in the dry-forest stand described here is general reproductive failure. However, undoubtedly there are some rare species in the forest that are replacing themselves, and there are probably some species that are everywhere rare. G. S. Hartshorn [in Tropiccrl Trees as Living Systems, P. B. Tomlinson and M. H. Zimmerman, Eds. (Cambridge Univ. Press, New York, 1978)l has suggested that "nonregenerating" rare species may simply require particular types of light gaps in order to regenerate, and that rarity i\ due to the infrequency of creation of such gaps.
54. Of 21 wind- or bird-dispersed species, 14 showed density dependence, but of nine mam- mal-dispersed species, only three showed density dependence.
55. Diffuse competition is a term coined by R. H. MacArthur [Geographical Ecology (Harper & Row, New York, 1972)l. It refers to the sum of competition from all interspecific competitors in the community acting on a given species. I have not yet analyzed the associationof species to check this possibility.
56, R. H. Whittaker, Science 147, 250 (1965).
57. Data of H. Klinge, as was reported by E. F. Bnlnig [Atnazoniana 4, 293 (1973)).
58. The curves for the Smoky Mountains are derived from analvsis of sinele. 0.1-hectare stands in the cove and spruce-fiyfdrests, respectively. Also, they represent all vascular plants, notjust woody plants; and the importance values are based on annuai net primary production. The re- spective dry-forest and rain-forest curves, however, are based on larger quadrats (13.44 and l .0 hectares), but only woody plants; and the re- spective importance values are based on basal area and above-ground biomass. The quantitative effects of these differing methods are difficult to assess, but fortunately the effects are partially canceling (larger plots mean more species, but eliminating nonwoody species means fewer species). If there is a greater percentage of nonwoody plants in temperate forests, the difference between the species richness of temper- ate and tropical forests may be somewhat underestimated in Fig. 8.
59. M. P. Johnson and P. H. Raven, Evol. Biol. 4, 127 (1970); S. J. McNaughton and L. L. Wolf, Science 167, 131 (1970).
60. R. M. May, in Theoretical Ecology: Pri/lciple.~ and Applicutions, R. M. May, Ed. (Saunders, Philadelphia, 1976).
61. H. Caswell, Ecol. Mo/logr, 46, 327 (1977).
62. R. MacArthur, Am. Nar. 94, 25 (1960): G. Sugihara (unpublished result) has also develooed a "sequeniial breakage" model that generates lognormal patterns.
63. Losses are governed by the hypergeometric distribution. Thus, for the ith species the probability of losing jindividuals, j 5 D 5 N,,, where N,, is the population of species i at the current time 1, is given by
64. A disturbance can be as small as the death of a single tree.
65. Recruitment to fill the D disturbance vacancies is governed by the binomial distribution. Let N',, be the number of the ith species at time t after disturbance. For the ith species, the probability of contributing m replacement indiv~duals, tn c D, is: K N ' -D)"-ln K-D K - D
66. It is easy to prove that rare species are more likely to go extinct per unit time than are com- mon species: If N,, 1 D, there is no chance that species i will go extinct in the next disturbance; but if IV,, 5 D, there is a nonzero chance of ex- tinction in the next disturbance. Therefore, the mean time to extinction of a species j with N,, > D must exceed that of a species i with N,, 5 D by at least the mean time it takes for N, to decrease to D. If a "rare" species is one for which N 5 D, then increasing D will increase the number of rare species going extinct per disturbance over successive disturbances.
67. In this example, I chose a large D simply to speed up the process of random-walk extinction. These transient distributions obey the "canoni- cal hypothesis" of F. W. Preston [Ecology 43, 185 and 410 (1%2)]. It has been shown (21) that the canonical lognormal is the result of impos- ing a fixed ceiling, K, on the total number of in- dividuals of all species in the community. This result has also been discovered independently by G. Sugihara (personal communication). The Monte Carlo simulations were performed at the University of Iowa Computer Center on an IBM 360170.
68. Because the species can random walk up or down in abundance, either outcome (extinction or complete dominance) is possible. I chose to illustrate a species with Kl2 individuals because it represents the abundance at which a species is equally likely to go to either outcome. It is also the abundance with the longest mean transient time, both outcomes considered.
69. The model is a Markovian random walk of the abundance of the ith species between 0 and K. Mean transient times from a starting abundance of M2 individuals were found from the fundamental matrix determined for oarticular values of D and K.
70. There is an added risk of extinction for rare species if they are more clumoed than common soe- cies, such that a single disturbance might kill'all individuals in a given local area. 71. The extinction of many temperate tree species is well documented in Europe (73).
72. For example, see M. L. Heinselman, Qrtut. Res. 3, 329 (1973); J. D. Henry and J. M. A. Swan, Ecology 55, 772 (1974).
73. M. B. Davis, Geosci. Man. 13, 13 (1976).
74. R. Foster, personal communication; S. P. Hubbell, personal observations.
75. B. S. Vuilleumier, Scietice 173, 771 (1971); J. E. Damuth and R. W. Fairbridge, Bull. Geol. Soc. Am. 81, 189 (1970); J. Haffer, Scietlce 165, 131 (1969); B. S. Simpson and J. Haffer, Annu. Rev. Ecol. Sy.vt. 9, 497 (1978).
76. The model in its simplest form as presented here corresponds to the "equal chance hypothesis" discussed by Connell (101, provided that per capita chances of reproduction or death are made the same for all species. For greater realism, species differences in dispersal, fecundity, competitive ability, and resistance to environmental stresses need to be treated.
77. Help from the following people was vital to the completion of this study; Jeffrey Klahn, George Stevens, Paul Opler, Richard Ferguson, Ronald Leisner, William Burger, Daniel Janzen, Joseph Connell, Leslie Johnson, Robin Foster, and Douglas Futuyma. Others have also made con- tributions which are appreciated. The study was
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