Fluctuating Vole Populations Research Articles

Nonlinear time‐series modeling of vole population fluctuations

AbstractA central goal of population ecology is to understand and predict fluctuations in population numbers. Until recently, much of the debate focused on the issue of population regulation by density‐dependent factors. In this paper, I describe an approach to nonlinear modeling of time‐series data that is designed to go beyond this question by investigating the possibility of complex population dynamics, characterized by lags in regulation and periodic or chaotic oscillations. The questions motivating this approach are: what are relative contributions of endogenous vs. exogenous components of dynamics? Is the irregular component in fluctuations entirely due to exogenous noise, or do nonlinearities contribute to it, too? I describe the philosophy and the technical details of the nonlinear modeling approach, and then apply it to a collection of time‐series data on vole population fluctuations in northern Europe. The results suggest that population dynamics of European voles undergo a latitudinal shift from stability to chaos. Dynamics in northern Fennoscandia are characterized by positive Lyapunov exponent estimates, and a high degree of short‐term (one year ahead) predictability, suggesting a strong endogenous component. In more southerly populations estimated Lyapunov exponents are negative, and there is no one‐step ahead predictability, suggesting that fluctuations are driven by exogenous factors.

Cryptic Relatedness and the Opportunity for Kin Recognition in Microtine Rodents

Charnov and Finerty (1980) suggested that vole population density can be regulated by aggressive interactions that change with the relatedness of neighbors. As density increases, the relatedness of neighbors will decrease because of dispersal. Presumably, unrelated individuals will behave less amicably (more aggressively) toward one another. As density increases, dispersal will cause the relatedness of neighbors to decrease and therefore increase the levels of aggression in the population. Increased aggressive levels are postulated to cause the crash of numbers seen in vole population fluctuations. Inherent in this mechanism is the ability for voles to distinguish, and respond differentially between relatives and nonrelatives. As we will show, for this hypothesis to be correct, it must take into consideration cryptic relatedness, relatedness in which the members of the dyad cannot, in fact, determine that they are related. Familiarity, based on association prior to weaning, is the proximate cue that microtine rodents use to assess kinship (Blaustein et al. 1987a,b, Ferkin 1990). Familiar nestmates are recognized as close relatives, whereas unfamiliar nestmates are not identified as relatives (Schadler 1983, Gavish et al. 1984, Boyd and Blaustein 1985, Ferkin 1989, Ferkin and Rutka 1990). To date, there is no evidence to show that microtine rodents employ phenotype matching, recognition alleles, or the major histocompatibility complex to assess kinship (Yamazaki et al. 1980). Genetic relatedness, per se, plays no apparent role in kin recognition for microtine rodents (Bekoff 1981, Blaustein et al. 1987a,b, Ferkin 1990). Thus, the ability to assess genetic relatedness of unfamiliar relatives appears unlikely for microtine rodents. The persistence of kin recognition is dependent on repeated social contacts with close kin (Ferkin 1988, 1990). Although kin recognition and kin-directed behaviors, such as reduced agonism, sharing of nest sites, increased nepotism, and the avoidance of inbreeding, can operate without kin recognition if related individuals interact frequently (Hamilton 1964), repeated contacts with close relatives may not be a feature common to most microtine social organizations (Wolff 1985, Cockburn 1988). The opportunity to identify and be recognized as a close relative will be affected markedly by the type of mating system, population turnover rate, dispersal tendency, nesting patterns, and parental care displayed in a particular species. Therefore, the opportunity to identify and be recognized as a close relative will be affected markedly between microtine species. Microtine rodents that do not have protracted interactions with close relatives may not be able to recognize unfamiliar close relatives.

Spatial dynamics in fluctuating vole populations.

Patterns and consistency of distribution, spatial and temporal components, and the extent of spatial density-dependence were compared between semi-cyclic and cyclic populations of the vole species Clethrionomys glareolus and Microtus agrestis in south-central and north Sweden. Cyclic populations were less clumped and only C. glareolus showed a consistency in distribution between years. Spatial variation contributed little to the distributions in cyclic populations while the spatial and temporal variations were of the same magnitude in the semi-cyclic populations. The latter populations could be subdivided into areas with different spatial and temporal components. The spatial density-dependence increased from increase to decline years in C. glareolus but not in M. agrestis, which differed conspicuously between reregions in population development. The data imply that spatial dynamics should be considered as much as temporal ones for non-cyclic populations, that the same regulating or limiting factors may be at work in both spatial and temporal components and that, in addition, social behaviour may be important in explaining spatial dynamics. However, the latter effects may be fairly species-specific.

Food limitation and social regulation in a red fox population

This study evaluates a conceptual model on functional and numerical response to short‐term fluctuating vole populations of a red fox (Vulpes vulpes L.) population in south‐central Sweden. The model assumes that this particular population is located in between socially regulated stable populations to the south and direct food‐limited populations to the north. The model predicts: (1) food availability as the primary factor for limiting fox numbers, causing reduced rates of reproduction and survival during years of low vole densities, and (2) density‐dependent regulation during years of increasing and high vole densities resulting in increased group sizes within territories of fixed dimensions. During 1973–1980 data were obtained from 1216 fox scats, 874 fox carcasses, 63 tagged foxes, nine radio‐collared females and from yearly den counts in an area of 130 km2, Eight predictions of the model were tested. These concerned the occurrence of small rodents in fox diet, fluctuations in the density of foxes, variations in the number of fox litters, the effect on reproduction of providing supplemental food during January–May, the proportion of vixens bearing a litter different years, dispersal of young males relative to that of young females throughout the vole cycle, and variations in mortality rates of young males and females. All tests were in favour of the conceptual model, and contradictory to alternative models.

More Thoughts on Vertebrate Predator Regulation of Prey

Previous articleNext article No AccessNotes and CommentsMore Thoughts on Vertebrate Predator Regulation of PreySam Erlinge, Gorgen Goransson, Goran Hogstedt, Goran Jansson, Olof Liberg, Jon Loman, Ingvar N. Nilsson, Torbjorn von Schantz, and Magnus SylvenSam Erlinge Search for more articles by this author , Gorgen Goransson Search for more articles by this author , Goran Hogstedt Search for more articles by this author , Goran Jansson Search for more articles by this author , Olof Liberg Search for more articles by this author , Jon Loman Search for more articles by this author , Ingvar N. Nilsson Search for more articles by this author , Torbjorn von Schantz Search for more articles by this author , and Magnus Sylven Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by The American Naturalist Volume 132, Number 1Jul., 1988 Published for The American Society of Naturalists Article DOIhttps://doi.org/10.1086/284842 Views: 3Total views on this site Citations: 21Citations are reported from Crossref Copyright 1988 The University of ChicagoPDF download Crossref reports the following articles citing this article:Anders Pape Møller, Timothy A. Mousseau Assessing effects of radiation on abundance of mammals and predator–prey interactions in Chernobyl using tracks in the snow, Ecological Indicators 26 (Mar 2013): 112–116.https://doi.org/10.1016/j.ecolind.2012.10.025Alexandre Millon, Jan Tøttrup Nielsen, Vincent Bretagnolle, Anders Pape Møller Predator-prey relationships in a changing environment: the case of the sparrowhawk and its avian prey community in a rural area, Journal of Animal Ecology 78, no.55 (Jul 2009): 1086–1095.https://doi.org/10.1111/j.1365-2656.2009.01575.xCornelia Kraus, Heiko G. Rödel Where have all the cavies gone? Causes and consequences of predation by the minor grison on a wild cavy population, Oikos 105, no.33 (Jun 2004): 489–500.https://doi.org/10.1111/j.0030-1299.2004.12941.xA. R. E. Sinclair Mammal population regulation, keystone processes and ecosystem dynamics, Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences 358, no.14381438 (Aug 2003): 1729–1740.https://doi.org/10.1098/rstb.2003.1359G. L. Blackwell, M. A. Potter, J. A. McLennan, E. O. Minot The role of predators in ship rat and house mouse population eruptions: drivers or passengers?, Oikos 100, no.33 (Apr 2003): 601–613.https://doi.org/10.1034/j.1600-0706.2003.11026.xTero Klemola, Miia Tanhuanpää, Erkki Korpimäki, Kai Ruohomäki Specialist and generalist natural enemies as an explanation for geographical gradients in population cycles of northern herbivores, Oikos 99, no.11 (Nov 2002): 83–94.https://doi.org/10.1034/j.1600-0706.2002.990109.xG.L Blackwell, M.A Potter, E.O Minot Rodent and predator population dynamics in an eruptive system, Ecological Modelling 142, no.33 (Aug 2001): 227–245.https://doi.org/10.1016/S0304-3800(01)00327-1Ilkka Hanski, Heikki Henttonen, Erkki Korpimäki, Lauri Oksanen, Peter Turchin SMALL-RODENT DYNAMICS AND PREDATION, Ecology 82, no.66 (Jun 2001): 1505–1520.https://doi.org/10.1890/0012-9658(2001)082[1505:SRDAP]2.0.CO;2Lisa M Sheffield, Jamie R Crait, W Daniel Edge, Guiming Wang Response of American kestrels and gray-tailed voles to vegetation height and supplemental perches, Canadian Journal of Zoology 79, no.33 (Mar 2001): 380–385.https://doi.org/10.1139/z00-220JAN LINDSTRÖM, ESA RANTA, HANNA KOKKO, PER LUNDBERG, VEIJO KAITALA From arctic lemmings to adaptive dynamics: Charles Elton's legacy in population ecology, Biological Reviews 76, no.11 (Jan 2007): 129–158.https://doi.org/10.1111/j.1469-185X.2000.tb00061.xT. LODE Functional response and area-restricted search in a predator: seasonal exploitation of anurans by the European polecat, Mustela putorius, Austral Ecology 25, no.33 (Jun 2000): 223–231.https://doi.org/10.1111/j.1442-9993.2000.tb00023.xStephen M. Redpath, Simon J. Thirgood Numerical and functional responses in generalist predators: hen harriers and peregrines on Scottish grouse moors, Journal of Animal Ecology 68, no.55 (Sep 1999): 879–892.https://doi.org/10.1046/j.1365-2656.1999.00340.xSusan E. Riechert, Louis Provencher, Kendra Lawrence THE POTENTIAL OF SPIDERS TO EXHIBIT STABLE EQUILIBRIUM POINT CONTROL OF PREY: TESTS OF TWO CRITERIA, Ecological Applications 9, no.22 (May 1999): 365–377.https://doi.org/10.1890/1051-0761(1999)009[0365:TPOSTE]2.0.CO;2Mark O’Donoghue, Stan Boutin, Charles J. Krebs, Gustavo Zuleta, Dennis L. Murray, Elizabeth J. Hofer FUNCTIONAL RESPONSES OF COYOTES AND LYNX TO THE SNOWSHOE HARE CYCLE, Ecology 79, no.44 (Jun 1998): 1193–1208.https://doi.org/10.1890/0012-9658(1998)079[1193:FROCAL]2.0.CO;2Lars Erik Lindell, Anders Forsman Density effects and snake predation: prey limitation and reduced growth rate of adders at high density of conspecifics, Canadian Journal of Zoology 74, no.66 (Jun 1996): 1000–1007.https://doi.org/10.1139/z96-113KAI NORRDAHL POPULATION CYCLES IN NORTHERN SMALL MAMMALS, Biological Reviews 70, no.44 (Nov 1995): 621–637.https://doi.org/10.1111/j.1469-185X.1995.tb01654.xLouis Provencher, Susan E. Riechert Theoretical comparisons of individual success between phenotypically pure and mixed generalist predator populations, Ecological Modelling 82, no.22 (Oct 1995): 175–191.https://doi.org/10.1016/0304-3800(95)92851-ME. Paradis, G. Guédon Demography of a mediterranean microtine: the Mediterranean pine vole,Microtus duodecimcostatus, Oecologia 95, no.11 (Mar 1993): 47–53.https://doi.org/10.1007/BF00649505P. L. Meserve, J. R. Guti�rrez, F. M. Jaksic Effects of vertebrate predation on a caviomorph rodent, the degu (Octodon degus), in a semiarid thorn scrub community in Chile, Oecologia 94, no.22 (Jan 1993): 153–158.https://doi.org/10.1007/BF00341311Steven L. Lima, Thomas J. Valone Predators and avian community organization: an experiment in a semi-desert grassland, Oecologia 86, no.11 (Mar 1991): 105–112.https://doi.org/10.1007/BF00317396Lennart Hansson Spatial dynamics in fluctuating vole populations, Oecologia 85, no.22 (Dec 1990): 213–217.https://doi.org/10.1007/BF00319403

Latitudinal Gradients in Predator-Prey Interactions, Cyclicity and Synchronism in Voles and Small Game Populations in Finland

Finnish game questionnaire data from 1964-83 have been used to study cyclicity and the synchronism between species in population fluctuations of voles and 10 small game species. All the species are cyclic in at least some provinces. The most clearly cyclic species are black grouse and hazel grouse, with 6-7 yr cycles. Stoat, weasel and voles are representatives of the 3-4 yr type. The observed cycle lengths are concentrated in years four and seven. The degree of cyclicity is much higher in northern Finland than in southern provinces. The fluctuations of predators and their prey are not simultaneous. However, there is synchronism between ecologically similar species or synchronism (with time lag) between a predator and its prey. The relatively weak synchronism of voles to other species, the disharmony in the distribution of cycle lengths and the contradictory predator-prey relationships suggest that the alternative prey hypothesis cannot explain the origin of, for instance, tetraonid cycles.

Do Predators Synchronize Vole and Grouse Fluctuations?: An Experiment

Synchronized fluctuations in vole and grouse populations in boreal Fennoscandia have been explained as the result of shifting predation pressure: the alternative prey hypothesis. We tested this hypothesis by supplying additional food to predators in only one of two 615 ha areas during the crash in vole density in 1985. As predicted from the hypothesis the proportion of young grouse in autumn was higher in the experimental area than in the control area. Moreover, the proportion decreased in the control area in comparison with the year before, but did not do so in the experimental area. Thus, with respect to synchronized short-term population fluctuations of voles and grouse, we find the hypothesis that predators take only a doomed surplus less probable. However, we cannot discard the hypothesis that grouse decline during years of strong vole decline for example because a poor quality or low quantity of food makes them more vulnerable to predators. Hence, predation is a necessary but perhaps not a sufficient factor in mediating short-term fluctuations from voles to grouse in boreal Fennoscandia.

The role of seasonality in vole cycles

We examined the assertion that seasonal fluctuations in food availability are necessary for vole cycles to take place by studying the dynamics of a California vole (Microtus californicus) population inhabiting a relatively aseasonal environment. That population was one of the most stable (noncyclic) microtine populations yet described. Reproduction was comparatively aseasonal but appeared to be suppressed at high population density. We suggest that the role of seasonality in vole population fluctuations deserves further study.

Synchronous short‐term population fluctuations of some birds and mammals in Fennoscandia ‐ occurrence and distribution

We examined population fluctuations of the voles Microtus agrestis L. and Clethrionomys glareolus Schreb. by biannual trapping; black grouse Tetrao tetrix L., mountain hare Lepus timidus L., and red fox Vulpes vulpes L. by using bag records from all over Sweden, questionnaires from south‐central Sweden, and population indices from Grimsö Wildlife Research Area in south‐central Sweden. Synchronous population fluctuations between voles as a group and the other species conformed to a 3–4 year periodic pattern in both autumn and spring populations only in northern and central Sweden (the boreal forest region). Spring populations of boreal forest grouse, hare, and probably also fox, lagged one year behind the voles. The northerly areas also formed geographical units of co‐fluctuation within each game species. Using our own data as well as reviewing previous studies in Fennoscandia, we conclude that synchronized 3–4 year population fluctuations of voles and medium‐sized herbivores are confined to the central and northern part of Fennoscandia, although voles may exhibit short‐term population fluctuations further south. The synchronizing link between the herbivores could be (1) food plant quality and/or quantity and/or (2) predation. We could not reject either of these two plausible mechanisms as a cause of interspecifically synchronous fluctuations.

A 19‐Year Study of Microtine Population Fluctuations Using Time‐Series Analysis

This report involves time—series statistical analysis (including concurrent physical and community variables) of the population dynamics of 4737 voles (Microtus californicus) trapped over 19 yr while emigrating from two study enclosures on a Northern California grassland. Population fluctuation of voles, as documented in the literature as well as in this study, generally cannot be described by periodic or regularly cyclic equations, but rather are either random or occasionally pseudoperiodic where autoregressive correlation explains irregular cycles on the basis of the population's previous demographic history. For these two California vole populations, an autoregressive component accounted for approximately a third of the total variability in the population, while random extrinsic environmental variation explained almost all of the remaining variation. Weather played a key determinant role, influencing microtine populations both directly, and indirectly through an effect on vegetation. The distinction between periodic, pseudoperiodic, and random fluctuations in vole populations, and indeed in wildlife populations in general, cannot be dismissed as merely a question of semantics, because each entails a specific ecologic interpretation. The demographic characteristics of microtine rodents, although fluctuating more dramatically, were similar to those of other rodents.

IS A NEGATIVE REGRESSION OF ΔP ON P EVIDENCE OF A STABLE POLYMORPHISM?

There has been much interest in inferring evolutionary processes from the temporal and spatial distribution of gene frequencies. A recent paper by Kohn and Tamarin (1978) and an earlier paper (Tamarin and Krebs, 1969) examined the temporal changes in allele frequencies in fluctuating vole populations. When the change in gene frequency between successive samples Api (= P,+1 Pt), was plotted against P,, the gene frequency in the ith sample, a significant negative linear regression was obtained. The authors infer that the predicted gene frequency ?, when Ap equals zero, estimates an equilibrium gene frequency (see Fig. 1). Furthermore, they conclude that ? is a stable equilibrium maintained by some form of balancing selection. A stable, nontrivial equilibrium is a predicted result from the standard deterministic model where selection favors the heterozygote (Li, 1955). In this case, the relationship between p and Ap near the equilibrium is negative (see Li, Fig. 1), such that Ap restores p to the equilibrium point. Here I explore other, equally likely reasons for expecting a negative relationship between Ap and p. The relationship between Ap and p depends on the underlying distribution of gene frequencies from which one samples. Consider a case where gene frequency is a random variable with a uniform probability distribution. All gene frequencies in the interval (0,1) are equiprobable and there is no equilibrium point. If gene frequency in the ith sample p,, is an independent sample from the uniform distribution, and Api = p,+? p,, then the conditional expectation of Ap, given pi is

Growth, Reproduction and Residency in a Declining Population of Microtus agrestis

SUMMARY (1) The population dynamics of Microtus agrestis (L.) inhabiting a young larch plantation in south-west Britain were studied from spring 1967 to spring 1969. As the larch trees grew they shaded out the ground layer vegetation and there was a resultant decline in vole density from 97/ha to 6/ha during the 2 main years of study. By the summer of 1972, this species had disappeared altogether. (2) Vole growth rates were significantly correlated with an index reflecting the abundance and rate of growth of the grasses which formed their main source of food. (3) Despite a good start to the breeding season in 1968, conditions deteriorated rapidly as a result of shading, and there was a considerable rise in perinatal mortality. (4) Rates of residency (minimum survival) were lowest during two main periods of the year. The first, in spring, involved mainly males, and coincided with the onset of sexual activity. The second, in summer, involved mainly females, and coincided with the replacement of overwinter residents by animals of the year. (5) Decreases in the rates of residency were correlated with increases in the distances moved between consecutive captures by male voles, reflecting a concomitant increase in the size of their home ranges. (6) Previously published records of male spring body weights from several parts of Britain were found to be significantly correlated with the standing crop of a meadow at Moor House Experimental Station and with the hay yields of the unfertilized control plots in the Rothamsted Park Grass Experiment. These correlations suggest a link between vole population fluctuations and climate. The effects of the quality or quantity of the spring food supply upon vole growth and reproduction could play an important part in this link.

Viability of Large‐ and Small‐Sized Adults in Fluctuating Vole Populations

Peak populations of many species of lemmings and voles contain adults 20—30% larger than adults found in low populations (the Chitty effect). What is the adaptive advantage of being large? We try to answer this question by analyzing the survival rates of large— and small—size adults of 4 Microtus species. Two alternative hypotheses are proposed to explain the adaptive value of large size. Large adults may be r—selected genotypes at an adaptive advantage in increasing populations. Microtus pennsylvanicus and male M. townsendii fit this alternative because the survival advantage of large size is positively related to population growth rate and negatively related to density. Alternatively, large adults may be α—selected genotypes at an adaptive advantage in high density populations when aggression is important. Microtus californicus and M. ochrogaster fit this hypothesis because in these species the survival advantage of large size is positively related to density and negatively related to population growth rate. Thus large body size may be adaptive for quite contrary reasons in different species of cycle voles and lemmings.

TEMPORAL PATTERNS OF ALLOZYMIC VARIATION IN FLUCTUATING POPULATIONS OF MICROTUS OCHROGASTER.

Microtine rodents undergo 2-4 year fluctuations in population density known as population cycles. A genetic-behavioral mechanism that is based on an rand Kselection argument has been proposed by Chitty (1960, 1967) for controlling these cycles. During the phase of increasing numbers when mutual interference is minimal, genotypes with a high reproductive effort have a selective advantage, whereas during the peak phase when mutual interference is intensive, there is selection for aggressive behavior. Although aggressive genotypes have an advantage with respect to intraspecific competition for food and space, their fitness is reduced in other ways which sets the stage for the decline phase. Thus, Chitty's hypothesis assumes that natural selection coupled with changes in the genetic composition of the population is the driving force behind population cycles. The results from studies monitoring allozymic variation in fluctuating vole populations are compatible with Chitty's hypothesis. Semeonoff and Robertson (1968) observed changes in gene frequency at a plasma esterase locus in a declining population of Microtus agrestis. Canham (1969) found that heterozygote fitness at transferrin and albumin loci was correlated with density in Peromyscus maniculatus and Clethrionomys gapperi populations. Changes in gene frequency at transferrin (Tamarin and Krebs, 1969) and leucine aminopeptidase loci (Gaines and Krebs, 1971) were correlated with

Variation in the size of foxes in Scotland

Foxes become larger from south to north in Scotland, independently of the climate, the prey taken or the productivity of the areas in which each population lives. In areas where fluctuating vole populations are important as food, foxes born in high vole years are no larger than those born in poor years. However foxes became smaller after a severe winter in north-east Scotland, followed by a gradual increase in the size of each year class through a series of years with mild winters and expanding rabbit populations. It is suggested that food availability is determining the average size of foxes through selection, and the north/south cline in size is the result of increased hunting hours at higher latitudes during winter. The selective advantage of different size animals under conditions of different food availability is discussed.

Genetic Changes in Fluctuating Vole Populations

Genetic Changes in Fluctuating Vole Populations

GENETIC CHANGES IN FLUCTUATING VOLE POPULATIONS.

Microtine rodents undergo periodic fluctuations in population density (Krebs et al., 1969). Ecologists have searched unsuccessfully for the driving force behind these fluctuations, invoking environmental factors such as weather (Elton, 1929), nutrition (Pitelka, 1964), disease (Elton, 1925), and predation (Pearson, 1966). What has been conspicuously ignored in most studies on the regulation of rodent numbers is the genetic composition of the population. This study is a continuation of previous efforts (Krebs et al., 1969; Tamarin and Krebs, 1969) combining population genetics and ecology to provide predictive insight into population fluctuations of small rodents. The purpose of the study was threefold: (1) to demonstrate how the genetic architecture of a population changes with different phases of a population fluctuation, (2) to correlate genetic changes with demographic events occurring in the population, and (3) to establish the temporal and spatial replicability of these genetic changes. Several attempts have been made to monitor genetic changes in fluctuating rodent populations. Canham (1969), studying transferrin and albumin polymorphisms in fluctuating populations of Peromyscus maniculatus, Cleithrionomys rutilus, and C. gapperi, found that heterozygote fitness

The Role of Reproduction and Mortality in Population Fluctuations of Voles (Microtus)

Ecological MonographsVolume 28, Issue 1 p. 79-109 Article The Role of Reproduction and Mortality in Population Fluctuations of Voles (Microtus) Robert S. Hoffmann, Robert S. HoffmannSearch for more papers by this author Robert S. Hoffmann, Robert S. HoffmannSearch for more papers by this author First published: 01 January 1958 https://doi.org/10.2307/1942276Citations: 42AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume28, Issue1January 1958Pages 79-109 RelatedInformation