Oecologia (1993) 95:581-591

Oecologia 9 Springer-Verlag ~[993

Density dependence tests, are they? Henk Wolda 1, Brian Dennis 2 1 Smithsonian Tropical Research Institute, Unit 0948, APO AA 34002-0948, USA 2 University of Idaho, Dept. Fish and Wildlife Resources, Moscow, ID 83843, USA Received: 17 December 1992 / Accepted: 12 July 1993

Abstract. A large number of time series of abundances of insects and birds from a variety of data sets were submitted to a new density dependence test. The results varied enormously between data sets, but the relation between the frequency of statistically significant density dependence (SSDD) and the length of the series was similar to that of the power curve of the test, making the results consistent with the hypothesis of the densitydependent model being universally applicable throughout the data used. Pest and non-pest species did not differ in the incidence of SSDD. The more sampling error present in the data, the higher the percentages of SSDD. This was expected given that the power of the test increases with increasing sampling error in the data. Many of the data used here, as well as in the literature, clearly violate the basic assumption of the test that the organism concerned should be univoltine and semelparous. Yet the incidence of SSDD was the same in univoltine as in bi/polyvoltine species and the same in semelparous organisms as in birds that are reproductively active in more than one year. The seasonal migrant Autographa gamma in Britain and Czechoslovakia and even rainfall data were found to have SSDD. Statistical significance, however, does not automatically lead to the conclusion of density-dependent regulation. Any series of random variables which are in a stochastic equilibrium, such as a series of independent, identically distributed, random variables, is typically described better by the alternative (density-dependent) model than by the null (densityindependent) model. Significant test results were often obtained with sloppy data, with data that clearly violate the basic assumptions of the test and with other data where an interpretation of the results in terms of densitydependent regulation was absurd. Given the fact that other explanations have to be found for significant test results for all these cases, mechanisms other than regulation may very well be applicable too where the data are entirely appropriate for the test. The test is simply a data-based choice between a model without and one with Correspondence to: H. Wolda

a stochastic equilibrium. A time series as such does not contain any information about the causes of the fluctuation pattern, so that one cannot expect statistics to produce such information from that time series. A significant test result using suitable data is entirely consistent with the hypothesis of density-dependent regulation, but also with any other suitable hypotheses. Because the test results were generally consistent with the hypothesis of a universal applicability of the density-dependence model, a negative test result may only mean that the time series was not long enough for the density dependence that was present to become statistically significant. Positive results are difficult to interpret, but so are negative results. A final decision needs to be based not so much on the test result as on much detailed information about the population concerned. Because the "densitydependence test" does not test for the presence of the mechanism of density-dependent regulation and because of the loaded, multiple meanings of the term "densitydependence", calling the test a "test of statistical density dependence" may be preferable. Key words: Density dependence tests - Time series Insects - Birds - Regulation of numbers

Ever since the hypotheses of regulation of animal numbers by density-dependent processes were developed (Volterra 1931; Nicholson 1933; Nicholson and Bailey 1935) ecologists have tried to test them. As a result, evidence of the occurrence of density-dependent processes abounds in the literature, but in many cases questions remained concerning the capacity of those processes to produce the hypothesized regulation of numbers (Reddingius 1971; Reddingius and Den Boer 1989). Some investigators accepted the hypothesis on purely theoretical grounds (Royama 1977; Berryman 1991), but factual evidence remained scarce, and was often controversial or unconvincing (Andrewartha and Birch 1954, 1984; Den

582

Boer 1986, 1987, 1990, 1991; Den Boer and Reddingius 1989; Hassell et al. 1989). The result of all factors and processes acting on the number of individuals in a population is the number of individuals in the next generation. Therefore, in several studies tests for overall regulation were conducted using time series of animal abundances, hoping to finally lay the controversy surrounding the regulation hypothesis to rest. The results varied. In some cases only a small proportion of the series tested showed statistically significant density dependence (SSDD), in others that proportion was high. Part of the reason is that the probability of detecting SSDD was found to increase with increasing length of the time series (Hassell et al. 1989) as the power of at least some tests increases with series length (Solow and Steele 1990). Hanski (1990) and Hanski and Woiwod (1991) attributed the often low incidence of SSDD in the literature to the prevalence of forest pest insects in ecological studies, and explained his finding of high percentages of SSDD by most of the series used being on nonpest, non-outbreak species. Stubbs (1977) found SSDD in more permanent habitats to be less strong than in more temporary habitats. Holyoak and Lawton (1992) detected SSDD among bracken insects more often in open than in woodland habitats, and more often among univoltine insects. Longer series are better, but as to what constitutes an acceptably long time series there is no agreement. Hassell et al. (1989) found that the probability of detecting SSDD reached 100% after 15 years, at least in univoltine insects. Solow and Steele (1990) stressed that except in exceptional circumstances it is necessary to observe at least 30 generations. Holyoak and Lawton (1992) used series of only 8 and 12 years length. Woiwod and Hanski (1992) found an initial increase in the percentage of their series that showed SSDD but a levelling off after 20 years. They also found that this level may be lower than 100% so that in some (small) proportion of their data SSDD may not be found irrespective of the length of the series. There are two major problems with these densitydependence tests, one ecological and one statistical. Once a test finds SSDD, caution needs to be applied before firm ecological conclusions can be reached. The test and its interpretation need to be considered separately. A reliable statistically significant result does not necessarily prove the existence of density-dependent regulation. One must carefully examine all the information at hand about the species concerned before concluding for or against attributing an ecological meaning to the statistical result (Den Boer and Reddingius 1989; Den Boer 1990). As Dennis and Taper (1993) remind us, statistics draws a careful distinction between a statistical hypothesis (a candidate statistical model) and a scientific hypothesis (an explanation of some phenomenon), Also, the value of the existing statistical tests themselves is in doubt. There are not one but several tests for density dependence in time series. Den Boer (1990) used three different tests, Woiwod and Hanski (1992) also three, and Holyoak and Lawton (1992) used five different tests for direct density dependence. The results often vary widely

between tests. Cautionary notes have been published concerning such tests (Reddingius and Den Boer 1989; Solow and Steele 1990; Solow 1990, 1991; Reddingius 1990; Vickery and Nudds 1991 ; Dennis and Taper 1993). The statistical theory behind the tests used up to now is not as well understood as one would like, the behavior of the tests was not always well studied, and some tests are known to produce unreliable results. Any results from these tests are, therefore, highly questionable, and do not inspire too much confidence. One can be conservative and accept only the results that are significant in all tests, or that are significant only in the test with the least number of significant cases, or one can be generous and accept all results that are significant in at least one of the tests (Holyoak and Lawton 1992). Neither procedure is satisfactory. Existing tests for delayed density dependence are also highly questionable (Dennis and Taper 1993). Fortunately, this statistical problem, at least for direct density-dependence tests, now no longer exists. Recently a new density-dependence test, the parametric bootstrap likelihood ratio (PBLR) test, has been developed (Dennis and Taper 1993), a likelihood ratio test based on a stochastic model. Because it is a likelihood ratio test, because the behavior of the test has been thoroughly studied, and because the theory behind likelihood ratio tests has been studied over several decades and is thus well understood, it is the best kind of test one can hope for, given the fact that the information contained in a mere time series of animal abundances is limited. With this new test the statistical problems referred to above seem to have been solved, so that we can now examine the ecological problems without the lingering doubts about the statistical results. In the present paper we will test a variety of data sets for density dependence and discuss some of the results, such as those discussed above, in the light of this new technique and with new data. It will be emphasized that strong caution needs to be applied when drawing ecological conclusions from the statistical test results and a strict distinction will be maintained between statistical results and the ecological interpretation.

Methods T h e test

The PBLR test is described in detail in Dennis and Taper (1993) where its power is examined and the effects of sampling errors evaluated. The test compares the density-independent model: ln(Nt+l/Nt) = ao+Et, with the density-dependent model: ln(Nt+ 1~NO = al + blNt + Et where Nt is the number of individuals in year t and Et is a normally distributed variable with mean zero and a variance of ~2. The null hypothesis is that the density dependent model does not describe the observed pattern better than the density independent one. This means that He: b1=0 is tested against Hi: b1<0. The test is one-sided, as cases where bl > 0 are not considered of interest at this moment. As the test involves taking logarithms of the Art values, a decision had to be made on what to do with series that

583 include one or more zeros. This problem is at present under investigation, but the results thus far strongly suggest that replacing each zero by 0.5 gives the best evaluation of the data. Zeros may occur even in usually abundant species. Using only series without zeros strongly selects against the more variable species to a to us unacceptable extent, and zero-replacement values lower that 0.5 tend to distort the mean and the variance of In(Nt) as well as the variance of ln(N,+l/ln(Nt)) to an unacceptably large extent. Using the ln(Nt + z) transformation, where z is any positive value, for instance 1 or 0.5, introduces an unnecessary extra bias into the data. Among the choices available, we selected the least unacceptable one and we replaced any zero values in the data by 0.5. The modified test described by Dennis and Taper (1993) was used to deal with interrupted time series.

Data used The data used, both on insects and on birds, came from a variety of sources. With a few exceptions only uninterrupted time series of at least 20 years were used and only those that had an average of at least five individuals (insects) or two individuals (birds) per year (called here "common" species). Each time series was tested in its entirety, as well as the first 15, the first 20, and so on, years to test the effect of the length of the series. The sets are:

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to us by Pier den Boer of the Wijster Biological Station of the Agricultural University Invertebrates collected by sweeping and by Berlese funnels in the William Trelease Woods, Illinois (see Kendeigh 1979). There are 15 series of 38 years, a total of 20 series over 35 years, 23 over 30 years and 36 over 20 years. Original data kindly made available to us by the librarian of the University of Illinois at Urbana Migrating Lepidoptera in Great Britain, 21 species over 25 years and 20 over 30 years. Data (from 1933 onwards) from Williams (1971) Rice pests in various sites in Japan for 5 species. There are 152 time series of I5 years, 96 of 20, 56 of 25, 45 of 27 and 5 of 30 years. Data kindly made available to us by Akio 0take of the Fruit Tree Research Station at Yatabe (Ibariki). See also 0take (1966a, 1966b, 1978); 0take and Kono (1970) Agricultural pests at 3 sites in Tasmania. There are 27 series of 15 years and 13 of 24 years. Data kindly made available by E. J. Martyn, Dept. of Agriculture, Hobart, Tasmania Migrating moths, Netherlands. 10 species with 23 years of data, 13 with 15 years. Data collected by B.J. Lempke published in Williams (1971) Mosquitos collected with light-traps in 2 sites in Delaware (U.S.A). 28 series of 20 years. Data from Darsie et al. (1953)

Birds: Insects: A. Moths obtained by light-trap in Brno, Czech Republic. 112 common species collected over 28 years. Data kindly made available to us by Jaroslav Marek of the Agricultural University in Brno B. Homoptera obtained by light-trap in a tropical forest on Barro Colorado Island, Panama. Data cover 17 years and only the 69 most common species (> 1000 individuals in 17 years) were used, except for the seasonal vs aseasonal comparison where the most extreme cases were selected out of all species with at least 5 individuals per year. Data collected by the senior author (see Wolda 1987; Wolda and Broadhead 1985) C. Moths obtained by light-trap in Stratfield Mortimer, England. 112 common species monitored over 20 years. Data are part of the Rothamsted Insect Survey and are kindly made available to us by Ian Woiwod of the Rothamsted Agricultural Station D. Moths obtained by light-trap at Ruzyn~, near Prague, Czech Republic. Data on 19 economically important species covering 24 years, plus 110 species for which this 24 year series was interrupted during 3 years. Data kindly made available to us by Ivo Novak of the Research Institute for Crop Production in Prague. See Novak (1983) E. Moths of economic importance obtained by light-traps in various sites on the British Isles by the Rothamsted Insect Survey and published in their annual reports. There are 38 series of 23 years and 67 total of 20 years. The reports kindly given to us by L. Roy Taylor and Ian Woiwod of the Rothamsted Agricultural Station F. Moths obtained by light-trap near Cesk6 Bud~jovice, Czech Republic. 121 common species were monitored over 20 years. Data kindly made available to us by Karel Spitzer of the Entomological Institute in Cesk6 Bud~jovice. See also Rejm/mek and Spitzer (1982) G. Aphids of economic importance obtained by suction-traps in various sites of the British Isles by the Rothamsted Insect Survey and published in their annual reports. The reports kindly given to us by L. Roy Taylor and Ian Woiwod of the Rothamsted Agricultural Station. There are 32 time series of 32 years, 81 of 24 and 201 of 20 years H. Carabid beetles collected by pitfall traps in various sites in Drenthe, The Netherlands. There are 17 series of 31 years, 47 of 24 years and 103 total of 20 years. Data kindly made available

A. Ringed passerine birds in Sweden, 34 common species over 24 years. Data from Osterlof and Stolt (1982) B. Breeding birds in William Trelease Woods, Illinois. 17 species for 38 years, 19 species for 35 years. See Kendeigh (1982). Original data kindly made available to us by the librarian of the University of Illinois at Urbana C. Wintering birds in William Trelease Woods, Illinois. 16 common species for 40 years. See B D. Breeding birds in Robert Allerton Park, Illinois. 22 common species for 24 years. See B E. Breeding birds in Meijendel, Netherlands. 43 common species for 18 years. See Van Dongen and Ros (1973), Croin Michielsen et al. (1974), Regensburg and Wanders (1978) F. Birds trapped on the Kursh sand spit in Lithuania. 41 species over 17 years. Data from Dolnik and Paevskii (1980) G . Breeding rare birds in Britain. 27 species for 16 years. Data as published annually in issues of British Birds

Results

Basics, exemplified by Panamanian Homoptera F o r the 69 m o s t c o m m o n species o f H o m o p t e r a m o n itored over 17 years, 19 (27.5%) showed significant density d e p e n d e n c e at the P < 0 . 0 5 level, a n d 30 at the P < 0.10 level. A f t e r the first 10 years these n u m b e r s were o n l y 4 (5.8 %) a n d 12 respectively. A m e t a t e s t showed the overall d i s t r i b u t i o n o f P - v a l u e s to be highly significantly different f r o m r a n d o m , even after 10 years, so t h a t at least some o f the significant results are real. The increase in the p e r c e n t significant with i n c r e a s i n g length o f the series s h o u l d come as n o surprise as the p o w e r o f the P B L R test increases in the same way. H o w e v e r , ind i v i d u a l time series do n o t necessarily b e c o m e m o r e a n d m o r e significantly d e n s i t y d e p e n d e n t as the series gets longer. A plot o f Paa (P-value p r o d u c e d b y the density d e p e n d e n c e test) after 10 years a g a i n s t Pad after 17 years, even t h o u g h the c o r r e l a t i o n was significant, showed a

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Fig. 4. The relationship between the p-value resulting from the density-dependence test and the trend of abundance on time, expressed here as the absolutevalue of the correlationcoefficient.The 69 most common species of Panamanian Homoptera wide scatter of points (figure not shown). Of the four species that had a significant Pad value after 10 years, only one (Anormenis nigrolimbata) was still significant after 17 years. The abundance patterns of these four species is given in Fig. 1. Patterns that started as fluctuations at a certain level developed a clear trend or some large fluctuations at some time after the first 10 years, resulting in non-significant Pdd values at 17 years. The actual abundance patterns for the eight most significant and the eight least significant species are given in Fig. 2. This illustrates that a pattern with repeated returns to some intermediate level produced a significant result, as one might expect, while patterns with a clear trend or with large long-term fluctuations did not. The hypothesis tested is H0: bl = 0 versus H1 : b~ < 0. One might thus expect the actual value of bt to be more negative the lower Pdd- However, although the series with the most negative bt values tend to have low Pdd values, b~ values near zero can go together with low Pdd values just as easily (Fig. 3). This is entirely consistent with the properties of the test discussed by Dennis and Taper (1993). The power of the test is strongly affected by al (=speed of return after disturbance) but little by bl (= - a l / M , where M = the abundance level to which the system returns). There is a strong tendency for relatively high a~-values to be associated with low Pad values (Fig. 3). The return tendency the tests looks for is expressed in the a~ rather than in the bt parameter. Holyoak and Lawton (1992) found no relationship between the existence of trends in abundance and the Pdd values. Woiwod and Hanski (1992), on the other hand, with their much larger data set, found trends in abundance tend to be associated with non-significant Pdd values. Fig. 2 supports that finding. For all 69 Homoptera species, the relationship between Pdd and trend, expressed as the absolute value of the correlation coefficient between the logarithm of abundance and time, was significantly positive (Fig. 4). The closer the correlation coefficient was to zero, the lower Pdd. The relationship was weak, however, (r= 0.369) and remained so if other pa-

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Fig. 5. The incidenceof statisticallysignificantdensitydependence in a variety of insect data sets. Percentages calculated at the full length of the series and also after the first 15, 20, ... years.The letters at each line correspond to the upper case letters describing the source of the insect data in the 'data' section of the text. Power curves at two values of the parameter al and at the arbitrary values of other parametersused in Dennis and Taper (1993, Fig. 4) are also given

rameters for linear trend such as ao were used. A given linear trend is no guarantee of significant or non-significant density dependence. However, this was mostly due to the fact that fluctuation patterns were often not well described by a linear trend. The combination of low trend and non-significant Pdd values (points at extreme left in Fig. 4) occurred in widely fluctuating populations that happened not to show a trend in the study period (Fig. 2J). The combination of high trend and significant Pdd (points at extreme lower right in Fig. 4) occurred when the series was generally nicely zigzagging, but happened to have one or two low values at the beginning, or end, of the series and one or more high values somewhere near the other end (Fig. 2C, D). The relationship between "real" trend and Pdd was in fact stronger than the low, but significant, r-value of 0.369 suggests.

General comparisons The PBLR test was applied to a number of insect datasets, for the moment ignoring the requirement of the organisms being univoltine and semelparous. We used the entire length of the series as well as the first 15, 20, an so on years if the series were long enough (Fig. 5). We did the test also for the first 10 years of the series of the BCI Homoptera (B) and the Brno moths (A), but because of the low incidence of statistically significant density dependence (SSDD), presumably because 10-year series are too short for the test to have any power, those calculations were not repeated for the other data sets. There was a tendency for the incidence of SSDD to increase with increasing length of the series, paralleling nicely the relation of test power and series length given by Dennis and Taper (1993). The precise values making up the power curves varies with the actual parameters for

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each series, but two o f Dennis and Taper's power curves, using their parameter values, are reproduced in Fig. 5 to show their general shape. There were large differences between data sets in the incidence of SSDD. The highest percentages were found in the Rothamsted aphids (G) and Rothamsted moths (E), approaching the percentages found by Woiwod and Hanski (1992), although the data we had access to were only a small, selected, subset of those analyzed by them. The lowest percentages were found in the Dutch carabids (H). For some bird data the same procedure was applied (Fig. 6). Increases in the incidence of density dependence with increasing series length are slight. When plotting the same data for the percentage with Pdd<0.10, the increases are much more apparent. Again the percentages

of SSDD vary between data sets, but they are in the same range as those found for insects. Figure 7 gives the abundance patterns for the noctuid moth Autographa 9amma, This species was monitored by the Rothamsted insect survey over at least 20 years in eight localities (A-H) for which we have access to the data. In seven of these eight sites significant or even highly significant test results were found. In two of the three sites (I-K) in Czechoslovakia the results similarly were highly significant. A 67-year dataset on rainfall from Barro Colorado Island, Panama, presented in Fig. 8, shows highly significant "density dependence" in spite of a significant decreasing trend over time (Windsor 1990; Windsor pets. COmlTL).

587 Discussion

Power curves Figure 5 shows that the lines indicating the relation between the percentage of statistically significant density dependence (SSDD) cases and the length of the series nicely parallel power curves of the test. At each series length the corresponding point on a power curve indicates the percentage of SSDD one might expect, given that the density dependent model describes all data better. The curves of Fig. 5 and 6 so strongly resemble power curves that the results are consistent with the hypothesis that the density-dependence model fits the data better in most, if not all, cases. The fact that the percentages of SSDD found are below 100% could be mostly, if not entirely, due to the fact that the time series are simply not long enough. The results do not, of course, prove that the density-dependence model should be universally chosen over the density-independent model, but the results are not inconsistent with such a choice either.

Pest species Hanski (1990) and Hanski and Woiwod (1991) hypothesize that the reason why they found a much higher incidence of SSDD among the Rothamsted moths and aphids than is usually reported in the literature was that all species available were used, while in the literature a strong emphasis is placed on species of economic importance, especially forest pests. According to this hypothesis, pest species may have a relatively low incidence of SSDD. This hypothesis is not supported by the present results, at least not for agricultural pests (Fig. 5). The Rothamsted moth data (E) and aphid data (G) available to us only concern species considered important enough to be mentioned in the Rothamsted Insect Survey Bulletins, mostly because the species are of economic importance. Nevertheless, with this restricted number of species we find relatively high to very high percentages of SSDD, just as Woiwod and Hanski (1992) found for all species caught in the traps. Rice pest insects in Japan (K), moths of economic importance in Prague (D) and agricultural pests in Tasmania (L) are right in the middle of the other data sets that are all on insects not chosen for their economic importance. After 20 years the 19 pest species from Prague had 47.4% of SSDD. A further set of 110 interrupted time series on moths from the same trap, not selected for economic importance, were tested with the missing-data modification of the PBLR test (Dennis and Taper 1993) and here, after the equivalent of 20 years, the incidence of SSDD was 48.2%, almost exactly the value for the pest species given above. The present data do not suggest that pest status has any effect at all on the incidence of SSDD.

Data quality The best insect data used, those closest to actual population counts (Baars 1979), are Den Boer's data on Dutch carabid beetles. The corresponding time series have by far the lowest incidence of SSDD (Fig. 5h). At the other extreme, the highest percentages of SSDD are found in British aphids (G), the sloppiest data of them all. Aphids have not one but several generations per year, the suction traps used only monitor the winged forms of the local population and contain a possibly high percentage of long-distance migrants. The relation between the suction trap counts and the local population may thus be rather vague. The rest of the data are mostly obtained by lighttraps and mostly concern moths. The relation between light-trap catches and the local population probably varies a great deal, but may, on the average, be better than the aphid data and not as good as the carabid data. The frequencies of SSDD similarly are intermediate. For the bird data (Fig. 6), series B, D and E refer to breeding bird censuses, data that come close to actual population counts, while the other series have a much more tenuous relation with real population sizes, being birds ringed (A) or trapped (F) at a certain site or counts of birds in a wintering area (C). The "better" data, the actual population counts, show a clearly lower incidence of SSDD than the lower-quality time series. This result is not surprising. Dennis and Taper (1993) found that the power of the test increases with increasing sampling variance. The sloppier the data, the higher the incidence of SSDD expected at a given length of the time series. The reason is that the fundamental "observation" in density dependence tests is not a single population abundance at time t, but rather a transition from an abundance at time t to an abundance at time t + 1. Variability, whether natural or from sampling, has the effect of amplifying any return tendencies in the transitions. This effect of increasing variability on power is limited; there is a point of high variability at which power does begin to drop off (Dennis and Taper 1993).

The number of generations per year It is high time that we stand back from all these test results and contemplate what we are doing. First of all, all density dependence tests, including the PBLR test, strictly apply only to a first order Markovian time series, i.e., it is assumed that the number of individuals at time t (N0, given N t _ t , does not depend on abundances in previous years (Nt-2, Art-3 etc). The species need to be univoltine and semelparous for such an assumption to be strictly applicable. However, of the published studies very few (Den Boer and Reddingius 1989) adhere to this rule. Several studies cheerfully test for density dependence in species that are bi- or polyvoltine (e.g. Woiwod and Hanski 1992; Holyoak and Lawton 1992). Thus far, in this paper, we have done the same. Holyoak and Lawton (1992) found a higher incidence of SSDD among univoltine insects than in insects with other life histories.

588 Woiwod and Hanski (1992) expressed their surprise that in aphids the incidence of density dependence is very high, in spite of there being many generations per year. These authors assume that part of the explanation of the high percentages of SSDD in aphids might be that delayed density dependence in these circumstances may have been expressed as direct density dependence in their analyses. With p generations per year, the alternative density dependent model is

Table 1. Effect of the number of generations per year on the incidence of statisticallysignificantdensitydependence

= al+b I ~ Nt_l,j+Et_l,

Series length 17 years for the PanamanianHomoptera, 20 years for all others

In lj~lNt_ 1,j -

j=l

where Ntj is the abundance of thej ~ generation in year t. In a density dependence test applied to such data the "net reproduction" is the ratio of abundances between years and not between generations as one would like it to be. The "direct" density dependence Woiwod and Hanski refer to is an effect of the total number of winged aphids, summed over all generations in one year, on the same total number next year. Whatever the ecological mechanism causing the result is, it is hardly classical "direct" density dependence. Does the number of generations per year affect the incidence of SSDD? We selected some of the most seasonal insect species collected on Barro Colorado Island, and some of the most aseasonal species. Some of these were cockroaches, most were Homoptera. Details on the life history of these species are mostly unknown, but one may safely assume that many, if not all, of the highly seasonal species were univoltine and semelparous. The aseasonal species undoubtedly had several generations per year. Yet the incidence of density dependence in both categories was roughly the same (Table 1). The moths collected in three sites in the Czech Republic that are known to have just one generation per year were compared with the species that have two or more generations per year, using information from Novak (pers. comm.), Novak and Severa (1981), and Rejm~nek and Spitzer (1982). In all three cases the incidence of SSDD is somewhat lower among univoltine insects, but not significantly so. Holyoak and Lawton's finding (1992) of univoltine species having a higher incidence of SSDD was not duplicated. The statistical outcome of the test, at least in our examples, is not affected by the species having one or more generations per year, although a possible ecological interpretation depends very strongly on whether or not the species are univoltine. Another serious deviation from the basic Markovian rule is presented by long-lived organisms, such as many mammals, birds and also some insects, that live and reproduce in more than one year. Dennis and Taper (1993) applied the PBLR test to grizzly bears and elk and gave some arguments why such data can, in certain circumstances, still be used. In cases where it is difficult to maintain that abundances before year t-1 (Nt-2 etc) have no effect on the distribution of Nt, one would have

Spp. Panama, Homoptera

%Pdd< 0.05

Very seasonal 23 17.4

Univoltine 74 46.0 Prague, moths Brno, moths 72 26.4 Cesk6 Bud~ovice, 72 26.4 moths

Spp.

%Pdd< 0.05

Very aseasonal 46 13.0 Bi/Polyvoltine 28 57.1 37 32.4 38 28.9

to modify the test, as Dennis and Taper (1993) described, to include effects of those previous years properly. The percentages of SSDD obtained with data on birds here (Fig. 6) were very similar to those obtained with insect data (Fig. 5), suggesting again that statistically it did not make any difference whether or not species were longlived and reproduced in more than one year. The ecological interpretation, on the other hand, depends very much on this aspect of the species' life history.

Seasonal migrants Figure 7 shows, for the noctuid moth Autographa gamma, significant test results in seven out of eight British localities and two out of three Czech localities. The fluctuation patterns tended to be very zigzag and showed good "return tendencies", the quality the test looks for. However, this species cannot and does not survive the winter in Britain and has to immigrate from France each year. The individuals present at any one site in one year were unlikely to be the ancestors of the individuals present in the next year, unless they had a good homing instinct. Unless one accepts a series of assumptions, the numbers caught in one year, an index of the total number present summed over both generations, had nothing whatsoever to do with the local abundance of the species next year. One would have to assume that each year the immigrants came from the same area on the continent, that the moths in that entire area formed just one population, and that the migrants reaching Britain were a constant proportion of the individuals in that population, so that the immigrants totalled over all sites are a good index of the continental population size. Furthermore, the immigrants would have to be distributed in the same proportion over the individual sites for local data to reflect the total abundance. The average correlation between the time series of the eight British sites was 0.412 (range 0.120-0.835, n=28). We refrain from indicating statistical significance of correlation coefficients in these time series). Some sites were rather well correlated, others were not. This suggests, given also the fact that this species has two generations per year in Britain (Novfik and Severa 1981; Williams 1958), that local conditions largerly determined the development of the popu-

589 lation at each site. It seems likely that the direct effect of local population size in one year on that in the next is nil or nearly so. There was nevertheless a high incidence of SSDD, but ecologically a significant test result can hardly be interpreted as demonstrating regulation by densitydependent processes. In the Czech Republic the species is a mixture of local resident populations and a strong influx of seasonal immigrants. The test results there (Fig. 7I-K) are just as difficult to interpret ecologically as the British ones.

Rainfall data One last example is given in Fig. 8. The density-dependent model described the data far better than the densityindependent model (P<0.0001), but the data did not describe the fluctuation pattern of the abundance of some organism, but that of annual rainfall on Barro Colorado island. An ecological interpretation in terms of density-dependent regulation is hardly appropriate. For some meteorological reason annual rainfall at a particular site tended to be in the same range year after year, but the autocorrelation in the time series tended to be zero. Rainfall in one year had no predictive value for that in the next year. The data points in the time series can best be interpreted as random samples out of a gamma or a lognormal distribution. In such a time series values near the mode are much more likely than values near the tails of the distribution, so that extreme values are likely to be followed by values closer to the mode, which was interpreted by the test as a return tendency, producing a significant result. We repeated the test for a simulated series of data taken from such a distribution and, as expected, the result was highly significant. If Fig. 8 did describe the number of individuals in a population, an ecologist would have been impressed by the abundance returning to a certain range and an explanation for this fact would certainly be required. However, that explanation would not be density-dependent regulation. The key characteristic a statistical density-dependence test looks for in data is a return tendency. In the PBLR test the return tendency is expressed as a simple linear dependence of the average change in log-population abundance on population abundance: E In

] I N = nt = a+bnt

When b < 0, this average change is negative if n~> - a / b , and positive if n~< - a / b . The result is a tendency for log-abundance to change toward ln(-a/b). This return tendency, when combined with random shocks represented by Et, produces a statistical equilibrium. That statistical equilibrium is manifested through a stationary probability distribution of population abundance. A sequence of independent, identically distributed, (i.i.d.) random variables has a return tendency. Suppose Xo, )(1, X2,..., is such a sequence from a distribution with a mean of g. The expected value of the change, X~+t - XI, given X~= xt, is

E(Xt+ l--Xt

[ X t = xt) -~" E ( i Y t + l ) - - x

t :

~-- x t

This average change is negative or positive depending on whether xt is less than or greater than ~. It is not surprising that rainfall data or simulated i.i.d, observations pass density-dependence tests. Autocorrelation, or lack thereof, is not a necessary ingredient in statistical density dependence. I.i.d. observations are statistically density dependent in the sense of return tendency. Such processes are in statistical equilibrium and have a stationary probability distribution. The autocorrelation in the rainfall data is near zero, yet rainfall is essentially a process in statistical equilibrium. Statistical density dependence can have many types of autocorrelation. This is easy to illustrate with the model of Reddingius (1971) used by Pollard et al. (1987). That model is similar to the model of the PBLR test, except that the density-dependence term is proportional to In(N0 in stead of to Nt:

lnI~t l = a2+b21nNt+Et If b2 = - 1, the term ln(Nt) disappears from both sides, and ln(N1), ln(N2) .... are just a sequence of independent, identically distributed normal variables. Because ln(N~) is a simple linear autoregressive process, the correlation of ln(Nt+ 1) with In(NO can be negative, positive, or zero, depending on the value of b2 (see Pankratz 1983). The wide variety of autocorrelation patterns in this and higher-order models was discussed by Royama (1977, 1981). Conclusions

"Statistical density dependence" as defined presently by the alternate hypotheses of various density-dependence tests, is a return tendency in a stochastic process in the sense of conditional mean change. A "test of statistical density dependence" is a data-based choice between two models: a stochastic model without and a stochastic model with statistical density dependence. "Statistically significant density dependence" is a possible outcome of that choice. These statistical notions must be carefully distinguished from the scientific questions at issue in the density dependence debate. "Ecological density dependence", or "causal density dependence" (Royama 1978), or "regulation" (Reddingius and Den Boer 1989) is a return tendency in population abundance coupled with a scientifically defensible identification of a regulatory mechanism. With the outcome of SSDD in a data set, the investigator's work has not ended, but has just begun. Correlation, as the saying goes, is not cause. A fluctuation pattern in long-term data on population abundance cannot, in and by itself, reveal population regulation mechanisms. "Density-dependence" tests fit a null model and an alternative model to the data. The tests look for a "return tendency", a tendency for population sizes to return to some intermediate range of values, that results in a long-term statistical equilibrium of population size. Abundance fluctuations with a zigzag pattern tend to be described better by the alternative (density-

590

dependent) hypothesis, while more irregular patterns or patterns with trends tend to be described equally well or better by the null hypothesis (Figs. 1, 2, 7 and 8). If, for whatever reason, a fluctuation pattern tested does not show a clear trend, and especially if there is a zigzag pattern, test results tend to be significant statistically. The density-dependence model may describe the data better than the null model even if the patterns are based on data with a high sampling error, on time series that violate basic assumptions of the test, or on other data where an interpretation of the results in terms of densitydependent regulation are clearly absurd, such as was shown for a migrant insects and rainfall. The observed similarity between power curves of the test and the relation between frequency of SSDD and the length of the time series (Figs. 5, 6) was entirely consistent with the hypothesis of universal applicability of the density dependence model, suggesting that most persisting population time series will ultimately display SSDD if monitored long enough. The outcome of density-dependence tests require careful interpretation. A significant test result with appropriate data is consistent with the hypothesis of density dependent regulation. One may find that hypothesis the most likely explanation, especially if one knows more about the population than just the abundance over the years, but the test by itself does not constitute proof of such regulation. A time series as such does not contain information about the causes of the observed fluctuation pattern, so no statistical test is capable of finding those causes. Without any good additional information, any other mechanism that could produce the fluctuations observed is just as likely an interpretation as is regulation. We have given ample evidence that, for data where density-dependent regulation clearly is not the appropriate explanation, the density-dependence model often provided the best description of the fluctuation pattern. Whatever the explanations may be, there does not seem to be any reason why time series of population abundances of univoltine semelparous insects should be exempt from explanations other than regulation. Not being able to think of another such explanation in a particular case may produce a tendency to favor the regulation hypothesis, but one should realize that this hypothesis has not been proved. Nature may not feel herself limited by our lack of imagination. In view of this discussion, the term "density-dependence test" may be a misnomer. "Test of statistical density dependence" may be a better term (cf. Royama 1977), if statistical density dependence is defined as "a fluctuation pattern showing a clear return tendency." Acknowledgements.Weexpress our gratitude to all those mentioned in the text 'data' section who kindly gave us access to their data. The Panamanian insect monitoring program was supported by the Environmental Sciences program of the Smithsonian Institution. Miguel Estribi and Satumino Martlnez sorted the insects collected. References

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