Detection of density-dependent effects on caribou numbers from a series of census data

The m a i n objective of this paper is to review and discuss the applicabi l i ty of statistical procedures for the detect ion of density dependence based on a series of annual or mult i -annual censuses. Regression models for w h i c h the statistic value under the n u l l hypothesis of density independence is set a priori (slope = 0 or 1), generate spurious indications of density dependence. These tests are inappropriate because l o w sample sizes, high variance, and sampling error consistently bias the slope w h e n applied to a finite number of popula t ion estimates. Two distribution-free tests are reviewed for w h i c h the rejection region for the hypothesis of density independence is derived intr insical ly f r o m the data through a computer-assisted permutat ion process. T h e " r a n d o m i z a t i o n test" gives the best results as the presence of a p r o n o u n ced trend in the sequence of p o p u l a t i o n estimates does not affect test results. T h e other non-parametric test, the " p e r m u tat ion test", gives reliable results o n l y if the popula t ion fluctuates around a long-term e q u i l i b r i u m density. B o t h procedures are applied to three sets of data (Pukaskwa herd, A v a l o n herd, and a hypothet ical example) that represent quite divergent popula t ion trajectories over t ime. K e y w o r d s : density dependence, census, randomizat ion test, permutat ion test Rangifer, S p e c i a l Issue N o . 7: 3 6 4 5


Introduction
Population studies of ungulates generally aim at identifying causes of population fluctuations, and density-dependent effects that lead to population regulation (Messier 1991a, b).The first objective requires the investigation of processes that quantifiably influence the population rate-of-increase, hence revealing their limiting effect.The second objective is more specific; it addresses density dependence of dominant population processes, such as food competition, prédation, parasitism, and dispersal, to assess their regulatory effect on animal numbers (Fowler 1987;Messier 1989;Sinclair 1989).Density dependence may be revealed by analysis of changes in sources of mortality with population density (Sauer and Boyce 1983;Messier andCrête 1984,1985;Skogland 1985Skogland , 1986;;Freeland and Choquenot 1990).Alternatively, density dependence can be assessed at the population level using a series of census data (Vickery andNudds 1984; Gaston and Lawton 1987;Pollard etal. 1987;Reddingius and den Boer 1989).
The population dynamics of caribou or reindeer (Rangifer tarandus) has been reviewed by many authors in recent years (Bergerud 1980(Bergerud , 1983;;Leader-Williams 1980;Skogland 1985Skogland ,1986Skogland , 1990;;Messier etal. 1988).Food exploitation, predation, and winter snow accumulation have been identified as primary limiting factors, although most authors stressed that the respective impacts of these agents on population growth likely differ according to caribou densities, presence of alternate prey, geographic region, and environmental factors (e.g., Van Ballenberghe 1985;Bergerud and Ballard 1988).However, the density relationships of dominant causes of mortality, primarily those involving biotic interactions, are still poorly documented (Messier et al. 1988).This lack of information continues to restrict our capacity to empirically understand the demography of caribou in North America, particularly the mechanisms involved in the regulation of caribou numbers.
The principal objective of this paper is to review and discuss the applicability of statistical methods for the detection of density dependence based on a series of annual or multi-annual censuses.Such analyses would be warranted when the density relati-

Tests of density dependence
The hypotheses An animal population is said to be density independent if its growth rate is independent of the density of the population itself.Let Nt be the size of an animal population at a specific time in the annual cycle.
A simple model of density independence is: where Xx = In (7Vt ) (t = 1,2,n), is an independent normal random variable with mean zero and variance a 2 , and r denotes a "drift" factor.A population governed by [1] follows a random walk through time with and average drift of r; there is no tendency for the population to return to a long-term equilibrium value.
An animal population is said to be density dependent if its growth rate is correlated with its size.The correlation must be negative to create population stabilization.A model of density dependence frequently cited in the literature takes the following form: Figure 1.Test of density dependence proposed by Morris (1959Morris ( ,1963)), when applied to a series of censuses (Nt).It is hypothesised that ß = 1 for a density independent (DI) population, and ß < 1 for a density dependent (DD) population.
where J3 is a constant reflecting the degree of density dependence, and r, Xt, and q conform to model [1).
Here, the population growth rate depends upon Xt when JJ # 1.If J3 = l,thenmodel2 converges to model 1.A population governed by [2], when (3 < 1, will tend to fluctuate around an expected value of e^-P) (Fig. 1), which can be taken as the ecological carrying capacity (KCC, Macnab 1985).The variable r represents the population rate-of-increase assuming no density dependent effects, or rmax as defined by Caughley (1977: 53).

Regression models
Regression analysis has been used by many authors to analyse population data (review in Ito 1972, Slade 1977).A first approach, proposed by Morris (1959Morris ( , 1963)), consisted of regressing ln(Nt+1) on ln(Nt) for a series of annual censuses (Fig. 1).Densityindependent populations should generate a slope (B) of one, and a y-intercept equal to r (the exponential rate-of-increase).Density dependence should be indicated by |3 < 1.Here, (3 is estimated by b, the slope of the regression line computed by standard least squares procedure (Sokal andRohlf 1981: 468).Maelzer (1970), St. Amant (1970), andIto (1972) 1977).Varley and Gradwell (1960) presented another method for analysing serial census data which was expanded later by Krebs (1970), Watson (1970), Podoler andRogers (1975), andManly (1977).This method consists of expressing each source of mortality (frequently called "submortality") as the difference of log10 of population size before and after the submortality has acted.Each submortality, expressed here in ^-values for a number of years, can then be plotted against log population size before its action to assess the degree of density dependence (Podoler and Rogers 1975).For age-structured populations, the analysis can be restricted to a given age class (Sinclair 1973;Clutton-Brock et al. 1987;Albon et al. 1987) or to animals of all ages (Clutton-Brock etal. 1985).However, a problem arises when the study animal has a complex life cycle.In such cases it is often impossible to sequence mortality agents through time because they may act simultaneously (a notable difference compared to many invertebrate species; Varley et al. 1973).Nonetheless, total mortality (&total) from t to t+1 can be plottet against log(Nt) to reveal overall density dependence (Fig. 2; Ito 1972).Note that a regression expressing the rate of population growth against log population size is mathematically equivalent to a regression involving &total> assuming that the loss of breeding potential is treated as a submortality (Kuno 1971;Clutton-Brock etal. 1985;Messier 1991a).
There is a serious empirical difficulty in applying the approach of Varley and Gradwell.The authors assumed that a density independent population should be indicated by a slope of zero when ktou[ values are plotted against log population sif e.Like the model of Morris described above, the slope is biased by data that contain low sample sizes and high variance (Ito 1972), thus providing erroneous evidence for density dependence.The fact that one cannot derive the lvalue for the null hypothesis greatly hampers statistical testing for density dependence using key factor analysis.
Other regression statistics have been used to determine density dependence.These include (1) the slope of the principal axis, (2) the slope of the standard (reduced) major axis, (3) a comparison of the slope of a double regression [ln(Nt+i) on ln(Nt), and ln(Nt) on ln(Nt+1)), and ( 4) the coefficient of first order serial correlation (see Varley and Gradwell 1963;Varley etal. 1973;Bulmer 1975;Slade 1977).
However, recent Monte Carlo simulations have demonstrated that these statistics remain largely inappropriate to reveal density dependence (Slade 1977;Vickery and Nudds 1984;Pollard et al. 1987;Reddingius and den Boer 1989).

Non-parametric models
Pollard etal.(1987) suggested a simple, distributionfree approach for the detection of density dependence based on a series of annual censuses.The method uses the correlation coefficient, or the slope of the regression line, between the observed rate of population growth and population size (Fig. 3).A distinct feature of the proposed test is that the correlation coefficient under H0 (i.e., model 1) is derived intrinsically from the data by a randomization process.
The rationale of the randomization test of Pollard Rangifer, Special Issue No. 7, 1991

Applications
In this section, I apply the randomization test and the permutation test to three sets of data, two obtained from the literature and one hypothetical.Field data were from the Avalon caribou herd in Newfoundland (Bergerud 1971;Bergerud et al. 1983;Mercer et al. 1985;S. Mahoney, pers. comm.) and from the Pukaskwa herd in Ontario (Bergerud 1989; A. T. Bergerud, pers. comm.).The fictitious data set was generated by a computer simulation on the basis of model ( 2) where «Q = 10, r = 0.4, 6 = 0.9, and e with a mean of zero and a of 0.05.Here, my intention is to analyse extremes of population trajectories (Fig. 4), from fluctuations within density bounds (Pukaskwa herd) to unlimited growth (Avalon herd).The hypothetical example illustrates a case of an expanding population for which the respective role of density dependence and environmental vaga- ry on population growth requires statistical assessment (Fig. 4).Population estimates are summarized in Appendix I.
One minor modification to the randomization test was required to accommodate multiyear censuses.In these cases dx relative to a multiyear interval was divided by the number of years in that interval to calculate correctly the exponential rate-ofincrease.Such modification would affect the variance of dx values, but not the underlying relationship between dx and Xx As the randomization test is a distribution-free statistical analysis (Pollard et al. 1987), any differences in the variance among dt values, when annual and multiannual censuses are treated simultaneously, should not affect the test results.Note, however, that the original dt values were used to generate the permuted sequences of Xt.

Pukaskwa herd
The Pukaskwa herd ranged from 12 to 31 animals during the period 1972-1991 (Fig. 4).The density independent model was rejected at P < 0.05 by both the randomization and the permutation test (Table 1).It is, therefore, quite unlikely that this series of population censuses could have originated simply from rando m fluctuations.The evidence for density dependence is also reflected by the decline of dx with Xx (Fig. 5).

Avalon herd
The Avalon herd increased from 71 to 5782 animals during the period 1956-1990 (Fig. 4).Contrary to the previous example, there was a marked trend in the observed data set, typical of a population experiencing unlimited growth.The probability of rejecting the null hypothesis of density independence varied from 0.09 to 0.14 for the randomization test (Table 1).Consequently, the herd may have experienced reduced growth in recent years, but the change was not strongly expressed statistically.
The permutation test gave quite a different result.
None of the permuted series of dx produced a range of Xx values more extreme than that observed in the original sequences of Xx.In fact, this example points to a major weakness of the permutation test; the test loses its power when the population shows a pronounced trend over time.Indeed, when all dx values are positive, as it is in this example, permutation of dx values has no effect on the L statistic.

Example herd
The hypothetical example was constructed arbitrarily to mimic a herd recovering from a catastrophic  1).
The permutation test, however, is again plagued with low power.The probability to reject HQ is only 0.77 in spite of a clear trend in declining population growth with population size (Fig. 5).This reinforces the previous statement that the permutation test is largely ineffective when the population undergoes substantial growth without fluctuations around an equilibrium value.Gradwell's method), or because of low power when the population undergoes substantial growth or decline (Bulmer's test).Many authors have shown the inefficiency of these procedures (Maelzer 1970;St. Amant 1970;Kuno 1971;Ito 1972;Slade 1977;Royama 1977;Vickery andNudds 1984;Gaston and Lawton 1987;Pollard et al. 1987).
The development of distribution-free, nonparametric tests by Vickery and Nudds (1984, not reviewed here), Pollard et al. (1987), and Reddingius and den Boer (1989) represent important contributions to population ecology.In these procedures, the rejection region for the hypothesis of density independence is defined from the data through a computer-assisted randomization process.However, these approaches are not without problems.For example, Pollard etal. (1987)

Limitations of the randomization test
There are a number of factors that should be considered while interpreting results from the randomization test.I should remark, however, that none of them seem to create systematic biases that would invalidate the procedure.

Serial correlation in e
An assumption of the randomization test is that ct be a sequence of independent normally distributed variâtes, representing the stochasticity element of the system.However, e.t may be subject to serial dependence if, for example, where 6( < 1) is the serial correlation constant and iut a sequence of independent normally distributed variâtes.Maelzer (1970) showed that any serial cor-

Time delay
Often, the rate of population growth at time t is dependent not only on current population size (Xt), but also on some previous population size (Xt_i,Xx_2,etc.).In these circumstances, fluctuations in animal numbers are not a realization of a piece of first-order Markov chain (Reddingius 1971).The implications of population models that follow a second or upper order process, on the sensitivity of the randomization test, have not been evaluated.Second order process, where dt is a function of Xt and Xt_i, often exhibit cyclic fluctuations in numbers (Royama 1977).This implies that plots of dt on Xt will be characterised by ellipsoidal (counterclockwise) patterns with various degrees of compression towards the major axis (May et al. 197'4).Thus, we can safely generalise that the presence of a time delay would restrict the applications of the randomization test because of a poorly defined relationship between dt and Xt (Ito 1972;Royama 1977;Hanski and Woiwod 1991).
Table 1.Results of the randomization and the permutation test using three statistics ( 7 (1987) developed their statistical procedure with the assumption that the relationship between dx and Xt is linear.However, I suspect, without presenting a proof, that the randomization test can be applied on the basis of a curvilinear model when inspection of residuals along the regression line reveals a poor fit (e.g., Pukaskwa herd, Fig. 5).
As for the linear model, ryx of the curvilinear regression model (more appropriately R&f) calculated from the original sequence of Xt, can be compared with R values obtained from permuted sequences of dv Here, the hypothesis of density independence should imply that all correlation constants (Bj, B2, etc.) equal zero, assuming infinite sample sizes.For example, applying a quadratic model to the Pukaskwa data (Fig. 5) appreciably improves the fit (R<ix -0.82, compared to 0.59 for the linear model), with a rejection of H0 at P < 0.01.

Final remarks
Density dependence is achieved by a complex of factors whose collective action creates bounded population fluctuations (Berryman etal. 1987;Berryman 1991;DeAngelis and Waterhouse 1987).The question is not whether density dependence exists, for without it, the recognised persistence of most natural populations would be inexplicable (Royama 1977).Rather, the goal is to assess (1) in what range of densities do feedback mechanisms operate, and (2) what are the population processes involved?Any test of density dependence based on a series of census data cannot address the later question.For example, the abrupt decline in the rate-of-increase of the Pukaskwa herd at elevated densities (Fig. 5) may be due to food competition, emigration, or some form of interaction between the two factors.An appropriate test, however, would differentiate between a period of unlimited growth and a period during which mechanisms of population regulation are instrumental in stabilizing numbers.
Unlimited growth, often associated with a given range of densities, does not imply that prevailingpopulation processes are largely density-uninfluenced.
It is important to stress that two density-influenced factors may have antagonistic actions on population growth.For example, at Isle Royale, wolf preda¬ tion and food competition exert opposite influences on moose during periods of moderate density (Messier 1991a).The net effect of two population processes with opposite actions is to make the rate-ofincrease largely insensitive to changes in density (i.e., weak density dependence), a type of interaction associated with the unnecessary concept of "density-vague" population regulation (Strong 1984(Strong , 1986)).
Some ecologists object to the notion of "equilibrium" that underlies most tests of density dependence (Wolda 1989).A long-term equilibrium density is simply a mathematical abstraction illustrating the fact that a population trajectory over time would tend to converge toward that equilibrium (Berryman 1991).The modern view of population dynamics recognises the lability of equilibrium points due to stochastic effects, and the fact that equilibria can be unstable or multiple (DeAngelis and Waterhouse 1987;Berryman et al. 1987;Sinclair 1989).In that perspective, one of the basic questions in studies of population dynamics is not simply to determine whether the density of animals is regulated or not, but to assess the relative importance of densitydependent and density-independent processes in changes of population size over time (Schaefer and Messier 1991).
onship of individual causes of mortality cannot be assessed due to limited information.Yet, the detection of overall density dependence would imply that one or a number of mortality agents reduce the po-pulation rate-of-increase at elevated densities, and vice versa.Mechanisms of population regulation are exhibited only when density dependence actually operates, not during periods of unlimited growth.Therefore, such tests may suggest the most appropriate time period to initiate a demographic study aimed at revealing feedback mechanisms in caribou population dynamics.
[1] and [2] essentially form the null and the alternative hypothesis to test for the presence of density dependence at the population level.Specifically, we are asking the following question: does the sum of negative feedback mechanisms affecting population growth outweigh the sum of the positive feedback mechansisms, thus creating population regulation (Berryman et al. 1987; Berryman 1991)?

Figure 3 .
Figure 3. Illustration of the randomization test proposed by Pollard et al. (1987) where the annual exponential rate-of-increase is plotted against log population size (Nt).Density dependence (DD) should be indicated by a slope signifincantly lower than the slope associated with a density independent (DI) process.The density distribution of slopes under the hypothesis of density independence is derived from random permutations of the annual estimates of the population rate-of-increase.

Figure 4 .
Figure 4. Fluctuations over time of caribou numbers for the Pukaskwa herd in Ontario, the Avalon herd in Newfoundland, and the computerconstructed example.
Fig' ;ure 5. Relationships between the exponential rate-ofincrease and log population size for the three populations illustrated in Fig. 4.
hindered by the lack of a suitable statistical procedure.All proposed regression models for which the statistic values under HQ are set a priori (e.g., b = 0 or 1) lead to incorrect test results because of biases in the estimator of |3(Morris's method, Varley and Until very recently, assessment of density dependence from a series of population estimates has been Measurement errors Errors of measurement are known to bias the slope of the regression line between dt and Xt Maelzer Rangif er, Special Issue No. 7, 1991 1970; Kuno 1971; Royama 1977).However, such a departure will be present in the slope calculated from the original sequence of Xt, and in the slopes obtained from permuted series of Xt.As the randomization test is based on the comparison of b § with the distribution of lvalues computed by permutation, the test results should be relatively insensitive to sampling error.In fact, if we assume that errors of measurement are comparable among Xt values, then the overall variance of Xt (e) amounts simply to the sum of the true variance and the sampling variance.As shown by Pollard etal.(1987), the randomization test is not affected by differing values of e, except that high overall variance makes the detection of density dependence more difficult.
relation in et, and by extension in dt, biases the regression slope bbecause changes mXt no longer follow a first-order autocorrelation model, but instead follow a second-order autocorrelation model (Royama 1977; Reddingius 1990).However, the deviation of b from the true slope B is a function of the sign of the serial correlation parameter (8); negative values of Bdecrease ^whereas positive values increase b.A carryover effect of environmental influences, if occurring at all in ungulates (Picton 1984; Messier 1991a), should be associated with positive values of 9. Thus, following a severe winter the rate-ofincrease one year later should be somewhat lower than it would have been if the winter had been average.The net result of serial correlation among et should, therefore, positively bias b calculated from the original sequence of Xt, but not b values computed from permuted sequences of Xt (because here the serial correlation among et vanishes).The above argument suggests that the randomization test would underrate the intensity of density dependence if serial correlation occurs.However, a rejection of HQ would reinforce the conclusion that density dependent mechanisms actually operate.
), when applied to caribou census data of Fig. 4. Twere(l) the product-moment correlation coefficient (r^) and (2) the slope (b) of the regression line between population size (loge-values, Xt) and the exponential rates of growth (dt = X[+1 -Xt), and (3) the logarithmic range (L) of Xt values (Xm!lx -Xmm).Results include the observed statistic values (7* = r0, b0, and LQ) calculated from the original sequence of Xt values, and the mean statistic values of 500 permutated series of dv The estimate of .Pis also given, i.e., the chance to refute H0 (density independence hypothesis) while H0 is actually true.