Testing for density dependence allowing for weather effects

A test for density dependence in time-series data allowing for weather effects is presented. The test is based on a discrete time autoregressive model for changes in population density with a covariate for the effects of weather. The distribution of the test statistic on the null hypothesis of density independence is obtained by parametric bootstrapping. A computer simulation exercise is used to demonstrate the gain in statistical power by allowing for weather effects. Application of the method to time-series data on three species of butterflies and two species of songbirds showed stronger evidence of density dependence than two standard tests. Received: 4 October 1996 / Accepted: 4 August 1997     

Tests for density dependence

Several statistical tests for density dependence have been proposed in the literature, and so in any practical case the question poses itself which one of these tests to choose. This paper offers a few remarks additional to those made by Fox and Ridsill-Smith (1995) and others. Parametric statistical tested are based on a fully specified mathematical model. Examples of such tests are Bulmer's (1975) first test, and the test of Dennis and Taper (1994). Distribution-free tests are based on far less stringent assumptions. An example of such a test is the one proposed by Pollard et al. (1987). The choice between parametric tests can best be made by considering which one of the underlying mathematical models ist most plausible. If all models are almost equally plausible, considerations of computational requirement and ease of application may be important. Strong doubts concerning the plausibility of mathematical models may lead one to prefer a distribution-free test. An important feature of any test is its power, i.e. the probability of its rejecting the null hypothesis when this hypothesis is not true. Other things being equal, tests are preferable when they have superior powers. But power of a test depends on the true state of nature, and the only way to study power quantitatively is by assuming some mathematical model as approximately representing this true state. As any mathematical model can at best only be an approximation to the situation in nature, a mathematical model and the statistical tests based on it should be robust against small deviations from model assumptions. Solow (1990) showed that Bulmer's test is not robust with respect to the assumption that the residuals in the underlying autoregression model be stochastically independent. Contrary to what was suggested by Fox and Ridsill-Smith (1995), who misinterpreted some statements in Reddingius (1990), the present author thinks this is a serious shortcoming of this test since an ecologist cannot assume a priori that important density-independent ecological factors are not somehow serially correlated. Moreover, he is rather sceptical about the usefulness of statistical tests for density dependence. They have contributed more to misunderstandings than to a significant increase in ecological insight. In any case, statistical tests are designed to test hypotheses that are stated before data are collected, and the question which test to use also has to be answered before the data have been collected. Designing and using statistical tests a posteriori to detect things in data mainly leads to confusion and controversy.     

Models for testing

Summary This is a comment on a note by Solow (1990). It is shown that Solow's simulation results indicate that Bulmer's test for density dependence is non-robust to a particular kind of second-order Markovity that might well be overlooked by an ecologist. It is suggested that Solow's claim that Bulmer's test is insensitive is not wholly justified. Some scepticism concerning the applicability of statistical testing theory to animal population data is expressed.Communication no. 410 of the Biological Station, Wijster     

Spatial scale and the detection of density dependence in spruce budworm outbreaks in eastern North America

Using two tests for direct density dependence and standard techniques of time series analysis, we identified density dependence in defoliation time series of the spruce budworm across its outbreak range in eastern North America over the years 1945–1988. We carried out analyses for the entire region and for grid cells of defoliation maps at five spatial scales created by aggregating the smallest grid cells. The rate of detection of direct density dependence, as assessed by two previously published methods, decreased with increasing spatial scale. Using both methods, density dependence was detected more frequently at the periphery of the outbreak range, where defoliation rate was lower. This result suggested that density-dependent regulation may be stronger in those areas. The first order autoregressive process was the basic model for defoliation dynamics overall and the most common model across spatial scales. Second-order processes were encountered much less frequently, and those commonly identified as resulting from delayed density dependence generally occurred across spatial scales at a rate expected by chance alone. Our results were similar to those of other published studies, which have found the detection of density dependence to decrease at larger spatial scales. The results also reinforced the importance of considering spatial scale when diagnosing population processes using time series of abundance for single species. Received: 26 December 1999 / Accepted: 17 March 2000     

Detection of density dependence from annual censuses of bracken-feeding insects

Summary A variety of techniques were used to test for density dependence in 32 time series from bracken-feeding insects. Seventeen taxa (primarily species, but including some pooled data from two or more closely related species whose larvae could not be distinguished in frond surveys) occurred on an open site; a woodland site held 15 taxa. For series of 12 years, collected on the open habitat, direct density dependence was detected by one or more of the techniques in 10 (58.8%) of 17 taxa, compared to only 5 (33.3%) of 15 taxa with time series of 8 years in length from the woodland habitat. Delayed density dependence was detected in 6 cases for the open site and in no cases at the woodland site. Either direct or delayed density dependence was found in 13 (76.5%) of 17 taxa for the open site and 13 (86.7%) of the 15 taxa which occurred on both sites. Although these results suggest a high frequency of density dependence in the species making up the bracken insect community, results from individual tests were extremely variable. Density dependence was detected least often by Vickery and Nudds' (1984) test, and most frequently by Varley and Gradwell's (1960) test, although the latter is prone to high rates of detecting spurious density dependence. Direct density dependence was detected most frequently in taxa that were univoltine and did not have delayed diapause, i.e. in those taxa whose life-histories conform most closely to the assumptions of the models underlying the analyses. Delayed density dependence occurred more frequently in species with more complex life-histories at the open site (taxa that were either bivoltine or multivoltine, or had delayed diapause). The results are consistent with the view that that the bracken herbivore assemblage consists of populations which are independently regulated by density dependent processes, although the present analyses suggest that we cannot rely on these tests to firmly show whether density dependence is present or not in an individual time series of the lengths considered here.     

Seeing the trees for the wood: random walks or bounded fluctuations of population size?

Summary It is often claimed that the fluctuation of numbers in field populations is fundamentally different from random walks of densities, in that population size is kept between certain positive limits. To test this hypothesis patterns of fluctuation in field populations were compared with random walks of density of about the same duration. It was found that the boundaries (Log-Range) between which numbers fluctuate in field populations increase with time to about the same extent as in comparable random walks of density. Moreover, deviations of the trend of numbers over years (Average lnR) from zero trend in populations of 62 (carabid) species were just those expected for simulated random walk runs, with the median value of Var(lnR), and different values for mean population size that cover the possible range of survival times for these species. This means that the null hypothesis that in the field numbers would fluctuate as random walks of densities could not be rejected. Although it is not very probable that field populations fluctuate exactly like random walks of densities, random walk models appear to mimic the fluctuation patterns of field populations sufficiently closely to explain what happens in nature, and to deny the need for regulation. The same conclusion was drawn in earlier studies where statistical tests were applied to fluctuation patterns of field populations (Den Boer and Reddingius 1989; Den Boer 1990a). Random walks of densities do not exclude the possibility that local populations can persist for some centuries.Communication No. 435 of the Biological Station Wijster     

Modelling the movement of a soil insect

We use a linear autoregressive model to describe the movement of a soil-living insect, Protaphorura armata (Collembola). Models of this kind can be viewed as extensions of a random walk, but unlike a correlated random walk, in which the speed and turning angles are independent, our model identifies and expresses the correlations between the turning angles and a variable speed. Our model uses data in x- and y-coordinates rather than in polar coordinates, which is useful for situations in which the resolution of the observations is limited. The movement of the insect was characterized by (i) looping behaviour due to autocorrelation and cross correlation in the velocity process and (ii) occurrence of periods of inactivity, which we describe with a Poisson random effects model. We also introduce obstacles to the environment to add structural heterogeneity to the movement process. We compare aspects such as loop shape, inter-loop time, holding angles at obstacles, net squared displacement, number, and duration of inactive periods between observed and predicted movement. The comparison demonstrates that our approach is relevant as a starting-point to predict behaviourally complex moving, e.g. systematic searching, in a heterogeneous landscape.     

A test of statistical techniques for detecting density dependence in sequential censuses of animal populations

Summary Principal and reduced major axes, and Bulmer's (1975) tests have been suggested as methods for detecting the presence of density dependence in a series of population censuses that are unsuitable for analysis by alternative means e.g. by k-factor analysis. These alternative methods are tested using census data, some of which are previously unpublished, from natural populations known from independent evidence to be subject to density dependent processes. All the methods fail to detect density dependence reliably, irrespective of sample size and the dynamics of the population. We conclude that none of the methods tested is sufficiently reliable to be useful as a test of density dependence in sequential censues of animal populations.     

How can we model selectively neutral density dependence in evolutionary games

The problem of density dependence appears in all approaches to the modelling of population dynamics. It is pertinent to classic models (i.e., Lotka-Volterra's), and also population genetics and game theoretical models related to the replicator dynamics. There is no density dependence in the classic formulation of replicator dynamics, which means that population size may grow to infinity. Therefore the question arises: How is unlimited population growth suppressed in frequency-dependent models? Two categories of solutions can be found in the literature. In the first, replicator dynamics is independent of background fitness. In the second type of solution, a multiplicative suppression coefficient is used, as in a logistic equation. Both approaches have disadvantages. The first one is incompatible with the methods of life history theory and basic probabilistic intuitions. The logistic type of suppression of per capita growth rate stops trajectories of selection when population size reaches the maximal value (carrying capacity); hence this method does not satisfy selective neutrality. To overcome these difficulties, we must explicitly consider turn-over of individuals dependent on mortality rate. This new approach leads to two interesting predictions. First, the equilibrium value of population size is lower than carrying capacity and depends on the mortality rate. Second, although the phase portrait of selection trajectories is the same as in density-independent replicator dynamics, pace of selection slows down when population size approaches equilibrium, and then remains constant and dependent on the rate of turn-over of individuals.     

Elevational variation in density dependence in a subtropical forest

Density‐dependent mortality has been recognized as an important mechanism that underpins tree species diversity, especially in tropical forests. However, few studies have attempted to explore how density dependence varies with spatial scale and even fewer have attempted to identify why there is scale‐dependent differentiation. In this study, we explore the elevational variation in density dependence. Three 1‐ha permanent plots were established at low and high elevations in the Heishiding subtropical forest, southern China. Using data from 1200 1 m² seedling quadrats, comprising of 200 1 m² quadrats located in each 1‐ha plot, we examined the variation in density dependence between elevations using a generalized linear mixed model with crossed random effects. A greenhouse experiment also investigated the potential effects of the soil biota on density‐dependent differentiation. Our results demonstrated that density‐dependent seedling mortality can vary between elevations in subtropical forests. Species found at a lower elevation suffered stronger negative density dependence than those found at a higher elevation. The greenhouse experiment indicated that two species that commonly occur at both elevations suffered more from soilborne pathogens during seed germination and seedling growth when they grew at the lower elevation, which implied that soil pathogens may play a crucial role in density‐dependent spatial variation.     

Effects of delay,truncations and density dependence in reproduction schedules on stability of nonlinear Leslie matrix models

Understanding effects of hypotheses about reproductive influences, reproductive schedules and the model mechanisms that lead to a loss of stability in a structured model population might provide information about the dynamics of natural population. To demonstrate characteristics of a discrete time, nonlinear, age structured population model, the transition from stability to instability is investigated. Questions about the stability, oscillations and delay processes within the model framework are posed. The relevant processes include delay of reproduction and truncation of lifetime, reproductive classes, and density dependent effects. We find that the effects of delaying reproduction is not stabilizing, but that the reproductive delay is a mechanism that acts to simplify the system dynamics. Density dependence in the reproduction schedule tends to lead to oscillations of large period and towards more unstable dynamics. The methods allow us to establish a conjecture of Levin and Goodyear about the form of the stability in discrete Leslie matrix models.This research was supported in part by the US Environmental Protection Agency under cooperation agreement CR-816081     

Parasitoids of the goldenrod gall moth: effects of scale on spatial density dependence

Summary A parasitoid assemblage consisting of four hymenopteran species caused larval and pupal mortality of the host Gnorimoschema gallaeosolidaginis. In the absence of discrete host patches, the relationship between densities of hosts and parasitized hosts was examined on scales of 0.25, 1, 4, 16 and 25 m² within a 400 m² plot and between ten 16 m² plots within a 1·3 km² area. Within the 400 m² plot, positive density dependence was observed on scales of 1, 4 and 16 m². Rates of parasitism were found to be density independent across the ten 16 m² plots. The nature of parasitoid and host spatial patterns were examined at the 400 m² plot by using Goodall's (1974) paired-quadrat variance function, and by fitting the observed distribution of galls and parasitized galls to the Poisson and negative binomial on three scales. Postive density dependence at the 400 m² plot occurred in the context of a host density gradient and may indicate aggregation of search time by parasitoids. However, significant responses on scales of 4 and 16 m² may only reflect the sum of parasitoid responses on lower scales because of this gradient. Predominance of significant responses on a 1 m² scale may be related to the high degree of variation in host density, and limits to active searching range of parasitoids.     

Parameter estimation in a generalized discrete-time model of density dependence

Flexible discrete-time per-capita-growth-rate models accommodating a variety of density-dependent relationships offer parsimonious explanations for the variation of population abundance through time. However, the accuracy of standard approaches to parameter estimation and confidence interval construction for such models has not been explored in a generalized setting or with consideration of limited sample sizes typical for ecology. Here, we use simulated data to quantify the relative effects of sample size, population perturbations, and environmental stochasticity on statistical inference. We focus on the key parameters that inform population dynamic predictions in a generalized Beverton–Holt model. We find that reliable parameter estimation requires data spanning ranges where both low and high density dependence act. However, the asymptotic distribution of the likelihood ratio test statistic can be fairly accurate for constructing confidence regions even when point estimation is poor. Consideration of the joint profile likelihood surface is shown to be useful for assessing reliability of point estimates and dynamical population predictions. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Density dependence in an island population of silvereyes

The population of silvereyes Zosterops lateralis chlorocephalus , on Heron Island, Great Barrier Reef has been monitored accurately since 1965. Between 1979 and 1993, the breeding success of all birds was determined by monitoring nests. The population fluctuated between 225 and 483 individuals. Four cyclones led to substantial mortality. As this data set is long-term, has little observation error, and is from an effectively closed population, it provides an unusual opportunity to examine density dependence in reproduction or mortality. Using a stochastic logistic model, we found clear evidence of density dependence in adult population size. Logistic regression suggested that fledgling survival decreased with the numbers of birds attempting to breed. There was also some suggestion that adult survival might be density dependent. The fitted stochastic logistic model predicts negligible risks of extinction for this population, in contrast to the predictions of a published population viability analysis. Whilst our statistical model including density dependence may provide better predictions of the "usual" behaviour of a population than a population viability analysis, we suggest that caution should be exercised when statistically fitted models are used to predict the behaviour of the population at extremes, such as near extinction.     

Stochastic density dependence in population size of a benthic clonal invertebrate: the regulating role of fission

The influence of environmental variation on the demography of clonal organisms has been poorly studied. I utilise a matrix model of the population dynamics of the intertidal zoanthid Palythoa caesia to examine how density dependence and temporal variation in demographic rates interact in regulating population size. The model produces realistic simulations of population size, with erratic fluctuations between soft lower and upper boundaries of approximately 55 and 90% cover. Cover never exceeds the maximum possible of 100%, and the population never goes to extinction. A sensitivity analysis indicates that the model’s behaviour is driven by density dependence in the fission of large colonies to produce intermediate sized colonies. Importantly, there is no density-dependent mortality in the model, and density dependence in recruitment, while present, is unimportant. Thus it appears that the main demographic processes which are considered to regulate population size in aclonal organisms may not be important for clonal species. Received: 18 August 1999 / Accepted: 29 October 1999     

The effect of visual density on the fecundity of the guppy,Poecilia reticulata

Synopsis This paper examines the effect of visual density on the brood size of female guppies,Poecilia reticulata. Previous laboratory studies of guppies showed that the fecundity of females decreases as density increases. In the present study, the effect of visual density was examined under conditions where detrimental factors such as interaction of individuals, waste substances, and dissolved oxygen were constant. The brood size of females at visual densities of 4 or more individuals was significantly larger than at zero visual density. There was a significant positive correlation between brood size and visual density. Sex ratio did not significantly affect brood size. Guppies can recognize surrounding densities and change their own fecundity in response to changes in density.