Environmental noise and the population renewal process

Transcription

1 Journal of Negative Results Ecology & Evolutionary Biology vol. 3: 1 9 Helsinki, 2 October 2006 ISSN Environmental noise and the population renewal process Veijo Kaitala* & Esa Ranta Integrative Ecology Unit, Ecology and Evolutionary Biology, Department of Biological and Environmental Sciences, P.O. Box 65 (Viikinkaari 1), FI University of Helsinki, Finland (corresponding author s veijo.kaitala@helsinki.fi) Received 31 March 2005, accepted 14 June 2006 In our scenario density dependent population renewal is disturbed by an external modulator yielding a noise-modulated population time series. The task is to assess the presence of the noise in the population signal. Often in the real world very little is known on the population renewal process, or the noise influencing the process, only their joint outcome is known in terms of population time series (blurred with sampling errors). In this research, climate indices, such as the North Atlantic Oscillation, are taken as proxies of the external modulator. In our reply to the criticism raised by Scott and Grant (2004) we first review achievements of visibility research and then place their critique against the findings in visibility research. The agenda of visibility research is to seek the conditions (population processes, noise structure that are components less well known in real life) under which the presence of an external modulator is visible in the population time series. It turns out that the external noise is visible in the population dynamics only when the underlying deterministic kernel of the dynamics is locally stable. Four different ways of scoring the noise visibility give largely matching results under temporally structured noise. Introduction Understanding the causes and consequences of population fluctuations is the major concern of contemporary population ecology. The debate has been intensive on the relative merits of density-dependent and density-independent population regulation (Andrewartha & Birch 1954, Lack 1954, Murray 1993, Turchin 2003). The concern of climate (or weather) influencing population dynamics dates back to the climate control theory, CCT (Bodenheimer 1938). Motivated especially by the research on insect populations and game animal dynamics, the CCT advocates that population fluctuations of many species can be explained by weather/climatic factors (for recent references, see, e.g., Stenseth Editor in charge of this manuscript: Mike Fowler et al. 1999, Saether et al. 2000, Holmgren et al. 2001, Jaksic 2001, Ottersen et al 2001, Post et al. 2001, Blenckner & Hillebrand 2002, Jonzén et al. 2002b, Barbraud & Weimerskirch 2003). The most fabulous example being the proposed causal link between sunspot cycle-derived weather conditions and the dynamics of the snowshoe hare and the Canada lynx (Elton 1924, Sinclair et al. 1993, but see Ranta et al for a different view). In general, in the frame of the CCT, studies concerning the link between weather and population fluctuations are based on observed correlations between some weather variables or climate indices and long-term population data. So is the case also in the recent incarnation of this research theme with the anticipated global climatic change (examples listed by Ranta

2 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 2 et al. 2000, 2006, Stenseth et al. 2002). The recent search for a link between climate indices, such as the North Atlantic Oscillation, the North Pacific Index and the El Ninõ Southern Oscillation, is dominated by scoring correlations between various population time series and some a priori selected climate index (for references, see Ranta et al. 2000). Once a correlation, significant in statistical terms, is found the results are reported. The general bias to publish only statistically significant results makes one wonder in how many cases detecting a correlation between a population time series and a climate index failed, hence resulting in no publication. In this paper we determine the visibility of the noise simply by correlation. We acknowledge, however, that slightly different approaches exist to tackle the problem of finding the skeleton of the population regulation. For example, in many ecological time series analyses the weather signal is used as a covariate, and is excluded or included according to some model selection methods, such as stepwise regression or the Akaike s Information Criterion (e.g., Stenseth et al. 1999, Sæther et al. 2000, Post et al. 2001, Post & Forchhammer 2001, Jonzén et al. 2002b, Barbraud & Weimerskirch 2003). We do not consider these approaches here, because we specifically address using correlation as the detection method. Furthermore, using the alternative methods for finding the skeleton of the population regulation leads ultimately to the same problems related to spurious correlations as discussed in this paper. We skip, however, the topic of proving this here. Here we shall recapitulate visibility research addressing the question: Under what circumstances is the signal of external perturbation visible in noise-modulated population data? We then contrast the criticism raised by Scott and Grant (2004) with the findings of the visibility research. A brief history of Visibility research The agenda in visibility research differs to an extent from the empirical hunt for the correlation between long-term population data and climate indices. The starting point is simple, as all three relevant components of the process are known (Ranta et al. 2000). The population renewal process obeys a known density dependent function, and the process is disturbed with a noise signal with known structural properties. When the noise-modulated population data are collected, the task is to identify the presence of the noise in the noise-modulated population signal. To our knowledge, the complete list of articles published in the domain of visibility research is as follows: Ranta et al. (2000), Kaitala and Ranta (2001), Lundberg et al. (2002) and Jonzén et al. (2002a). The noise detection methodology in these articles has been cross correlation where the noise signal and noise-modulated population data have been correlated with various time-lags. A fresh addition in the non-crowded field of visibility research is the contribution by Scott and Grant (2004). Ranta et al. (2000) explored the visibility of external noise (structured noise with short- and long-term components) using Ricker dynamics and linear second order autoregressive process, AR(2), as models for population renewal. High correlation (with time-lag 1) was observed only in conditions where the equilibrium population size of the deterministic population dynamics was locally stable (Edelstein-Keshet 1988), that is, when we look at the equilibrium state of the deterministic kernel of the population dynamics in the absence of external disturbance. The visibility of the noise in the periodic and complex region of the renewal process improves a bit if the modulating noise is very strong (has a wide range). Kaitala and Ranta (2001) explored noise visibility (cross correlations with lag 1) using an age-structured population renewal process for iteroparous and semelparous breeders. Their noise had an autocorrelation structure, blue noise with negative autocorrelation, white with zero autocorrelation and red with positive autocorrelation. The noise was not visible with semelparous breeders. With iteroparous breeders noise is visible in the dynamics when the deterministic kernel of the renewal process produces locally stable population dynamics, when the noise is red and when the noise influences the youngest age group only. Otherwise, detection of the noise

3 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 3 in the population signal is difficult. Lundberg et al. (2002) studied Ricker dynamics, split into births and deaths (Ripa & Lundberg 2000), as the skeleton of population renewal process. The noise (full range of autocorrelation structure from 1 to 1) affected either births, deaths, or both. Regardless of noise colour and what process it was influencing, it was visible only when the underlying population dynamics were locally stable. The merits of Jonzén et al. (2002a) will be discussed in the next section. The biological filter research by Laakso et al. (2001, 2003, 2004), though touching upon the question how environmental noise modifies the population renewal process, is a bit off the mainstream of visibility research. Here the assumption is that biological processes, e.g., growth, act as filters of environmental noise. Laakso et al. (2001, 2003) conclude that the outcome, in terms of noise visibility, is much dependent on the kind of the filter (sigmoid, saturating, optimal). In their recent research these authors found that population dynamics, when influenced by filtered noise, increase extinction risk when population growth rate is low (i.e., under locally stable dynamics), much of the details being dependent on the kind of the filter (Laakso et al. 2004). Summarising, it appears that noise is more easily visible in the noise-modulated population data if the underlying renewal process yields locally stable (not periodic or chaotic) dynamics. This conclusion is not very specific to the details of the renewal kernel....but don t forget Royama and Roughgarden Two often forgotten players in developing methodology for visibility research are Royama (1981, 1992) and Roughgarden (1975). Royama (1981) addressed the question where one suspects a given climatic factor (or any other density-independent factor) as an agent perturbing an observed population time series (see also Royama 1992: pp ). Let X t denote the observed (noise-modulated) time series (in logarithms) and µ t the noise. Further, let k denote differencing with the following operator, where k is the order of the difference. For example, in ecology the rate of change in population density from time t to time t + 1 is R t = X t+1 X t = 1 (X t ). Royama (1981) proposes the following ways of scoring the correlation between X and µ: (i) COR(X,µ), (ii) COR[ 2 (X),µ], (iii) COR[ 1 (X), 2 (µ)] and (iv) COR[ 3 (X), 2 (µ)]. The biological interpretation of (i) and (ii) are natural and understandable, but that of (iii) and (iv) remain a bit obscure. Method (i) has been used in visibility research. Royama s selection for the population renewal process is the stationary linear second-order [AR(2)] density-dependent process and the external noise is composed of independent, identically distributed random numbers, i.e., with white noise (no autocorrelation). Under these conditions Royama (1981) finds analytical solutions for all four methods (i iv) of scoring visibility. His major conclusion is that the presence of non-correlated noise becomes more easily visible when scoring methods (ii), (iii) and (iv) are applied. Let us emphasize that since Royama (1992), there has been increasing concern that many biological populations obey non-linear dynamics. Furthermore, it is known that many climate and weather variables are not white noise (e.g., Burroughs 1992). The contribution by Roughgarden (1975) to visibility research is in that he showed that a first order autoregressive process, AR(1), when disturbed with another AR(1) process, generates an AR(2) process. Post and Forchhammer (2001) recapitulated the finding. This is to say that parameter values from a noise modulated AR(1) process are not the same as the parameter values used to generate the autoregressive data. Jonzén et al. (2002a) extended this approach to the linear AR(2) process. They concluded that the parameter estimates of noise-modulated AR(2) process are far off the parameter values of the AR(2) process. Also the process order of noisemodulated AR(2) process is often something else than 2. The message of Roughgarden (1975) and Jonzén et al. (2002a) simply is that the reconstruction of the underlying renewal process is a difficult, if not impossible task. Thus, with noise modulating the renewal process, one will often incorrectly identify the underlying biology (using standard methods of model fitting).

4 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 4 Exercises With all these in mind, we did the following exercise. The population renewal process was taken either as the deterministic skeleton of the AR(1) process (Royama 1992) or the Ricker model X t + 1 = (1 + a 1 )X t + a 0 (1) X t + 1 = X t exp[r(1 X t )] (2) where (1 + a 1 ) is the autoregressive parameter (ranging between 1 and 1) a 0 is a constant, and r is the population growth rate (r is also the bifurcation parameter of the Ricker model, with r < 2 emergent dynamics are locally stable, 2 < r < yields periodic fluctuations while larger values of r yield chaotic dynamics). The two processes, equations (1) and (2), are then perturbed with noise so that X t + 1 = X t + 1 µ t, (3) where µ t is approximately an AR(1) process with serial correlation (for generating temporally structured noise, see, e.g., Ranta et al. 2006). For the autocorrelation we used = 0.7 (blue noise), = 0 (white noise) and = 0.7 (red noise). The values of the autoregressive noise were generated from the truncated range 1 w to 1 + w, 0 < w < 1 (we used w = 0.1). Our task is to score the visibility of the noise in the noisemodulated population dynamics, equation (3). For this purpose, we use the correlation tools i iv by Royama (1981, 1992). In addition, we also fit the renewal models, equations (1) and (2) to the noise-modulated data to asses the AR(1) and Ricker model parameter values used to generate the population data perturbed by the noise. Obeying Roughgarden (1975) we also fitted the linear second order autoregressive process, AR(2), to the AR(1) dynamics modulated with AR(1) noise. By doing these exercises we will answer to a greater extent the criticism raised by Scott and Grant (2004). The results of these exercises are straightforward. With correlated noise ( = 0.7 and = 0.7) the visibility [using method (i)] of the noise in the modulated population data is found for the AR(1) process (Fig. 1a, j). This holds true for the Ricker dynamics only in the range of r yielding locally stable dynamics (Fig. 2a, c, e). For the AR(1) renewal, with poor visibility, we have good chances of recovering the parameter values generating the population time series (Fig. 1f), but not when the visibility of the noise is good (Fig. 1b, j). For the Ricker dynamics parameter r, recovery is reasonably good in the region of periodic and complex dynamics (Fig. 2b, d, f, note that although the fitted values locate close to the 1:1 line, their 95% confidence limits did not overlap with the values of r used in the simulations). With both AR(1) and Ricker kernels we can say that when visibility is good we have poor parameter recovery. When the environmental noise has an autocorrelation structure, fitting AR(2) process parameters to the noise modulated data yields either negative (when 0.7) or positive (when = 0.7) values for (1 + a 1 ), the direct density dependence component (Fig. 1c, k). The coefficients of determination, R 2, attain rather high values (Fig. 1d, h, l) thus strongly suggesting that the noise-modulated process indeed is an AR(2) dynamics, as described by Roughgarden (1975). Comparing the match between the different noise-detection methods (correlations i iv) shows little improvement in visibility when the modulating noise itself has an autocorrelation structure (Fig. 3a, b, e, f). The results match Royama s (1981, 1992) suggestions only with white noise ( = 0, Fig. 3c, d). It also seems that, with the exception of white noise, the different renewal kernels yield differing visibilities when using the four visibility-scoring methods. Discussion Scott and Grant (2004) argue that the study by Kaitala and Ranta (2001) is limited in that only one pattern of density dependence and one detection method were used. They suggest that our conclusions are in part a consequence of our choice of population model and in part a consequence of using relatively weak or inappropriate statistical methods. This is true to the extent that Kaitala and

5 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 5 Fig. 1. First order linear autoregressive process, AR(1) is used as the kernel of population renewal. The process is disturbed with external noise having temporal structure [serial correlation being either 0.7 (a d) 0 (e h) or 0.7 (i l)]. Panels a, e and i give the correlation (high positive and negative values are an indication of good visibility) between the noise and noise-modulated population dynamics against various values of a 1, the AR(1) parameter. Panels b, f, and j give fitted values for the AR(1) parameter against the actual values of a 1 used. After Roughgarden (1975) an AR(1) process when disturbed by an AR(1) process is an AR(2) process. Thus, for the noise modulated AR(1) process we fitted the two AR(2) parameters (panels c, g, k), corresponding coefficients of determination (in %, key inserted in panel d) are also indicated (panels d, h, l). Ranta (2001) used only one model category, the Leslie matrix projection method, but different models have been used in our other visibility research articles (Ranta et al. 2000, Laakso et al. 2001, 2004, Lundberg et al. 2002). The conclusion thus far holds that using method (i), the presence of noise is visible under a locally stable kernel of the population dynamics with temporally structured noise. Our present results of comparing the different noise-detection methods suggest that correlating noise-modulated population data with the noise (with lag 1) is a relatively good method once the noise is temporally structured. Only under white noise we can duplicate Royama s (1981, 1992, recall that Royama used a linear kernel for the population renewal process topped off with white noise) results that methods (ii) and especially (iii) and (iv) are superior to (i). This is fine, but there is good evidence that environmental noise is not white. Many time series on climate show positive serial correlation, while a few are negatively autocorrelated (e.g., Burroughs 1992, Halley 1996). Scott and Grant (2004) suggest that the interplay between environmental stochasticity and density dependent population growth means that there is no single best method to detect the influence of environmental forcing, even when population dynamics are approximately linear. We agree to the extent that the exception is locally stable dynamics. Then the visibility of the noise is often reasonably good. Scott and Grant (2004) argue that by using correlation we are making the assumption that there is a linear relationship between the noise and the noise-modulated population data. This

6 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 6 Fig. 2. Non-linear Ricker dynamics is used as the kernel of population renewal. The process is disturbed with external noise having temporal structure [serial correlation being either 0.7 (a, b) 0 (c, d) or 0.7 (e, f)] The lefthand column of panels (a, c, e) gives the correlation between the noise and noise-modulated population dynamics against various values of population growth rate r (from 1 to 3.5 with a step of 0.5) used to generate the population data. The right-hand column of panels (b, d, f) give fitted values for the Ricker parameter against the actual values of r used. The dotted line is the 1:1 match. is a valid point, but it can be avoided by using non-parametric correlation coefficients, such as Spearman s rank correlation or Kendal s τ. Statistical knowledge (e.g., Siegel & Castellan 1989) tells us that with long-enough time series (i.e., large-enough sample sizes) results of parametric (linear assumption) and non-parametric (no assumption of linearity) correlation coefficients will eventually converge. Using piecewise time series analysis techniques (Tong 1990) is likely to introduce problems when addressing the possible non-linear relationship between the noise signal and population data. This is because these techniques will alter any autocorrelation structure in the noise data. As a new method for detecting the fingerprint of noise in population data, Scott and Grant (2004) suggest that for some patterns of population dynamics, environmental impacts are more readily detectable by correlating running average environmental conditions with the population time series or by correlating the first differences of the population time series with environmental noise. We caution against following this suggestion. A moving average is an effective transformation to smoothen out jagged time series data to bring out a trend (Box & Jenkins 1976). It is well known (with weather data: Burroughs 1992) that this transformation also creates artificial cyclic patterns that do not necessarily exist in the original data. Changing cycles into noncyclic time series data by the moving average transformation is known as the Slutzky effect [those interested on this effect are referred to

7 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 7 Fig. 3. Royama s (1981, 1992) suggestions to detect the fingerprint of noise in the noise-modulated data (y-axis) graphed against the correlations between the population data and noise (x-axis). The operator k is differentiation [the exponent k gives the process order, e.g., R t = 1 (X)], COR(x,y) is the Pearson correlation between x and y. AR(1) and Ricker dynamics are treated separately for various serial correlations, = 0.7 (blue noise), = 0 (white noise), = 0.7 (red noise). an example elaborated by Royama (1992: pp )]. Scott and Grant (2004) argue that populations with different vital rates and patterns of density dependence can respond differently to environmental forcing. We agree fully with this, as well as with the suggestion that a range of statistical methods may be necessary to detect the impacts of the environmental forcing. Up to this point we have deliberately been using the correlation method (i) as it has been the method often used when exploring the possible impact(s) of climate change, often the North Atlantic Oscillation as a climate proxy, on population data (references in Ranta et al. 2000). Our present results indicate that method (i) does not compare badly with the three other methods proposed by Royama (1981, 1992) to detect the impact of temporally structured noise in population data. We conclude by agreeing with Scott and Grant (2004) that new tools are badly needed to analyse the interplay between populations and their environment. However, a common problem in ecological data interpretation is that population and abiotic time-series are often short and temporally autocorrelated (Halley 1996). We are inevitably facing the general problem of separating demography from environment. For example,

8 JNR EEB vol. 3 Kaitala & Ranta: Environmental noise and the population renewal process 8 positively autocorrelated time series have three possible explanations (Jonzén et al. 2002a): (i) time-lags in population regulation (e.g., delayed density-dependence) generates a red population time series (Kaitala & Ranta 1996), (ii) the positively autocorrelated environmental variability modifies the population time series (Roughgarden 1975, Lawton 1988, Pimm & Redfearn 1988, Kaitala et al. 1997), or (iii) a combination of both. Thus, we may often face a situation where we have valid alternative process explanations for the same observed time series. Different model structures, however, represent different assumptions about the demographic structure of the population (Royama 1992, Stenseth et al. 1999). Ultimately, healthy theoretical modelling is necessary to all ecological interpretation and understanding (e.g., Berryman 1999). References Andrewartha, H. G. & Birch, L. 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