Steinar Engen, Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Fax: +47 72 55 33 14; E-mail: email@example.com
1A central question in population ecology is how to estimate the effects of common environmental noise, e.g. due to large-scale climate patterns, on the synchrony in population fluctuations over large distances. We show how the environmental variance can be split into components generated by several environmental variables and how these can be estimated from time-series observations.
2With a set of time-series observations from different locations not necessarily covering the same time span, it is shown how the spatial autocorrelation of the residual variance component, not explained by the covariates and corrected for demographic stochasticity, can be estimated using classical multinormal theory.
3Some previous results on spatial scaling in continuous linearized models on log scale are extended to also provide the scaling for the residuals. This is shown to be close to the spatial scaling of the autocorrelation in the environmental noise and only weakly affected by migration.
4The logistic model of local population dynamics with the NAO index as the only covariate is fitted to 22 populations of the Continental great cormorant Phalacrorax carbo sinensis. The spatial scale of the environmental noise is estimated to be about 155 km. The NAO index alone accounts for about 10% of the total environmental variance and nearly all of the regional environmental variance (long-distance environmental autocorrelation).
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Vertebrate populations often show synchronous population fluctuations in space. It has been well known for several decades (Elton 1924) that the populations of some species such as the snowshoe hare (Lepus americanus) or the arctic lemming (Lemmus norvegicus) in many areas show similar fluctuations at sites separated by hundreds of kilometres (Stenseth et al. 1999). Recently, comparative analysis involving several species living in the same area have in general demonstrated large interspecific variation, even among closely related species, in the spatial scaling of the synchrony in population fluctuations (Lindström, Ranta & Lindén 1996; Paradis et al. 1999, 2000; Cattadori, Merler & Hudson 2000). We still do not understand the mechanisms generating such interspecific differences in synchrony.
Analyses of linear or log-linear models have shown that the autocorrelation in the fluctuations in log size of two populations with no migration between them equals the autocorrelation in the environmental noise (Moran 1953; Lande, Engen & Sæther 1999). This also applies for between-years changes in log population sizes. Some evidence suggests that such a Moran effect can synchronize population fluctuations (Grenfell et al. 1998). Lande et al. (1999), Bjørnstad & Bolker (2000), Kendall et al. (2000), Ripa (2000), Engen (2001) and Engen, Lande & Sæther (2002a,b) extended these results to also include migration and showed that even short-distance migration may contribute substantially to an increase in the scale of population synchrony, especially if population-density regulation is weak. Extensive simulation studies of discrete time models have supported these results (Ranta, Kaitala & Lundberg 1997b; Ranta, Lindström & Helle 1997a). Thus, these analyses demonstrate that factors affecting environmental variation in space and time as well as processes acting within the populations themselves (e.g. strength of density regulation) influence the strength and scaling of synchrony in population fluctuations. Geographical patterns in population fluctuations often are studied using the Mantel test (Koenig 1999). Two matrices are computed: a distance matrix of the distances between all possible pairs of sites and a correlation matrix where the elements are the correlation coefficients of annual variation in population sizes at all possible pairs of study sites. A standard way to present the data is to plot the correlation coefficient computed from pairs of time-series observed during the same period against the distance between the different pairs of study plots. How rapidly the correlation decreases with distance indicates the spatial scaling of the synchrony in the population fluctuations. Although several modifications of this method have been suggested (Bjørnstad, Stenseth & Saitoh 1999a; Peltonen et al. 2002), a major problem with these non-parametric approaches is to link theory with data (Bjørnstad, Ims & Xavier 1999b). Another problem with the kind of non-parametric approach described above is that demographic stochasticity in population fluctuations is not taken into account. Demographic stochasticity is dominating compared to environmental stochasticity when local populations are small (Lande, Engen & Sæther 2003), and has by definition (Engen, Bakke & Islam 1998) no spatial scaling. Consequently population synchrony, which can only be affected by environmental stochasticity, is expected to increase with population size.
Here we provide a model that utilizes knowledge about the demographic variance and incorporates the possibility of a decrease of synchrony with population size due to demographic stochasticity. Our approach is especially useful for fully censused populations in which we can ignore sampling errors in population estimates and work with the full likelihood function for the problem. One advantage of this, in addition to the possibility of including demographic stochasticity, is that we can include any observed population size in the analysis and do not need to pay attention to the fact that the time-series only partly overlap in time. Rather than using raw estimates of correlations we work directly with the likelihood function using the complete set of residuals obtained after fitting models to the local populations.
The correlations then enter the model through the parameters of a spatial autocorrelation function. A third advantage with our approach is that we can assume that the local populations possibly have different dynamics. We do this by fitting stochastic population models to each local population, and include in the spatial analysis only the environmental component of the residuals, correcting for expected change in population size for different values of population sizes the previous year. Because theoretical as well as empirical analyses have focused previously on the synchrony of (log) population sizes or between years change in (log of) these (see Ranta et al. 1997b and Bjørnstad et al. 1999), we also include a theoretical analysis of the spatial scaling of residuals in a continuous homogeneous model in space and time with migration.
We apply this method to estimate the pattern of synchrony in the fluctuations of breeding populations of the Continental Great Cormorant Phalacrocorax carbo sinensis in Europe. We focus especially on how a large-scale climate phenomenon, the North Atlantic Oscillation (Hurrell et al. 2003), affects the spatial scaling of population synchrony.
single location analysis
Writing Nt for the population size at a given location, the dynamics are modelled by:
(eqn 1 )
where rt is the growth rate varying stochastically between years and g(·) is an increasing function defining the local density regulation. For population fluctuations driven by demographic and environmental stochasticity we have where are the environmental and demographic variance, respectively (May 1973, 1974; Turelli 1977; Engen et al. 1998; Lande et al. 2003). The demographic variance can be estimated by a simple sum of squares from individual data on survival and reproduction (Sæther et al. 1998; Lande et al. 2003). When such data are available, one will usually have data from a large number of individuals, giving accurate estimates of the demographic variance (Sæther et al. 1998). In the following we therefore consider the demographic variance to be known. We shall use the logistic form of density regulation, g(N) =βN2. Then, for small and moderate population fluctuations a simple first-order approximation gives the model:
where Xt= ln Nt while Ud and Ue are independent variables with zero mean and unit variance. Our main goal is to investigate the environmental noise terms Ueσe at different locations, including the population synchrony it generates. The temporal and spatial stochastic fluctuations in this term may be generated partly by other quantities such as, for example, the local abundance of other species, the local temperature and the NAO index. We introduce such quantities generally as random effects writing:
(eqn 4 )
where yi,t is the above random covariate number i at time t, U is another normalized variable and σ2 is the component of the environmental variance that can not be explained by fluctuations in the covariates. This leads to the relation so that the covariates together explain a fraction:
(eqn 5 )
of the total environmental variance. This decomposition of the environmental variance is also valid if the covariates are subject to noise with temporal autocorrelation. In this case the total environmental noise term Ue may also have a temporal autocorrelation. However, this does not affect our approach, which only assumes that the residual noise term obtained after correcting for the effect of the covariates is generated by a white noise process.
Assuming that Xt+1 = lnNt+1 is normally distributed when conditioned on Nt, and writing f(x; µ, σ2) for the normal probability distribution with mean µ and variance σ2, the log likelihood function takes the form:
(eqn 6 )
where Yt denotes the vector of covariates, v(N) = σ2 +/N, and the mean is the appropriate modification of eqn 2:
(eqn 7 )
The sum in eqn 6 is taken over those years for which the population size in the previous year is known. This means that if there are time gaps in the data of more than one year, we ignore the information contained in the population change over this gap, which in any case will be small.
Now, assuming known demographic variance, all other unknown parameters are estimated by numerical maximization of log likelihood, and uncertainties are evaluated by parametric bootstrapping involving simulating the time-series using the initial population size and the estimated parameters.
The spatial analysis will be based on studying the residuals obtained from fitting the model to time-series observations at each location z,
(eqn 8 )
where Ê denotes the estimated expected value. We use the normal approximation and choose a parametric form for the spatial autocorrelation of the U, of the form:
(eqn 9 )
where h(z) decreases from 1 to 0 as z increases from 0 to infinity. We use the exponential function h(z) = e–z/l and the Gaussian form which both correspond to positive definite autocorrelation functions. Using the same type of spatial autocorrelation function for the population sizes, Lande et al. (1999) and Engen et al. (2002a,b) applied the standard deviation of the function h(z) scaled to become a distribution as a measure of spatial scaling of population synchrony. Hence, the parameter l in the above functions is actually this measure of spatial scaling defined for the residuals.
Data from different locations often are available over different but partly overlapping time periods. However, for each year we do have a set of residuals with zero means, estimated standard errors, and correlations defined by eqns 8 and 9. For locations z1 and z2 the correlation is:
(eqn 10 )
For a given set of parameters (ρ0, ρ∞, l) defining the spatial synchrony, we have a complete description of a multivariate normal distribution each year, but possibly with different sets of locations at different years. The complete likelihood function is found by multiplying together the functions for each year or actually adding the log likelihood contributions for each year. Finally, this is maximized numerically to give estimates for (ρ0, ρ∞, l). Generally, the more the time-series overlap, the more information the likelihood function contains about the correlation parameters (ρ0, ρ∞, l). In order to obtain any information at all, we must require that at least two of the series are observed at least at one common time step. Otherwise the correlation parameters will be absent from the likelihood function and hence cannot be estimated.
The sampling properties of the estimates are found by parametric bootstrapping (Efron & Tibshirani 1993), that is, by simulating the whole set of local time-series, using the initial values of the data and the estimated parameters. The residuals must be simulated from the appropriate multinormal model defined by the autocorrelation function and the distance matrix.
The multinormal likelihood function can easily be calculated numerically using a lower triangular linear transformation, the Choleski decomposition (Ripley 1987). The same representation also gives a fast method for stochastic simulations required to perform the bootstrapping. This is explained in detail in Appendix I.
In Appendix II we extend some previous results for the scaling of spatial autocorrelation functions in a model that is linearized on the log scale. Lande et al. (1999) showed that the scaling for the population size defined as the above standard error derived from the autocorrelation, in a linear model allowing small and moderate fluctuations, was Here le is the scaling for the environmental noise, is the variance of the migration distance in a given direction (the p indicates the distribution of migration distance), m is the migration rate and γ is the strength of local density regulation. Engen (2001) derived the same result for a similar model on the log scale that allowed large population fluctuations, but with continuous migration modelled as a diffusion with lp approaching zero and m approaching infinity keeping . Engen et al. (2002b) showed for the same model that the scaling of the autocorrelation for the yearly differences of the log population sizes was close to + M/2 for γ < 1. Using the same model and the same approach it is shown in Appendix II that the scaling for the residuals in this model is:
(eqn 11 )
which turns out to be close to + M for γ < 1·15. Expressed by the mean yearly migration distance d̄, this is approximately + 0·64d̄2.
Two subspecies of great cormorant breed in Europe: the Atlantic great cormorant P. c. carbo and the continental great cormorant P. c. sinensis.
Their breeding ranges are mainly non-overlapping (Van Eerden, Koffijberg & Platteeuw 1995), although in some areas individuals of both subspecies are found where they may even form mixed subspecies colonies (Carss & Ekins 2002). After the decline in persecution during the 1960s and 1970s, a rapid increase in numbers of continental great cormorant occurred (Van Eerden & Gregersen 1995), whereas in recent decades a decline in the rate of increase has been recorded over large parts of its breeding range (Bregnballe et al. 2003). We extracted censuses of fluctuations in the size of colonies of the continental great cormorant in Sweden and in continental Europe (Fig. 1). The colonies are censused by counting the number of active nests. The breeding habits of this species make it likely that inaccuracy in the population estimates is small. Only colonies that were censused for 15 or more years were included, in order to avoid large uncertainty in the parameters that were estimated separately for each location.
The NAO-index is based on the difference of normalized sea level pressures (SLP) between Ponta Delgada, Azores, Portugal and Stykkisholmur, Iceland from 1864 to 2002 for the winter period December–March (Hurrell 1995). The NAO is a global climate phenomenon (Visbeck et al. 2001; Hurrell et al. 2003). In general, high values of NAO are associated with strong wind circulation in the North Atlantic causing an increase in temperatures and precipitation in northern Europe, but dry weather in the Mediterranean region (Hurrell 1995; Mysterud et al. 2000). In contrast, negative values of NAO are associated with decrease in temperature and precipitation in north-western Europe (Hurrell 1995; Mysterud et al. 2000; Catchpole et al. 2000).
The demographic variance was estimated from data on individual variation among females in their fitness contributions to the following generations in the population at Vorsø in Denmark (Fig. 1) using methods in Engen et al. (2005) for age-structured populations assuming age of maturity to be 2 years (Frederiksen & Bregnballe 2001). This is a well-studied colony where 12 024 chicks have been colour-ringed during the period 1977–2000 (see Frederiksen & Bregnballe 2000 and Schjørring, Gregersen & Bregnballe 2000 for further description). The estimation method is of the same type as the method used for estimating environmental variances in age-structured populations (Tuljapurkar 1982; Lande & Orzack 1988; Engen et al. 2005), except that the variances in vital rates are the within-years variation among individuals in survival and reproduction (see further description of estimation methods in Sæther et al. 2004; Engen et al. 2005). These variances are approximately inversely proportional to the total population size, and the constant factor of proportionality is the demographic variance of the process. This estimate is typical for bird species with age of maturity at 2–3 years (Lande et al. 2003; Sæther et al. 2004). We subsequently assume that this represents the demographic variance in all colonies.
An examination of the population estimates showed large intercolony variation in specific growth rate, ranging from 0·11 to 1·41 (Table 1). As the rate of return to K in the logistic model is equal to r (Lande et al. 2003), this also shows that the strength of density regulation around the carrying capacity differs among the colonies, although the uncertainty in the estimates of both those parameters was often large (Table 1).
Table 1. The estimated parameters (± SD) for the different colonies of the cormorant. r is the specific growth rate and K is the carrying capacity, assuming a logistic model for the density regulation, βNAO the regression coefficient of change in population size on NAO, is the component of the environmental stochasticity explained by variation in NAO, is the residual component unexplained by NAO and is the total environmental variance. The figures in brackets refer to the location in Fig. 1
Variation in NAO explained a significant proportion of the annual variation in the change in population size of 14 colonies (Table 1). All together, βNAO was positive in 16 of 22 colonies. On average, variation in NAO explained 9·3% of the environmental variance , ranging from 0 to 63%.
To examine the strength and scaling of the synchrony in the population fluctuations we fitted spatial autocorrelation functions of the Gaussian and exponential forms. Both models gave similar results (Fig. 2). For instance, at a distance of 5 km the correlation coefficient of the residuals was 0·197 and 0·225 in the Gaussian and the exponential models, respectively. Furthermore, the estimated scale of the local component of synchrony in the population fluctuations was 159 km for the Gaussian and 152 km for the exponential model. Parametric bootstrap distributions of the estimated scale are shown in Fig. 3. Although the estimated scale is uncertain, we do find the order of magnitude, and can conclude that the scale is smaller than about 300 km. As expected from Table 1, variation in NAO explained a proportion of the synchrony in the population fluctuations. We see that the NAO increases the autocorrelations of the environmental part of the residuals equally at all distances (Fig. 2) as it induces a common covariance between any residuals. The long-distance spatial autocorrelation is small, and seems to be mainly an effect of a synchronization of population fluctuations generated by fluctuations in the NAO index.
Spatial autocorrelation in environmental noise as well as migration (Lande et al. 1999, 2003) and environmental heterogeneity (Engen et al. 2002b) will generate spatial autocorrelation in population fluctuations. This population synchrony can be observed for the total (log) population size and the yearly differences in (log of) these. In the present study we use the full likelihood function to estimate population synchrony. Because the dynamics differ among locations it is convenient to work with the residuals measured at each location after fitting a stochastic population model. We use the local estimates for environmental variances and correct for the demographic component which, by definition, has no spatial autocorrelation. Analytical results have been derived previously to describe how migration and local density regulation affect the synchrony of (log) population size in continuous models (Lande et al. 1999; Bjørnstad & Bolker 2000; Engen 2001; Engen et al. 2002a,b), as well as discrete time models (Kendall et al. 2000; Ripa 2000). Many of these results have been supported qualitatively by simulation studies (Ranta et al. 1997a,b; Ranta et al. 1998). An important lesson from these analyses is that the scale of population synchrony depends on which characteristic of the population fluctuations is compared. In general, migration has greater effect on the scale of the fluctuations in population size than the scale of the yearly differences in population size, which is close to the scale of the local environmental noise (Engen et al. 2002a,b; Lande et al. 2003). Here we extend these results to show that the scale of the residuals also is close to that of the environmental noise. The effect of migration is approximately twice as large as that obtained using differences, but still small unless in cases where the spatial scale of the environmental noise can be neglected (le≈ 0).
Several approaches have been used to analyse spatial patterns of population fluctuations. One approach (Bjørnstad et al. 1999; Peltonen et al. 2002) is to calculate the correlations (parametric or non-parametric) between (log-) population size or yearly differences for each pair of locations, using the time span where observations are available for both populations. A spatial autocorrelation function is then computed by smoothing and adopting a filtering technique that ensures that the obtained autocorrelations are positive definite. Uncertainties are found by bootstrapping, choosing observed pairs of time-series with replacement. This is a non-parametric approach that can be used to explore the shapes of spatial autocorrelation functions. However, especially when the time-series overlap only partly in time it may represent an inefficient use of the available information.
Another approach is to consider each observed time-series (with n observations) as a point in the n-dimensional space and perform a principal component analysis (Fromentin et al. 1997; Kendall et al. 1999; Lekve et al. 1999; Viljugrein et al. 2001). This is typically a purely statistical approach which is difficult to interpret biologically. The conclusions derived from such analyses are also strongly dependent on the experimental design of data collection rather than the population dynamics as the method does not include spatial coordinates for the observations. Another problem is that the method is applicable only if the time span is the same for all time-series.
The present approach is based on the residuals obtained by fitting stochastic population models separately to each location. Ideally, these will be temporally independent, which is not generally the case for the yearly (log-) differences in population size as their distribution depends on population size in the previous year, which will generally be dependent between locations. Although our theoretical result on the scaling of residuals given in Appendix II is valid only for the homogeneous models used to derive it, it still probably provides the correct order of magnitude for how the spatial autocorrelation is affected by migration and autocorrelation in the noise. Because we use the full likelihood for the complete data set, the method is statistically very efficient.
Earlier approaches do not make a distinction between demographic and environmental stochasticity. The models used are then based on the assumption that spatial autocorrelation only depends on the distance between locations and not on the population size. In the present approach, however, the effect of demographic stochasticity on the autocorrelation is incorporated into the model. Other approaches to this problem have always been based either on the assumption that the demographic variances are zero or that the population sizes are large enough to neglect demographic stochasticity. Even though large intraspecific variation may be found in (Sæther et al. 2003; Sæther et al. 2004), we have used a common estimate of obtained from one location only. We consider this to be a more realistic approach giving more correct estimates of population synchrony than assuming = 0. Furthermore, k̂ was larger than 1000 pairs in several of the colonies (Table 1), which makes the contribution from demographic stochasticity to the population fluctuations small (Lande et al. 2003). However, in cases where the local populations are small at some locations and larger at others, the effect of demographic variance may be considerable. In Fig. 4 we illustrate this strong effect of demographic stochasticity by plotting the estimated spatial autocorrelation functions for the residuals for different values of population size, using the estimated demographic variance and a value of demographic variance that is typical for birds (Sæther et al. 2004). We see that the synchrony decreases at a given distance with decreasing population size and increasing value of .
During recent decades, the numbers of the continental great cormorant have increased over large parts of its distributional range (van Eerden & Gregersen 1995; Bregnballe et al. 2003), due mainly to reduced human persecution. As a consequence, an increasing number of new colonies have been established (Bregnballe & Rasmussen 2000), often involving a redistribution of individuals after years with poor breeding success (van Eerden & van Rijn 2003). Because of this pattern of colony formation, many colonies will fluctuate independently in size. However, the pattern of growth of most colonies is density-dependent (Frederiksen, Lebreton & Bregnballe 2001) and can be described well by a logistic model of density regulation (Table 1). By analysing the residuals from this model we are able to demonstrate large-scale synchrony in the population fluctuations of the continental great cormorant (Figs 2 and 3).
Theoretical analyses (summarized in Lande et al. 2003) as well as extensive simulation studies (e.g. Lundberg, Ranta & Kaitala 2000; Ranta et al. 2000) have shown that dispersal and spatial autocorrelations in the environmental noise both affect the spatial scaling of population synchrony. For the continental great cormorant, we estimate the spatial scale of population synchrony to be less than 300 km (Fig. 3). No quantitative analysis has yet been published on the pattern of dispersal of the great cormorant. However, high site fidelity has been found in some colonies (Schjørring et al. 2000). Furthermore, variation in NAO was able to explain nearly all the population synchrony at large distances (Fig. 2). Large population growth rates were also often associated with positive values of NAO (Table 1). As positive values of NAO in this area in general are associated with mild winters (Hurrell 1995), this suggests that variation in climate conditions in the wintering areas may act as a synchronizing agent on the population fluctuations of the continental great cormorant. As individuals from several colonies often use the same wintering areas (Marion 1995; Frederiksen et al. 2002) this may explain the synchronization effect of this large-scale climate phenomenon at large distances (Fig. 2). However, the tendency for a smaller synchronizing impact of NAO among closely located colonies (Fig. 2) may also suggest that other environmental conditions or dispersal is important at small distances.
This work was supported by the Norwegian Research Council (project 142883/432 and 155903/720, Klimaeffekt-programmet) and the US National Science Foundation. We also thank SOVON, the Staatliche Vogelschutzwarten, Landesumweltämter, H. Engström, W. Knief, M. R. van Eerden and H. Zimmermann for allowing us to include their data on colony sizes.
likelihood and simulations for the multinormal distribution
Using Cholesky decomposition
Let (X1, X2, ... , Xn) be multinormally distributed with mean values EXi = µi, var(Xi) = and correlations corr(Xi, Xj) = ρij and write Yi = (Xi–µi)/σi for the standardized components. The Yi can be expressed as linear combinations of independent standardized normal variates U1, U2, ... ,Un
(eqn 12 )
where the coefficients aij defining a lower triangular matrix are determined by the correlation matrix only (Ripley 1987). Using the fact that corr(Yi, Yj) = corr(Xi, Xj) = ρij and writing up the correlations in terms of the aij we can easily solve for these coefficients. We first observe that a11 = 1 and subsequently for i = 2, 3, ... , n the coefficients with i as first index are:
Stochastic simulation can now be performed by first simulating the Ui, then calculating the Yi using eqn 12, and finally computing the Xi = σiYi + µi.
To find the likelihood function we need the distribution of the Xi which can be expressed as:
where φ(u) = (2π)−1/2 exp(–u2/2) is the standard normal distribution. Here the observations Xi are the given data and the independent standardized variables Ui must be calculated by first calculating the Yi = σiXi + µi and solving eqn 12 giving:
U1 = Y1/a11
and subsequently for i = 2, 3, ... , n,
The contribution to the log likelihood from the vector (X1, X2, ... , Xn), defined as a function of the unknown parameters to be estimated, is:
Finally, the contributions from all independent vectors of observed residuals through time are added to produce the complete likelihood function.
spatial scaling in the loglinear model
The following is based on the linear model with migration dealt with by Engen (2001) and later applied to derive several scaling results (Engen et al. 2002a,b). In the absence of migration and demographic stochasticity we consider the loglinear continuous time model:
where n(z, t) is the population size at location z at time t, r is the population growth rate at small densities, and γ is the strength of local density regulation. The derivative (d/dt)B(z, t) is a white noise process at location z with no temporal autocorrelation, actually the derivative of a Brownian motion (Karlin & Taylor 1981). This environmental noise has a spatial autocorrelation, called the spatial environmental autocorrelation, defined by ρe(z)dt = EdB(w, t)dB(w + z, t).
Adding density-independent migration to this model the equation for the densities takes the form:
giving for the log densities:
where is the stochastic growth rate. Here individuals are assumed to migrate at a rate m and the migration distance has distribution p(ξ) with zero mean. In the limit as m tends to infinity and the migration distance tends to zero one can show that the Fourier transform of the autocovariance function c(z, t) = cov[X(w, u), X(w + z, u+ t)], that is
takes the form:
where fe is the Fourier transform of (Engen 2001). In the limit the migration of each individual is described by a two-dimensional Brownian motion with zero mean. The parameter M is easily interpreted through the fact that the probability that an individual migrates a distance longer than d during a time interval of length t is exp[−d2/(2Mt)].
We now want to investigate the spatial scaling of the residuals obtained by considering the continuous model only at discrete times with time steps one, that is: