Biological filtering of correlated environments: towards a generalised Moran theorem

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1 Oikos 6: , 27 doi:./j x, Copyright # Oikos 27, ISSN Subject Editor: Owen Petchey, Accepted 9 January 27 Biological filtering of correlated environments: towards a generalised Moran theorem Jörgen Ripa and Esa Ranta J. Ripa (jorgen.ripa@teorekol.lu.se), Dept of Theoretical Ecology, Ecology Building, Lund Univ., SE Lund, Sweden. E. Ranta, Dept of Biological and Environmental Sciences, P.O. Box 65 (Viikinkaari ), FI-4 Univ. of Helsinki, Finland. Many species from diverse taxa are known to display synchronous fluctuations across vast geographical ranges. It is often thought that climate factors influencing the growth of conspecific populations are correlated over large distances and hence produce the synchronous population dynamics an effect known as the Moran effect. However, for species embedded in a food web the Moran effect needs not necessarily influence the focal species directly, but can act indirectly through other species. Such an indirect synchronization can also occur in an agestructured population, where the correlated environment of one age-class causes synchronous fluctuations of another. Here, we investigate this indirect Moran effect. We find first of all that synchrony is readily transferred through food webs or between age classes, which complicates the identification of the underlying synchronizing factor. Secondly, we find puzzling cases, where synchrony is enhanced as it is filtered through a food web or between age-classes. Our results also apply to systems of different species, but with closely matching dynamics. Since Elton (924, Lindström et al. 2) data have accumulated on spatial synchrony in abundance for populations from almost all major taxa (Ranta et al. 995, 999, Liebhold et al. 24). Three general explanations of population synchrony have been put forward. First, it is due to synchronous environmental forcing the so-called Moran theorem (Moran 953, Royama 992), or the Moran effect (Ranta et al. 995, 997). Moran (953) states that two (or more) populations sharing a common linear density-dependence in the renewal process that are disturbed with correlated noise will become synchronised with correlation matching the noise correlation. Second, population synchronicity is due to dispersal between populations that cause the synchrony (Elton 924, Elton and Nicholson 942, Hanski and Woiwod 993, Ranta et al. 995). In some cases dispersal can be ruled out because it is biologically impossible, as with the Soay sheep on Scottish islands (Grenfell et al. 998). In many other cases, it may be hard to tell which factor is more important dispersal or the correlated environment (Ranta et al. 995), although it can be shown that dispersal only synchronizes efficiently when local dynamics have long return times or are cyclic (Lande et al. 999, Bjørnstad 2, Kendall et al. 2, Ripa 2). Finally, mobile predators have been raised as candidate causes of synchrony (Elton 924, Butler 953, Watt 968, Ydenberg 987). A predator constantly seeking out the most abundant prey population will in the long run even out differences between prey population sizes. The Moran theorem first of all requires an, or a combination of, environmental variable(s), such as temperature or precipitation, which can show reasonable correlation across long distances (Koenig 22). Secondly, it has to be shown that the populations in question are strongly affected by that particular (combination of) environmental variable(s). Such analysis is afflicted with statistical problems, although there are examples where potential environmental factors have been singled out (Grenfell et al. 998, Cattadori et al. 2, Sæther et al. 24). We study here the case when the situation is even more complicated, i.e. when the synchronising environmental factor consists of populations of another species, living in the same environment and interacting with the populations of the focal 783

2 species. This other species may in turn be synchronised by synchronous weather factors, or possibly by dispersal or other means. Such an indirect Moran effect has been used recently to explain the synchrony of red grouse populations in Britain, through correlated environmental effects on a parasite (Cattadori et al. 25). In this study, we will take a closer look at the indirect Moran effect; the case of one species synchronised by correlated environments and interacting with another, the focal species, which experiences an uncorrelated environment. Under such scenario it may be difficult to detect what environmental factor is the ultimate cause of the observed synchrony, and this is for two reasons. (i) The effect is indirect as the environmental signal is biologically filtered through population interactions before it reaches the focal populations. (ii) There may be biological reasons why that particular environmental factor should not be important for the focal species, which implies that it will hardly be considered as a potential synchronising agent. The model we use is put forward as a small food web model, but is general enough that it can easily be interpreted as an age-structured population model, with possible interactions between the age-classes. Hence, our results apply also to indirect synchronizations between age classes within a single species. Below, we analyse to what extent spatial synchrony can spread through small communities, depending on spatial differences in population interactions and the level of environmental variability of different species in the community. We also discuss the possibility to identify the underlying cause of synchrony, i.e. the environmental factor ultimately responsible for the synchrony. The model We want to explore the general case of two small communities, each consisting of populations from the same two species, x and y. The two communities exist in two separate habitat patches, labelled A and B (Fig. ). We will assume that species x is the one of interest, i.e. the synchrony between the x-population in patch A and the x-population in patch B is what we want to explain. Further, we assume that the two communities have independent dynamics, in the way that there is neither interaction nor migration between populations from different patches. The only connection will be the correlation between the environmental fluctuations of the two y-populations. As a starting point, we will analyse general linear models of the A ε A (t) B ε B (t) x A (t) x B (t) y A (t) y B (t) ϕ A (t) ϕ B (t) ρ ϕ Fig.. A graphical description of the model. Two isolated patches A and B contain communities of two interacting populations, x and y. In each patch, population x is perturbed by environmental fluctuations o(t), whereas the environment of population y is denoted 8(t). The only connection between the two communities is the correlation r 8 between the environments of the y populations. 784

3 dynamics within each patch: x A (t)a xx x A (t)a xy y A (t) o A (t) y A (t)a yx x A (t)a yy y A (t) 8 A (t) Un A (t)an A (t)g A (t) (a) and x B (t)b xx x B (t)b xy y B (t) o B (t) y B (t)b yx x B (t)b yy y B (t) 8 B (t) Un B (t)bn B (t)g B (t); (b) where x A/B (t) and y A/B (t) denote population densities at time t, where the mean has been subtracted. Parameters a ij and b ij correspond to the interaction coefficients within (i /j) and between (i"/j) the populations. For instance, a xx relates to competition within the x-population in patch A. An a xx close to / implies the population has weak intraspecific competition, whereas a negative a xx implies strong, overcompensating intraspecific regulation. The interspecific coefficients, like a xy and a yx, can be interpreted as population interactions in the usual way, such that two negative coefficients indicate competition, and so on. o A/B (t) and 8 A/B (t) are the stochastic environments of the x and y populations respectively. For simplicity of the argument, we assume that the environments have zero means and no temporal autocorrelation. Further, we assume that they are all independent, with the exception of 8 A and 8 B, which are correlated: corr (8 A (t); 8 B (t))r 8 (2) To the right in Eq. are given the community dynamics in matrix notation, where n is a column vector of the population densities (x and y), g is a vector of the environmental factors (o and 8), and A and B are the system matrices. Thus, the elements of A are denoted a ij and the elements of B are denoted and b ij. The community dynamics in a single patch following Eq. is a first order vector autoregressive process, VAR() (Reinsel 997), also called a multivariate autoregressive process, MAR() (Ives et al. 23). Such a process is stationary if the system matrix is stable, i.e. the eigenvalues of the matrix have magnitude less than one (Reinsel 997). Apparent single-species dynamics The dynamics governed by Eq. is of order one, as there are no time lags influencing the renewal process. Hence, the future growth of the populations only depends on the current densities and the future environmental variability. It is worth noting, however, that if the census is done for only one species in each patch, it will appear as if it has delayed density dependence (Royama 992, Reinsel 997). For example, it is readily shown that the populations in patch A have apparent single-species dynamics according to: x A (t)(a xx a yy )x A (t)(a xx a yy a xy a yx )x A (t2) õ A (t) (if a xy ") (3a) y A (t)(a xx a yy )y A (t)(a xx a yy a xy a yx )y A (t2) where 8 A (t) (if a yx ") (3b) õ A (t)o A (t)a yy o A (t)a xy 8 A (t) (3c) 8 A (t)8 A (t)a xx 8 A (t)a yx o A (t) (3d) The equivalent expressions apply to patch B, only with b ij substituted for a ij (i,j/x,y). First of all, note that the density dependent parts in Eq. 3a and 3b are second order, autoregressive expressions, AR(2). Further, the coefficients of the density dependency are equal for the two populations. However, the environmental input, õ(t) and 8(t); differs between the two populations and is a blend of the two environments. More specifically, when analysed as single population dynamics each population appears to experience a moving average of its own environment, and on top of that the delayed environment of the other population (Eq. 3cd). It is important to note that this different filtering of the two environmental signals gives each population its unique dynamics, even though the density dependent deterministic parts of Eq. 3a b are exactly equal. Synchrony of identical communities We will use the Eq. 3ad to investigate to what extent the populations of species x are synchronised by the correlated environmental fluctuations of species y (Eq. 2). In the two following sections, we analyze the simplified case when the internal dynamics of the two patches are equal (a ij /b ij, i,j/x,y). First, we assume only one species is subject to environmental fluctuations, which simplifies to the original Moran theorem. In the second section below we allow for environmental disturbances of both species, which complicates the analysis, but we nevertheless reach some general results. Single species environment Here we assume the environment of y is much more variable than that of x, such that it is the totally dominating source of external variability. In other words, we assume Var[o A (t)]/var[o B (t)]/. If in addition the dynamics in the two patches are equal, it means x A and x B are governed by the same dynamics (Eq. 3a) and have equivalently filtered environmental 785

4 disturbances, namely the delayed 8 A and 8 B (Eq. 3c). In such a situation the Moran theorem applies, and the correlation (synchrony) of the two populations is equal to that of the environments, i.e. r x /corr[x A (t), x B (t)]/r 8. The same reasoning applies to the y- populations, which gives r y /corr[y A (t), y B (t)]/r 8. Thus, if the endogenous dynamics in the two patches are equal and the environment of species y is the totally dominating source of variability, then the two species will be equally synchronised and the population synchrony will equal that of the environment. This result is easily generalized to a community of n species where only a single species is influenced by environmental fluctuations. So how will the synchronisation appear, from a time series point of view? Is it possible to detect the source of synchrony (the environment of species y) by studying time series of species x? Equation 3c gives, as noted above, that the environment of species y (8) appears as lagged disturbances of species x. In other words, species x will appear as affected by the environmental factor 8, but with a lag of one year. Given that time series of 8 are at hand, such as weather data, the delayed dependency can be detected (Cattadori et al. 25). The spatial synchrony of the environmental factor 8 is easily confirmed, given time series data from different sites. Thus, the source of this kind of indirect synchronisation should be possible to detect if appropriate data is available. The largest difficulty would be to look in the right place the spatially correlated 8 is important for species y, but not directly so for x, so there are no immediate biological reasons to look for 8-dependence of x. Environmental fluctuations of both species Now consider the case when the environmental fluctuations of the x-populations [o(t)] are non-negligible. This will naturally decrease the synchrony of the populations, since o A (t) and o B (t) are uncorrelated by assumption. In Appendix, we present the full expressions for the synchrony of the two species. With a lot of algebra, it is also possible to prove that the x-populations will always be less synchronised than the y-populations (Appendix ), which is nevertheless quite intuitive. As the variance of o increases, the synchrony of the x-populations will decrease. At the same time, the dependency on the synchronous environmental factor 8 will be weakened, since o will constitute an increasing proportion of the environmental disturbances (Eq. 3c). We illustrate some of the results above in Fig. 2, where the synchrony of the x- and y-populations of an example community is plotted against the ratio of the variances of o(t) and 8(t). When Var[o(t)] is close to zero, i.e. on the y-axis, the synchrony of both species approaches that of 8(t). As the ratio increases, i.e. o(t) becomes an increasingly important source of variability, the synchrony of both species decreases, but the synchrony of species x has a steeper initial decline and remains smaller than that of y. Even though the x synchrony, r x, inevitably decreases as the ratio Var[o(t)]/Var[8(t)] increases, the decline is slower under some circumstances than others. For instance, a large amplitude of a xy will give a large contribution from 8(t) to the apparent environment of x (Eq. 3c). In other words, a strong interaction from y on x will make x much influenced by the environmental fluctuations of y, which is not surprising. Since.8 Correlation.6.4 corr(x) corr(y) Var(ε) / Var(ϕ) Fig. 2. The synchrony (correlation) of the x and y populations in two identical communities, as a function of the ratio between the variance of the x environment (o) and the y environment (8). As the uncorrelated x environment becomes progressively stronger, the synchrony of both species decreases, but the x-synchrony is always lower than that of y. Parameter values: a xx /b xx /.3, a xy /b xy //.6, a yx /b yx //.4, a yy /b yy /.5, r 8 /

5 the environment of y by assumption is strongly spatially correlated, a large influence on x will make x correlated, i.e., synchronous, too. Consequently, a large a xy will promote the synchrony of x. The other parameter important for the environmental influence on the dynamics of x is a yy, viz., the intraspecific density dependence of species y. In the expression for õ A (t) (Eq. 3c) a yy functions as a weighting in a moving average filtering of o A (t). The relationship between the sign and amplitude of a yy and the variability of x is quite intricate, and we refrain from a full analysis here. We will, however, get back to the dependency on a yy in the following part. Synchrony of non-identical communities We now relax one of the assumptions above and allow for differences between the two habitat patches such that the governing dynamics are different, i.e. A"/B (Eq. ab). As a rule of thumb, different dynamics renders a decreased level of synchrony (Ripa and Ives 23, Engen and Sæther 25, Royama 25, Hugueny 26). There may be exceptions, however, as we shall soon see. We begin with a very much simplified case with a one-way interaction and a single species environment. This simple setting nevertheless shows some interesting properties which also can be found in the more general case. One-way interaction and single species environment As a first example we study the case of a one-way interaction, i.e. when the focal species x is affected by the density of y, but not the other way around. This simplifies the Eq. ab in the way that a yx /b yx /. The y populations get totally independent dynamics, corresponding to first order autocorrelated processes, AR() (Box et al. 994): y A (t)a yy y A (t)8 A (t) and y B (t)b yy y B (t)8 B (t) (4a) (4b) We also assume the x populations in each patch have identical intraspecific competition, such that a xx /b xx. In other words, the remaining possible difference between the two communities is the intraspecific regulation of species y, a yy and b yy, as well as the effect of species y on x, a xy and b xy. We further make the weak assumption that the effect of species y on x is of the same nature in the two patches, i.e. a xy and b xy are of equal sign (and non-zero). A last simplifying assumption that the environmental fluctuations of species y totally dominate the communities, i.e. s 8 /, allows for an intelligible analytical solution to the synchrony of the different species. With some algebra (Appendix 2) it is possible to show: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( a 2 yy r y )( b2 yy ) r 8 (5a) a yy b yy and a 2 xx r x a yy b yy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( a 2 xx a2 yy )( a2 xx b2 yy ) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( a 2 yy )( b2 yy ) r 8 a yy b yy a 2 xx a yy b yy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ( a 2 xx a2 yy )( a2 xx b2 yy ) y (5b) The expression in Eq. 5a is always smaller in magnitude than r 8 (given a yy "/b yy ), which implies that the synchrony of species y is always smaller than the synchrony of its environment. Further, Eq. 5b can be rewritten (Kuckländer 26, Eq. 5.): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ( a 2 xx r x a2 yy )( a2 xx b2 yy ) a2 xx (a yy b yy qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi )2 r ( a 2 xx a2 yy )( a2 xx b2 yy ) y (5c) which astonishingly implies that the x populations are always more synchronized than the y populations, as long as a yy "/b yy and a xx "/, i.e. as long as the intraspecific competition differs between the two y populations, and as long as the x populations do not have perfectly compensating intraspecific regulation (remember a yy /a xx by assumption here). This result was given by Kuckländer (26) in a slightly different context and a slightly different interpretation. We use it here merely to point out that under some circumstances, like the assumptions given above, synchrony can actually increase as it filters through a food web. Full interactions The conclusions from above apply to some extent to the case when the species truly interact, i.e. when a xy and b xy are different from zero. It is possible to find examples where the x populations are more synchronized than the y populations, even when the uncorrelated environment of x has non-zero variance. The analysis, however, becomes very complicated and it is 787

6 hard to interpret the analytical solutions. Therefore, we finish by giving an example from a more standard ecological model. The model is non-linear, but we chose parameter values such that its linearized approximation predicts the synchrony enhancement noted above. Consider a stochastic, discrete time Lotka-Volterra competition model: x A (t)x A (t) exp r x;a x A(t ) a x;a y A (t ) o A (t) K x;a (6a) y A (t)y A (t) exp r y;a y A (t ) a y;ax A (t ) 8 A (t) K y;a (6b) where x A (t) and y A (t) denote the densities of the populations in patch A, r x/y,a the intrinsic growth rates, a x/y,a the competition coefficients and K x/y,a the carrying capacities of populations x and y, respectively. The terms o A (t) and 8 A (t) represent stochastic environments as before. The same equations apply to the dynamics of the community in patch B, only with all indices A changed to B. The intention here is to show an example of the interesting case when the x-species has a higher synchrony than the y-species, but which is still biologically plausible. One such example is given by the parameter values: r /.5 and a/.8 (all populations), K x / (both patches), K y,a /.85 and K y,b /.2. Thus, the only difference between the two patches is that the carrying capacity of the second competitor (species y) changes from.85 to.2. However, due to the strong competition this moderate change in carrying capacity leads to quite a large change in equilibrium densities and quite dramatic changes in the equilibrium intra- and inter-specific competition (see a sample simulation in Fig. 4). In Fig. 3, we have depicted the synchrony of species x and y in this system as functions of the ratio between the variances of the x and y environments. The solid lines give the results of simulations of the full non-linear model, Eq. 6a b, and the dashed lines indicate the prediction from the linearized system equations (the corresponding system matrix is presented in Ripa and Ives (23), the necessary calculations are described in Appendix ). As predicted, the synchrony of x is higher than that of y, at least as long as the environmental fluctuations of y are considerably stronger than those of x. Note that the linear approximation predicts the synchrony of the nonlinear system with relatively good accuracy, despite quite strong environmental stochasticity; s o 2 /s 8 2 /. in all simulations. The non-linear simulation results are generally lower than the linear prediction, which is in agreement with other studies on the effect of nonlinearities on population synchrony (Engen and Sæther 25, Royama 25). Fig. 4 shows a sample simulation when the x populations are the most synchronized (s o 2 / s 8 2 /.), which not only shows that the populations move far away from equilibrium population sizes (dotted lines), but also illustrates the highly correlated dynamics of the two x populations (top panel) and the little correlated dynamics of the two y populations (bottom panel)..8 Correlation corr(y) corr(x) -2-2 Var(ε) / Var(ϕ) Fig. 3. The synchrony (correlation) of two species in two competitive communities governed by discrete time Lotka-Volterra equations (Eq. 6a b) (solid lines), and the corresponding expected correlation calculated from the linearized model (dashed lines). The populations were first simulated in time steps and then the following time steps were used to calculate the correlations. The total environmental variance of the system, Var(o)/Var(8)/s o 2 /s 8 2, was kept constant at.. Other parameter values: r x,a /r x,b /.5, a x,a /a x,b /a y,a /a y,b /.8, K x,a /K x,b /, K y,a /.85, K y,b /

7 ..4 x A Population size, y A Population size, Time, t Fig. 4. Sample time series from the simulations in Fig. 3. Top panel: The x population sizes of patch A (solid line, left y-axis) and patch B (dashed line, right y-axis). The two y-axes are scaled such that the equilibrium population sizes coincide (dotted line). Lower panel: Simultaneous y population time series from patch A (solid line, left y-axis) and patch B (dashed line, right y-axis). Again, the y-axes are scaled such that equilibrium population sizes coincide (dotted line). Parameter values: s o2 /.99, s 8 2 /.99, i.e. s o 2 /s 8 2 /., s o 2 /s 8 2 /.. Other parameters as in Fig x B Population size, Population size, y B Discussion We have found that synchrony is easily transferred between populations in a community, or, as noted in the introduction, between age- or stage-classes in a structured population. An abiotic factor may synchronise a particular species (or age-class), but also other interacting species (age-classes) an indirect Moran effect. The effect is particularly strong if the communities have similar dynamics. Our results have direct implications for how to look for synchronising agents. The underlying biological mechanism by which a variable abiotic factor drives the system may be far from obvious if the direct effect is not on the focal species, but on another, interacting species. The indirect effect on the focal species will also be delayed (Eq. 3c d), which further complicates the identification of the synchronizing agent. We have also found that in some situations, synchrony is enhanced when transferred through a community, such that the indirectly synchronised species has a higher synchrony than the species that experiences the actual synchronous environment. This phenomenon is theoretically fascinating, but there are a couple of reasons why it should be of somewhat minor practical importance. First of all, the circumstances when it can arise are quite special. In the simplified case with a one-way interaction, a substantial synchrony enhancement requires identical intra-specific competition of the x populations, but substantial differences in the intra-specific competition of the two y populations. Second, it is somewhat difficult to generalize to a large number of geographically separated communities. In the simplified case all x populations need to have in principle identical intraspecific competition while the variability among the y populations regulation should be large. Although we have not investigated the full interaction case in any detail, we suspect the requirements for a substantial, detectable, synchrony enhancement are equally strict and hard to find in real systems. Our results are based on a linear model, but we showed that the linear model well predicted the behaviour of a standard non-linear community model. However, the generality of the linear model lends itself to a whole suite of biological interpretations. For instance, contrary to Moran s (953) assertion, it is not required that the two communities consist of the same species (i.e. the species do not necessarily need to have exactly matching density-dependence). One or both species can be different, which means our results apply to patterns of inter-specific synchrony as well, as long as two of the species are affected by the same environmental fluctuations. A good example of this comes from three species of forest grouse (capercaillie, 789

8 black grouse and hazel grouse) that are close to each other in terms of biology (Lindström 994). These species display synchronous dynamics both within species and across species (Lindén 988, Lindström et al. 995, 996, Ranta et al. 995, see also Moran 952, Cattadori et al. 2, Watson et al. 2). Crossspecies synchrony has also been shown for Lepidoptera (Myers 998) and freshwater fish populations (Tedesco et al. 24). The model could also, as noted above, describe an age-structured species, such that for instance x denotes adults and y indicates juveniles. With that interpretation, our results show how a synchronous environment of the juvenile stage can cause synchronous dynamics of the adults (Kaitala and Ranta 2, Scot and Grant 24). Naturally, the interpretation can be reversed such that a correlated adult environment synchronizes the dynamics of juveniles or eggs, a phenomenon which has been shown experimentally (Benton et al. 2). Throughout, we have assumed that the environmental fluctuations of the focal species, x, show no correlation between patches. This assumption is presumably rarely met in natural systems, since many important environmental drivers show high correlation over large distances (Koenig 22). For a complete understanding, therefore, it is necessary to sort out the simultaneous effects of several synchronous environmental factors, affecting one or several populations of a food web. Here, we have taken the first step towards such an understanding, but chose to make some simplifying assumptions to be able to derive comprehensible analytical results. We nevertheless acknowledge that there is more work to be done. We have shown that the Moran theorem works for communities of interacting species (or age-classes) as well as for the single species case. If two communities have identical dynamics and the environmental fluctuations of a single species dominate, the synchrony of all species will be equal to that of the dominating environmental signal. This has been anticipated based on empirical data on different grouse species (Moran 952, Lindén 988), as well as freshwater fish (Tedesco et al. 24), but never analysed except for highly nonlinear special cases (Cazelles and Boudjema 2). The fluctuations of the synchronizing agent are visible in the dynamics of all species, but the effect can be delayed. If the two communities are not equal, there are special cases when the indirectly synchronized species have higher synchrony than the species directly affected by the synchronizing agent, although we suspect such cases are hard to find in natural populations. We nevertheless have demonstrated how synchrony can filter through communities or food webs, leading to an increased understanding of this widespread phenomenon. Acknowledgements We thank Nina Kuckländer for valuable comments. J.R. is supported by the Swedish Research Council. References Benton, T. G. et al. 2. Population synchrony and environmental variation: an experimental demonstration. Ecol. Lett. 4: Bjørnstad, O. N. 2. Cycles and synchrony: two historical experiments and one experience. J. Anim. Ecol. 69: Box, G. E. P. et al Time series analysis: forecasting and control, 3rd ed. Prentice-Hall. Butler, L The nature of cycles in populations of Canadian mammals. Can. J. Zool. 3: Cattadori, I. M. et al. 2. Searching for mechanisms of synchrony in spatially structured gamebird populations. J. Anim. Ecol. 69: Cattadori, I. M. et al. 25. Parasites and climate synchronize red grouse populations. Nature 433: Cazelles, B. and Boudjema, G. 2. The Moran effect and phase synchronization in complex spatial community dynamics. Am. Nat. 57: Elton, C. S Periodic fluctuations in the numbers of animals: their causes and effects. Brit. J. Exp. 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10 a 2 2 (D ) a 2 xy (D ) ( a xya yx a xx a yy a 2 yy a3 yy a xx a yx a xy a 2 yy )c e ( a xy a yx a 2 xx a xxa yy a yy a 3 xx a2 xx a xya yx ) ( a xy a yx a 2 xx a xxa yy a yy a 3 xx a2 xx a xya yx ) a 2 yx (D )c e which, after multiplying by the (non-zero) denominators on each side and some algebra, yields: c e (D)(DT)(DT)(a xx a yy a xy a yx ) (A:2) where T/trace(A)/a xx /a yy. A solution to r x /r y for c e / requires that at least one of the other factors on the left hand side is zero. The stability criteria for a 2- by-2 system matrix in discrete time can be written: ½T½ BD (A:3a) and DB (A:3b) which means that the three first factors in Eq. A.2; ( D), (/D/T) and (/DT) are zero exactly on one of the stability boundaries given in Eq. A.3ab. The last factor in Eq. A.2, (/a xx a yy /a xy a yx ), requires some further investigation. It becomes zero as a xy a yx / /a xx a yy, which implies: D2a xx a yy (A:4) Inserting (A.4) into (A.3b) gives: a xx a yy B (A:5) Inserting (A.4) into (A.3a) gives: ½a xx a yy ½BD2a xx a yy (A:6) Eq. A.6 first of all implies that a xx a yy has to be positive, i.e. a xx and a yy are of equal sign. If they are positive, we get: a xx a yy Ba xx a yy (A:7) 2 The arithmetic mean of two numbers is always larger or equal to the geometric mean: a xx a yy p ] ffiffiffiffiffiffiffiffiffiffi a xx a yy ; 2 which contradicts Eq. A.7 since we have already concluded that stability requires a xx a yy B/, which pffiffiffiffiffiffiffiffiffiffiffiffi implies a xx a yy a xx a yy : The same sort of reasoning applies to the case when a xx and a yy both are negative. It follows, finally, that the last factor on the left hand side in (A.2) can not be zero as along as the stability criteria in (A.3) are met. In conclusion, Eq. A.2 is only fulfilled on or outside the stability boundaries of the system matrix, which means r x /r y (which led to A.2 in the first place) can not occur inside the stability region. Consequently, if r x B/r y for a single set of parameters inside the stability region it is true in the whole (continuous) region. A simple example is a / a 2 /a 2 /a 22 /, which gives r x / and r y /r 8. To summarize: Given two identical two-species communities without dispersal between them, where only one species has a correlated environment, the other species is always less synchronized. The only exception is when s x /, i.e. the species without environmental correlation has a constant environment, in which case both species are equally synchronized, r x /r y. Appendix 2. Correlation between non-identical communities Here we briefly describe how to calculate the correlations between the x and y populations governed by the dynamics in Eq. ab under the simplifying assumptions that the interaction is one-way (a yx /b yx / ), and the only difference between the communities is the intra-specific regulation of species y (a xx /b xx and a xy /b xy but a yy "/b yy ). The total system of four populations thus has the system matrix: a xx a xy a A yy B a xx a xy A b yy If it is further assumed that the environment of species y is the totally dominating stochastic fluctuation in each community, the covariance matrix of the four corresponding environments becomes: Var(8) r S 8 Var(8) B A r 8 Var(8) Var(8) Using the same methods as described above (Appendix ) to calculate population correlations gives Eq. 5ab in the main text. 792