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Visibility of demography-modulating noise in population dynamics

Per Lundberg and Jorgen Ripa, Dept of Theoretical Ecology, Lund Uni�., Ecology Building, SE-223 62 Lund,Sweden (per.lundberg@teorekol.lu.se). – Veijo Kaitala, Dept of Biological and En�ironmental Science, Uni�. ofJy�askyla, P.O. Box 35, FIN-40351 Jy�askyla, Finland. – Esa Ranta, Integrati�e Ecology Unit, Di�. of PopulationBiology, Dept of Ecology and Systematics, P.O. Box 17, FIN-00014 Uni�ersity of Helsinki, Finland

The abundance or density of all natural populationsis changing over time. A fundamental problem instudies of population renewal is to what extent theobserved changes reflect inherent, presumably density-dependent processes or whether they are largely inthe hands of environmental stochasticity. The debateis an old hat (e.g., Sinclair and Pech 1996), but it stillrevives now and then (Murray 1999, Turchin 1999)and has been revitalised by the recent research onhow changes in global climate may influence popula-tions (e.g., Alheit and Hagen 1997, Forchhammer etal. 1998, Tunberg and Nelson 1998, Post andStenseth 1999). The question of the visibility of de-mography and noise in ecological data does not onlyhave academic interest. It is also critical to all popu-lation management and conservation, as well as tohow we can detect and predict population responsesto environmental change. One problem is that rela-tively long time series of data are required for us tobe able to infer the underlying processes. Such dataare often rare in ecology and also hampered by errorsin sampling (Solow 1995). Second, the identity of theexternal noise is often not known. For example, is itthe long-term, region-wide changes such as the NorthAtlantic Oscillation (NAO) or the El Nino SouthernOscillation (ENSO) that is dominating (in a biologicalsense) the environmental variability, or is it moreshort-term and local stochasticity that is important?The answer to such questions calls for an understand-ing of how temporally structured (e.g., autocorrelated)environmental variability interacts with demography(e.g., Roughgarden 1975, Ives 1995, Ripa and Lund-berg 1996, 2000, Petchey et al. 1997, Ripa et al. 1998,Ripa and Heino 1999, Lundberg et al. 2000).

In our approach to this problem we assume boththe external disturbance and the modulated popula-tion dynamics to be known (Ranta et al. 2000,Laakso et al. 2001). Our task is now to find underwhat conditions the signature of the noise is visible inthe noise-modulated dynamics. Here, we shall startfrom the elementary parts of the renewal process ofany population, births and deaths. In an earlier study(Ranta et al. 2000), we made no assumptions aboutwhat parts of the demography that are affected byenvironmental noise. Likewise, the recent study byLaakso et al. (2001) focused on the correlation be-tween environmental noise and the resulting dynamicswithout distinguishing between different demographiccomponents. Here, we extend both those studies bybeing explicit about demography (yet omitting age orstage structure).

Noise, demography, and dynamics

The point of departure is the simple renewal function

Nt+1= f(Nt, mt) (1)

where N is population density and f is some functionmapping population density at time t to the next timestep. The renewal is hence density dependent and de-pendent on the environmental noise, m. We let thenoise term be a first-order autoregressive process(Ripa and Lundberg 1996)

mt+1=�mt+s�1−�wt+1 (2)

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where � is the autocorrelation parameter varying be-tween −1 (indicating negatively autocorrelated (‘blue’)noise) and 1 (indicating positively autocorrelated (‘red’)noise). wt is a random normal deviate with zero meanand standard deviation s. Hence, �=0 represents whitenoise. The square-root term in eq. (1) scales the vari-ance of m such that it is independent of the value of �(Ripa and Lundberg 1996).

Let us now make eq. (1) more explicit in terms ofbirths and deaths, and we have

Nt+1=Nt+Bt−Dt (3)

and when scaling births and deaths to per capita rates,�t=Dt/Nt and �t=Dt/Nt, we have

Nt+1= (1+�t−�t)Nt (4)

We let the per capita birth and death rates be bothdensity dependent and be dependent on environmentalvariability. This allows us to let births and deaths beindependently influenced by environmental noise. Wewill use the form of density dependence suggested byRipa and Lundberg (2000), such that eq. (4) becomesidentical to the well-known Ricker model. We hencehave

�t= [exp(r)−1]�

exp�

−rNt

Kt

�n�t=1−exp

�−r

Nt

Kt

�(5)

where r and K are parameters determining the shapeand magnitude of density dependence (see Ripa andLundberg 2000 for details). Substituting eq. (5) into eq.(4), we have Nt+1=Nt exp(r(1−Nt/Kt)), which is thedesired Ricker model. Note here that it is one of theparameters determining the per capita rates that isaffected by the environmental stochasticity; K. We letKt=K+mt, where K was set to 100. The parameter sin eq. (2) was then adjusted so that Kt fluctuatedbetween 50 and 100 (experimentation showed that theexact range of Kt is not critical for the results). We letthe noise affect either births, deaths or both. When oneof the processes was noisy, the other had Kt=K=100,and when births and deaths were affected simulta-neously, they had matching Kt.

We now used the above models to simulate popula-tion dynamics from which we were trying to find thesignature of the noise. This was achieved by first initiat-ing the renewal process by drawing N1 from a uniformdistribution of random numbers between 5 and 100.The simulations were run for 2000 generations, and thelast 1000 were used to calculate the relevant statistics.For this purpose we used the cross-correlation coeffi-cient (e.g., Chatfield 1984). The way we implement the

external forcing into the birth and/or death processmakes us suspect that the noise should be visible withlag −1 (Ranta et al. 2000; the signature, when visible,can also be seen with longer lags, but the correlationsare weaker).

Fig. 1. Cross-correlations (with lag −1; high positive valuesindicate here that the presence of the signal has been success-fully identified in the signal-modulated population dynamics)graphed against the noise colour � and the parameter r in theRipa-Lundberg modification of the Ricker equation. In (A)the noise is affecting births only, in (B) deaths only, while in(C) it affects with matching values both births and deaths.

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Visibility of the noise

The noise is visible only within a limited range of theparameter r (Fig. 1). When the parameter r exceedsapproximately 2, the cross-correlation between thenoise signal and the resulting dynamics starts to disap-pear. The loss of the visibility is, to some extent,affected by the colour of the noise: the visibility beingbetter for larger values of r with blue noise. Thedisappearance of the signature of the external forcing inthe noise-modulated dynamics takes place sooner(against increasing r) and is steeper if the noise modu-lates deaths only (Fig. 1A). When the noise influencesbirths or both births and deaths, the drop of visibility isshallower against r, and with blue noise it takes up tothe verge of chaotic dynamics until the visibility of thenoise is entirely lost (Fig. 1A, C).

Discussion

The possibility to detect the noise (i.e., the ‘environ-ment’) in ecological time series is contingent on anumber of factors. Apart from the magnitude (vari-ance), its own temporal structure (colour) relative tothe lag structure of the population itself is important(Roughgarden 1975, Royama 1981, Ripa and Heino1999, Ripa and Lundberg 2000, Turchin and Berryman2000). Although the elementary theory for this interac-tion was developed some time ago (Roughgarden 1975),it has largely been forgotten or ignored (Ripa andHeino 1999). The theory builds on linear analysis ofarbitrary, non-linear population models and it has beenshown that this approximation generally is very good(Ripa and Heino 1999). Here, we have used a fullnon-linear model, dissected into its demographic parts,and we can confirm that even under those circum-stances the problem of noise detection remains (Rantaet al. 2000).

As it turns out, knowing what basic part of thedemography that is most affected by the environmentalvariable in question does not help much in finding acorrelation between noise and dynamics, unless theskeleton dynamics is rather tame (i.e., r has not passedthe value 1.5). Interestingly, the colour of the noise hasonly minor influence on the correlation for low r-val-ues, especially if both births and deaths are noisy. Thisis a slightly different result from a previous study ofours (unpubl.) where blue (red) population dynamics(high r in this model) were positively correlated withblue (red) noise dynamics. In that study we used,however, a linear approximation instead of the truenon-linear and demographically more explicit modelused here. We therefore conclude that not only does itmatter exactly how the population model is formulatedand what assumptions are made about what part of thedemography that is affected, but also whether one

chooses approximations or not. This calls for cautionwhen working out the expected correlation between theenvironment and the population dynamics. Demo-graphic and environmental stochasticity are inevitablyintertwined with the density-dependent structure of agiven population (Lundberg et al. 2000, Ripa andLundberg 2000, Bjørnstad et al. 2001) and those rela-tionships have to be carefully worked out before anypredicted responses of the population to modulatingenvironmental variability are to be inferred.

Acknowledgements – Financial support for this study wasreceived from the Swedish Natural Science Research Council,the Swedish Research Council for Forestry and Agricultureand The Finnish Academy. Muikku at Ahvenranta made itpossible to complete this paper.

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