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Citation for published version

Bearup, Daniel and Petrovskii, S. and Blackshaw, R. and Hastings, A. (2013) Synchronized dynamicsof Tipula paludosa metapopulation in a southwestern Scotland agroecosystem: Linking patternto process. American Naturalist, 182 (3). pp. 393-409. ISSN 0003-0147.

DOI

https://doi.org/10.1086/671162

Link to record in KAR

http://kar.kent.ac.uk/64320/

Document Version

Author's Accepted Manuscript

Synchronized Dynamics of Tipula paludosa

Metapopulation in a South-Western Scotland

Agroecosystem: Linking Pattern to Process

Daniel Bearup, Sergei Petrovskii1

Department of Mathematics, University of Leicester,

University Road, Leicester LE1 7RH, U.K.

Emails: djb69@le.ac.uk, sp237@le.ac.uk

Rod Blackshaw

Centre for Agricultural and Rural Sustainability, University of Plymouth,

Drake Circus, Plymouth PL4 8AA, U.K.

Email: r.blackshaw@plymouth.ac.uk

Alan Hastings

Department of Environment Science and Policy, University of California at Davis,

Davis, CA 95616, USA.

Email: amhastings@ucdavis.edu

Keywords: metapopulation, synchronization, dispersal, Moran effect, T. paludosa

Publication type: article

Additional elements of the expanded online edition: Appendix

Figures to print in color: Figs. 1, 2 and 11

1Corresponding author. Phone/fax: +44 116 252 3916/3915, email: sp237@le.ac.uk

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Abstract

Synchronization of population fluctuations at disjoint habitats has been observed in

many studies but its mechanisms often remain obscure. Synchronization may appear as

a result of either inter-habitat dispersal or because of regionally correlated environmental

stochastic factors, the latter being known as the Moran effect. In this paper, we consider

the population dynamics of a common agricultural pest insect T. paludosa on a frag-

mented habitat by analyzing data derived from a multi-annual survey of its abundance

in 38 agricultural fields in South-West Scotland. We use cross-correlation coefficients and

show that there is a considerable synchronization between different populations across

the whole area. The correlation strength exhibits an intermittent behavior so that close

populations can be virtually uncorrelated but populations separated by distances up to

about 150 kilometers can have a cross-correlation coefficient close to one. In order to dis-

tinguish between the effects of stochasticity and dispersal, we then calculate a time-lagged

cross-correlation coefficient and show that it possesses considerably different properties

to the non-lagged one. In particular, the time-lagged correlation coefficient shows a clear

directional dependence. The distribution of the time-lagged correlations with respect to

the bearing between the populations has a striking similarity to the distribution of wind

velocities, which we regard as evidence of long-distance wind-assisted dispersal.

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1 Introduction

Understanding of population dynamics in complex environments has been one of the main

challenges both for theoretical and empirical ecology over the last few decades (Levin 1976;

Kareiva 1990; Lundberg et al. 2000). Environment is known to shape the geometry of

ecological interactions through a variety of specific spatial and spatiotemporal mecha-

nisms such as landscape structure (Pickett and Thompson 1978; Kaitala et al. 2001),

seasonality and solar cycles (Sinclair and Gosline 1997), and transient weather conditions

both on global and regional scale (Baars and Van Dijk 1984; Post and Forchhammer 2002;

Raimondo et al. 2004). The latter is usually regarded as environmental stochasticity or

noise; e.g. see Vasseur and Yodzis (2004) and references therein.

Landscape heterogeneity often results in a situation where populations of the same

species occupy disjoint habitats. Depending on the inter-habitat distance, individual

mobility and the nature of the environment between the habitats, e.g. how harsh it is, these

local populations may or may not interact with each other through dispersal. The classic

concept of metapopulation (Hanski and Gilpin 1991) refers to the case where the sizes

of local populations fluctuate independently, thus assuming that the dispersal coupling

between them can be neglected. In many cases, however, this is not the case and the

population fluctuations in different habitats appear to be, to a certain extent, correlated

(so that the metapopulation concept had to be updated accordingly, e.g. see Haydon

and Steen 1997; Sutcliffe et al. 1997). This phenomenon is known as synchronization,

and inter-habitat dispersal has been identified as a synchronizing factor (Liebhold et

al. 2004). There is considerable evidence that dispersal coupling by just a tiny fraction

of the population may bring population fluctuations into synchrony (Haydon and Steen

1997; Kendall et al. 2000; Ripa 2000).

Remarkably, dispersal coupling is not the only factor resulting in population synchro-

nization. The impact of spatially correlated environmental noise on disconnected popu-

lations can synchronize the population fluctuations too, the phenomenon being known as

the Moran effect (Moran 1953a,b; Royama 1992; Ranta et al. 1997). Having originally

been discovered theoretically (cf. “Moran’s theorem”), it has later been widely observed

in different taxa and in various environments (Liebhold at al. 2004). Synchronization of

population fluctuations can therefore be driven by the regional environmental stochastic-

ity, by the interaction between local populations through dispersal, or by a mixture of

both (Goldwyn and Hastings 2011). Without any impact from the Moran effect, dispersal

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is typically too weak a force to produce synchrony (Goldwyn and Hastings 2011), but the

Moran effect alone cannot produce very high levels of synchrony. What is less clear is the

relative importance of these two forces.

Synchronization has many implications across the whole range of ecological sciences.

Good understanding of patterns and mechanisms of synchronization is required in or-

der to efficiently manage issues arising in agro-ecology (Rosenstock et al. 2011), in pest

control (Milne et al. 1965; Blackshaw 1983; Williams and Liebhold 1995) and in nature

conservation programs (Earn et al. 2000). Identifying particular factor(s) resulting in

synchronization is therefore important. Indeed, linking an observed ecological pattern to

a specific process has been a major issue in contemporary ecology (Levin 1992). However,

since both dispersal and the Moran effect can have a similar impact on population dy-

namics, it is often very difficult to distinguish between them unless direct measures of the

effect of dispersal are possible. For example, observed synchronies in the yield of pistachio

trees must be due only to the Moran effect as there is no equivalent to dispersal in this

system and this is confirmed by models (Lyles et al. 2009). But a system where dispersal

can be eliminated as a force causing synchrony is rare, and differentiating the effects of

stochasticity from that of dispersal is sometimes regarded as one of the greatest challenges

to ecologists studying spatiotemporal population dynamics (Liebhold et al. 2004).

Another challenging problem is to identify the corresponding spatial scale of the mecha-

nisms involved. For species with low mobility, the scale of synchronization due to dispersal

is known to be smaller than the scale induced by the regional stochasticity. In particu-

lar, in a field study on butterflies, Sutcliffe et al. (1996) showed that the spatial scale of

dispersal coupling is on the order of 5 kms while population synchrony can be observed

on much larger distances of up to 200 kms. The larger spatial scale of synchronization

is therefore likely to be linked to regional stochasticity, although this may not always be

true if insect dispersal is assisted by the wind. Synchronization of population dynamics

has been observed for many other insect species (Baars and Van Dijk 1984; Hanski and

Woiwod 1993; Sutcliffe et al. 1996; Peltonen et al. 2002), although the specific factors re-

sponsible for synchronization were not always clear. Other striking examples of patterns

of synchrony come from the dynamics of childhood diseases (Rohani et al. 1999). This

study shows the importance of interactions between dispersal and dynamics since two

diseases, measles and pertussis, show very different spatiotemporal patterns even though

the dispersal parameters must be the same.

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In this paper, we consider how the spatial pattern observed in population dynamics of

an insect species dwelling on a habitat consisting of an array of agricultural fields can be

affected by the landscape properties and by the weather conditions. The focus of this study

is on T. paludosa which is a common pest in the British Islands and can cause significant

damage to agriculture (Blackshaw and Coll, 1999). For this reason, its dynamics has been

a focus of numerous field studies (Milne et al., 1965; Mayor and Davies, 1976; Blackshaw,

1983) as well as some theoretical work (Blackshaw and Petrovskii, 2007).

In order to address the issues of pattern, process and scale, we have analyzed annual

data on the population abundance of T. paludosa collected between 1980 and 1994 in a

few dozen agricultural fields across South-Western Scotland. Analysis of the time series of

population density obtained at different locations across the region shows that the fluctua-

tions in the abundance of T. paludosa are not independent. However, the synchronization

pattern that we have observed exhibits some rather counter-intuitive properties. There

is a considerable degree of synchronization between some of the fields but an absence

of synchronization between others. Note that the generic dependence of synchronization

on the inter-habitat distance is well known, with the degree of synchronization between

different populations usually decreasing with distance (e.g. Sutcliffe et al. 1996; also Lund-

berg et al. 2000; Liebhold et al. 2004). However contrary to this, synchronization between

T. paludosa abundances in different fields does not show any clear relation to the inter-

field distance. Furthermore, we show that the observed synchronization pattern has a

distinct directional aspect. In particular, while the north-west and south-east areas of

the region are on average strongly correlated, there is much less correlation between the

north-east and south-west areas. By linking this directional asymmetry to weather data,

we show that it is likely to result from wind-assisted dispersal. In order to distinguish the

effect of dispersal from that of stochasticity, we introduce a delay-based cross-correlation

coefficient and show that it exhibits a pattern of directional dependence very similar to

that of the wind velocity.

2 Methods

2.1 Species

Tipula paludosa Meig., the marsh cranefly, are found in cool temperate regions of northern

Europe and the USA (Jackson and Campbell 1975). T. paludosa is univoltine with peak

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adult emergence between mid-August and mid-September in the UK. Mature T. paludosa

are flying insects and thus are found in a range of environments. However, dispersal is

thought to be limited as females are poor fliers, emerging gravid and typically laying

eggs within twelve hours of emergence (Blackshaw and Coll 1999). The larval stages of

T. paludosa, known as leatherjackets, are soil dwelling and relatively lacking in mobility;

consequently they are restricted to the locality in which they hatch. They feed primarily

on the roots and stems of grasses and cereals, although they are also able to consume

a variety of other crops (Blackshaw and Coll 1999). Leatherjackets are considered an

agricultural pest, although it is relatively rare for them to destroy a sward. A substantial

body of work postulates that leatherjacket populations can be affected by environmental

conditions, specifically by the average rainfall prior to hatching. However, a significant

effect (that may even result in a population crash) is only seen when weather conditions

in September/October are exceptionally dry (Milne et al. 1965). On the other hand,

temperature was shown to have little effect on T. paludosa dynamics (Blackshaw and

Moore 2012). Correspondingly, other studies suggest that the environmental factors may

be less significant than the effects of population density, especially in a harsh environment.

In particular, contest competition through combat between leatherjackets is posited as

the mechanism for this population regulation (Blackshaw and Petrovskii 2007; Petrovskii

and Blackshaw 2003).

2.2 Population data

Populations of T. paludosa larvae in Scottish farmland were surveyed annually (usually in

January/February) between 1975 and 1994. The results as a whole remain unpublished,

a part of the data was earlier used by Blackshaw and Petrovskii (2007). Population

counts were obtained from soil cores extracted from individual fields (20 cores per field),

which is a common technique for soil zoology (cf. Mayor and Davies 1976). The details

of sampling (such as the time of sampling, the core’s volume etc.) were consistent across

the survey and as such all counts obtained are comparable. For each field, the mean

number of insects per core was calculated and we assume that these numbers provide

absolute estimates of the population levels at the time of sampling. A total of 83 fields

were sampled over the course of the survey period. However, most fields were sampled

for less than the full twenty years. In particular, sampling did not begin in the same year

for all fields. Furthermore, not all of the time series are complete for the period studied;

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for some years, the count is not available.

The purpose of this study is to investigate the synchronisation of local populations,

i.e. the populations in individual fields. In order to obtain a detailed view of the spatial

aspect, it is desirable to include as many fields as possible. On the other hand, the

accuracy of estimates of the degree of synchronisation will be dependent on the length of

the time series used. Consequently it is preferable both that time series cover the greatest

duration possible and that as many different time series as possible are used.

Given these limitations, it is not possible to use all fields surveyed. A compromise

between number of fields and length of time series has to be found. Consequently, we

restrict our analysis to a subset of 38 fields for which a complete fifteen year time series

between 1980 and 1994 is available. A map of the fields in question can be seen in Fig. 1.

Information about the minimum, maximum and time-average size for each of the local

populations (as represented by the mean value across the collected soil cores, see above)

is given in Tab. 1. It is readily seen that the populations exhibit considerable variability.

More detailed data on the T. paludosa population size in five particular fields over the

given period, 1980-94, are shown in Fig. 2. A visual inspection of the data reveals a

certain degree of correlation between the time series, such as, for instance, a decrease in

all five population sizes between 1987-88, a minimum population size in 1985 and 1993

(three out of five fields), an increase in the population size between 1993-94 (four out of

five fields), etc. A quantitative insight into this is made below.

2.3 Elimination of population density dependence

Ninety-seven percent of the populations included in the study (i.e. 37 out of 38 fields)

display significant population density dependence when subjected to the test outlined by

Pollard et al. (1987). The effects of this density dependence on local population dynamics

may obscure the effects of synchronisation between these populations. Consequently it is

desirable to eliminate these density dependent effects from the local population dynamics.

The following model:

∆Nt = Nt+1 −Nt = Nt

[

10α(

N

Nt

)−β

− 1

]

(1)

(where N is the average population across the entire study area and α and β are to be

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determined), which we write as

Nt+1 = f(Nt) = Nt · 10α

(

N

Nt

)−β

, (2)

was introduced by Blackshaw and Petrovskii (2007) to describe these effects.

The parameters α and β can be determined by linear regression of log((∆Nt/Nt) + 1)

against log(Nt/N) for a given time series. Density dependence is expected to be a species

property and as such the same parameters apply for each individual population. Thus

we determine these parameters for the time series obtained by computing the average

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Figure 1: Map showing the locations of fields, indicated by balloon markers, included in this

study. Numbers correspond to those in Tab. 1. The flag markers show the positions of weather

stations from which data were obtained. The arrows provide an approximate location of the

corner points of the plots used in Sections 3.4-3.5 (Figs. 8 to 11). Barren and rocky areas are

shown in various shades of brown, while green shading is an indicator of grass cover. Relative

height of terrain is indicated by the three dimensional effect.

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Table 1: Minimum, maximum and the time-average of population counts for each field. Field

numbering corresponds to that used in Fig. 1.

Num Grid Ref Min Max Average Num Grid Ref Min Max Average

1 NY 416779 0 16 4.33 20 NX 693537 0 16 4.07

2 NS 114668 2 41 12.33 21 NY 426783 0 16 6.80

3 NX 716619 0 13 3.67 22 NY 394758 2 21 7.07

4 NX 724573 0 15 3.93 23 NS 369400 0 25 6.00

5 NY 049748 0 22 4.73 24 NS 659469 1 48 9.53

6 NX 071534 2 23 6.93 25 NR 661199 0 16 5.60

7 NX 057556 1 24 6.53 26 NS 324683 0 20 3.93

8 NY 101852 0 11 2.20 27 NS 265446 0 32 11.53

9 NS 639691 0 23 4.13 28 NS 562727 1 17 6.33

10 NS 402392 0 13 3.53 29 NX 743601 0 17 4.60

11 NS 278435 1 20 8.07 30 NX 839734 0 20 4.47

12 NR 698229 0 12 6.33 31 NS 053638 1 41 12.27

13 NX 008680 0 33 5.53 32 NS 111703 1 32 8.20

14 NS 452152 2 15 7.73 33 NS 385235 1 36 7.93

15 NX 093520 0 8 4.07 34 NS 440289 0 20 6.00

16 NX 463378 0 24 9.47 35 NS 671956 0 34 6.73

17 NS 412331 0 27 6.60 36 NN 943236 0 8 2.33

18 NX 377452 0 11 4.53 37 NS 129682 0 23 5.67

19 NS 046674 1 24 8.87 38 NY 203711 0 24 6.47

population across all fields at each time point. The resulting data are plotted in Fig. 3.

Note that the resulting values of α and β correspond well to those obtained in previous

studies (Blackshaw and Petrovskii 2007).

Now for each time series of populations, X, we can compute the difference between the

observed and predicted populations for any time point as follows:

RXt = Xt − f(Xt−1), (3)

(where t = 2, 3, . . . , n) to obtain a time series of residuals, RX , one time step shorter than

the original observed data. These residuals measure the degree to which real populations

deviate from the levels predicted by the density dependent mechanism either by chance

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ea

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rja

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t co

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t

East most field

West most field

North most field

South most field

Center most field

Mean

0

5

1980 1982 1984 1986 1988 1990 1992 1994

Le

Year

Figure 2: Plot of population counts against time for five example fields. Fields were chosen

on the basis of geographic position; specifically, four are on the edges of the study area (field

no. 36 at the North, no. 21 at the East, no. 16 at the South and no. 25 at the West) and the

fifth is in the centre (field no. 14). The mean count across all thirty-eight fields in the study was

determined for each year and is also plotted (black line).

or due to underlying processes which are not described by internal population regulation.

A histogram of all such residuals was constructed, see Fig. 4 (left), which strongly

suggested that the residuals arise from a log-normal distribution. To confirm this intuition,

a Q-Q plot of the residual distribution against the log-normal distribution was constructed,

LNR = exp(N (1.9, 0.537))− 7.217; see Fig. 4 (right). It is clear that the majority of the

quantiles plotted lie on the line y = x so the residual distribution is approximately given

by the log-normal distribution stated.

Thus we can describe the population dynamics of these populations with the following

stochastic difference equation:

Nt+1 =

f(Nt) + ηR, if f(Nt) + ηR > 0,

0, otherwise,(4)

where ηR is a random variate drawn from the distribution LNR.

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−0.5 0.0 0.5

−1.

0−

0.5

0.0

0.5

1.0

log(

∆Nt

Nt+

1)

log(Nt N)

Figure 3: Log-log plot of the average per capita population increase against average

relative population (crosses). The solid line was obtained by linear regression analysis of

these data and has the intercept α = −0.0335 and the slope −β = 0.8903. The R2 value

of this regression is 0.42.

−10 0 10 20 30 40

0

20

40

60

80

100

Residual value

Fre

quen

cy

−10 0 10 20 30 40 50

−10

010

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Theoretical Quantiles

Sam

ple

Qua

ntile

s

Figure 4: Left: Histogram of residuals as obtained after removing density dependence

from the original data; see Eq. (3). Right: Q-Q plot of the residual distribution against

the log-normal distribution given by exp(N (1.9, 0.537))− 7.217.

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0625

0626

0627

2.4 Finding relationships between the local populations

We quantify the degree of synchronization between the populations of two fields (say,

X and Y ) by calculating the Pearson product-moment correlation coefficient of their

respective residual time series, RX and RY . This value is given by the following expression:

r0(RX , RY ) =

∑n

i=2(RX

i − µXR )(R

Yi − µY

R)√

(∑n

i=2(RX

i − µXR )

2) (∑n

i=2(RY

i − µYR)

2), (5)

where RXi and RY

i are the residual population densities at year i in fields X and Y ,

respectively, and µXR and µY

R are the sample means of the two time series, i.e.,

µXR =

1

n− 1

n∑

i=2

RXi , µY

R =1

n− 1

n∑

i=2

RYi . (6)

It is not immediately clear, however, what constitutes a statistically significant correla-

tion coefficient. Since we work with time series of finite length, and the data are affected

by stochastic factors, any given value of the correlation coefficient (5) may appear by

chance. One must therefore distinguish between the cases when high absolute values of

r0(RX , RY ) are superficial and the cases when these values are the result of actual syn-

chronization. In order to do so, the population data were subjected to a careful statistical

analysis; full details of the analysis are given in the online Appendix.

2.5 Assessing the effects of time delay

The correlation coefficient (5) is not capable of fully explaining all possible relationships

between two field populations. For example, if the fields are coupled by dispersal, then

the corresponding biological mechanisms may be subject to time delay. The population

census during the survey was done in winter, i.e. before the species enters its mobile

(flying) stage. Therefore, the effect of dispersal coupling, if any, will only be seen in the

next year census.

Obviously, the effects of delay are not taken into account by the standard correlation

coefficient r0. In order to identify such relationships (for a single generational delay of

one year), we introduce a delay-based correlation coefficient which is calculated between

two time series, A and B, as follows:

r1(A,B) =

∑n−1

i=1(Ai − µA,1)(Bi+1 − µB,1)

√

(∑n−1

i=2(Ai − µA,1)2

)

(∑n

i=2(Bi − µB,1)2)

, (7)

12

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0631

0632

0633

0634

0635

0636

0637

0638

0639

0640

0641

0642

0643

0644

0645

0646

0647

0648

0649

0650

0651

0652

0653

0654

0655

0656

0657

0658

0659

0660

0661

0662

0663

0664

0665

0666

0667

0668

0669

0670

0671

0672

0673

0674

0675

0676

0677

0678

0679

0680

0681

0682

0683

0684

where µA,1 and µB,1 are defined as follows:

µA,1 =1

n− 1

n−1∑

i=1

Ai , µB,1 =1

n− 1

n∑

i=2

Bi . (8)

We emphasize that, generally speaking, r1(A,B) 6= r1(B,A). Therefore, the delay-

based correlation coefficient separates the effect that the populations of field A have on

field B (described by r1(A,B)) from the effect that the population of field B may have

on A (described by r1(B,A)). In other words, it takes into account a possible asymmetry

in the inter-field coupling. Such asymmetry can occur, for instance, when insect dispersal

is assisted by the wind (Gatehouse 1997; Compton 2002) in case wind has a prevailing

direction. This kind of asymmetry reflects what is essentially a traveling wave as has been

observed in the dynamics of childhood diseases (Rohani et al. 1999).

It remains then to determine which time series this coefficient should be calculated

for. An immediate analogue of the comparison used in the undelayed case would use

r1(RX , RY ). This would then compute the degree to which deviations from internal

dynamics of one population affect deviations from the internal dynamics of another pop-

ulation. However, here we argue that this is not a meaningful measurement. Instead we

are interested in how the absolute population in one location (regardless of whether it

is higher or lower than internal dynamics would predict) may affect the population dy-

namics at another location. The strength of this relationship should be described by the

value of r1(X,RY )2. Note that in this case the asymmetry lies not only in the correlation

coefficient but in the series which are compared.

3 Results

3.1 The effect of distance on strength of correlation

Several earlier studies have shown that there exists a clear “synchrony versus distance

pattern” where the correlation between population abundances tends to decrease as the

distance between the populations increases (Sutcliffe et al. 1996; Lundberg et al. 2000;

Peltonen et al. 2002). A reasonable initial hypothesis therefore seems to be that popu-

lations in fields which are close together are more likely to exhibit synchronization than

populations that are spatially more separated. In order to investigate this hypothesis,

2Values computed in this way are referred to as r1 from now on.

13

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0701

0702

0703

0704

0705

0706

0707

0708

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0738

0739

0740

0741

the correlation coefficients obtained for each pair of fields are plotted against the distance

between the fields; see Fig. 5, top-left for r0 (no time delay), top-right for r1 (time delay

of one year).

A visual inspection, however, does not reveal any clear pattern. We observe that,

indeed, some fields are strongly synchronized up to distance of 150-170 kms; this hap-

pens both with and without time delay. There are also fields that are significantly anti-

Figure 5: Plots of correlation coefficients for pairs of distinct fields against distance between

those fields. The top-left plot shows all the correlation coefficients calculated without a time

delay (as given by Eq. 5), the top-right plot uses a time delay of 1 year (Eq. 7). The bottom row

show the correlation coefficients that are statistically significant at 5% level, left without time

delay, right with a delay of 1 year. The solid lines indicate trends predicted by linear regression

analysis.

14

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0748

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0750

0751

0752

0753

0754

0755

0756

0757

0758

0759

0760

0761

0762

0763

0764

0765

0766

0767

0768

0769

0770

0771

0772

0773

0774

0775

0776

0777

0778

0779

0780

0781

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0793

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0797

0798

correlated. On the other hand, for any inter-field distance (including the cases of apparent

proximity), there are fields that are not correlated at all. On the whole, the plots for r0

and r1 are broadly similar3, although stronger negative correlations are observed for the

time delay case. We therefore conclude that, as such, the absolute distance between fields

cannot be the only controlling factor in whether the populations of two fields synchronize.

Hence, more information about the ‘geometry’ of the environment has to be taken into

account.

This qualitative understanding can be made more rigorous using linear regression anal-

ysis. The results are plotted as solid lines in Fig. 5. Interestingly, the regression analysis

of correlation with respect to distance reveals different behaviour between the two cases.

In the no-delay case, the correlation strength shows a tendency to decrease as distance

increases. The gradient of the slope is small (approximately −0.0074), but statistically

significant with a p-value below 0.01. The correlation coefficient predicted by the best-

fitting line is about 0.2 at small distances and it approaches zero for distances on the order

of 300 kms. In the time-delay case, the correlation strength is not significantly different

from 0 for any distance range.

The existence of the inter-field coupling over the whole area becomes even more ev-

ident if the correlation-versus-distance analysis is restricted to statistically significant

values only. The results are shown in the bottom row of Fig. 5. In the no-delay case all

the significant correlation coefficients appear to be positive and have a relatively large

value between 0.5 and 1.0 (Fig. 5, bottom, left). In the time-delay case, all the statis-

tically significant correlation coefficients are large and negative (Fig. 5, bottom, right).

Surprisingly, in neither of the two cases does the correlation strength show any decay with

distance; on the contrary, the two best-fitting lines have gradients of 0.001 and −0.0029

respectively, but these values are statistically not significantly different from zero.

3.2 The effect of direction on strength and sign of correlation

In the previous section, we showed that the population dynamics of T. paludosa in the

study area is synchronized over large distances, although the inter-field distance alone

provides a rather poor description of the synchronization pattern, especially in the time-

delayed case (see Fig. 5, top, right). More details of the synchronization pattern can be

obtained if we consider the relative positions of fields whose populations are correlated.

3Note that the time delay case includes twice as many points as the no delay case.

15

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0803

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0841

0842

0843

0844

0845

0846

0847

0848

0849

0850

0851

0852

0853

0854

0855

Figure 6: Relative frequency of positive and negative correlations as a function of bearings

between fields. Left: positive correlation without time delay between residual populations,

Eq. (5). Right: negative correlation with a one year time delay between population and residual

population, Eq. (7). The radial distance between the center of the circle and the thick broken

line gives the relative frequency of positive (left) or negative (right) correlations (as a fraction

of unity) within the given bearing range; see details in the text.

This can be described in part by the directions of the lines connecting any pair of fields,

i.e. by considering the bearing of one field from another.

A quantitative insight into how relationships between populations vary with respect

to bearing can be made by considering the relative frequency of positive and negative

correlation values in a given bearing range. That is given a bearing range, 0-10◦ for

example, we divide the number of positive (or equivalently negative) correlation values

obtained in that range by the total number of correlation values obtained in that range.

The resulting histograms for correlations computed with and without a time delay are

shown on a unit circle in Fig. 6. Note that, in order to take into account the different

tendency observed in the delayed and non-delayed cases (as seen from Fig. 5, top), for

r0 we present the fraction of positive correlations obtained, while for r1 we present the

fraction of negative correlations obtained.

It is readily seen that, when no time delay is considered, the relative frequency of

positive and negative correlations is approximately independent of bearing, and positive

relationships (shown by the thick solid line in Fig. 6, left) are clearly more common over

16

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0868

0869

0870

0871

0872

0873

0874

0875

0876

0877

0878

0879

0880

0881

0882

0883

0884

0885

0886

0887

0888

0889

0890

0891

0892

0893

0894

0895

0896

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0901

0902

0903

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0905

0906

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0908

0909

0910

0911

0912

the whole circle. In contrast, when a time delay is introduced, the relative frequency

of positive and negative correlations is strongly dependent on the bearing between the

fields. Negative correlations (shown by the thick solid line in Fig. 6, right) prevail in the

range of bearings between south-east and due north, while in the remainder of the range

positive correlations dominate. We therefore conclude that the synchronization pattern

has a clear directional aspect.

3.3 Analysis of weather pattern

Weather conditions are known to have a significant impact on the population dynamics of

many insect species (Baars and Van Dijk 1984; Williams and Liebhold 1995; Raimondo et

al. 2004). In particular, the direction and strength of the wind were proved to be factors

strongly affecting dispersal of flying insects (Gatehouse 1997; Compton 2002). Aiming to

explain the observed directional asymmetry in the synchronization pattern, we therefore

now turn our attention to the weather pattern. Indeed, synchronization between local

populations is known to result from either the Moran effect or dispersal coupling between

habitats. It seems improbable that the regional stochasticity (e.g. short-term stochastic

variation in the local temperature and humidity) could possess a directional aspect. In

contrast, the existence of a prevailing wind direction, if any, obviously can make the

coupling asymmetric.

Measurements of wind velocity were recorded at four weather stations spanning the

study area (see Fig. 1), i.e. Dundrennan, Abbottsinch, West Freugh and Tiree. The time

series are obtained from a single year, 1987, which we assume to be representative for the

whole duration of the survey. The data cover a two month period, August to September.

This period corresponds to peak adult emergence of T. paludosa and thus is the time in

which interactions between dispersed populations are most likely. The data4 comprised

hourly measurements of the average wind speed and the wind direction.

Some signs of environmental heterogeneity within the study domain can be immediately

identified from the distributions of average windspeed; see Fig. 7. The weather stations at

West Freugh and Tiree tend to report higher windspeeds than the less exposed stations at

Dundrennan and Abbottsinch, presumably because higher windspeeds can be sustained

over the relatively flat surfaces of the surrounding seas and wide stretches of water. In

contrast, the wind direction appears to be less influenced by local terrain as is shown by

4Original data supplied by the Met Office; http://www.metoffice.gov.uk

17

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0952

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0956

0957

0958

0959

0960

0961

0962

0963

0964

0965

0966

0967

0968

0969

Tiree

Average windspeed (m/s)

Den

sity

0 5 10 15

0.00

0.10

0.20

0.30

West Freugh

Average windspeed (m/s)

Den

sity

0 5 10 15

0.00

0.10

0.20

0.30

Abbottsinch

Average windspeed (m/s)

Den

sity

0 5 10 15

0.00

0.10

0.20

0.30

Dundrennan

Average windspeed (m/s)

Den

sity

0 5 10 15

0.00

0.10

0.20

0.30

Figure 7: Histograms of average wind speed and wind direction over the study period at four

weather stations. Wind direction histograms are in polar form and are laid out in the same

order as those of average wind speed.

18

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1006

1007

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1011

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1013

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1016

1017

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1021

1022

1023

1024

1025

1026

the comparison between Fig. 7 and Fig. 1. The stations at West Freugh, Abbottsinch and

Dundrennan suggest that winds predominantly blow either due south or between south

east and north, at Tiree the separation between these two modes is less pronounced. In

general, there is very little wind blowing in the south to north-west sector. This is in

strikingly good agreement with the pattern shown in Fig. 6, right.

3.4 Spatial cross-correlations with no time delay

In Sections 3.1 and 3.2, we quantified the cross-correlations between fields by pooling

all correlations together. In particular, the existence of correlations between the local

populations (with no time-delay) across the region “on average” was shown by finding

the best-fitting line to the whole array of pairwise correlation coefficients versus distance;

Figure 8: The height of the bar at a given field indicates the number of populations with

residuals significantly correlated to that field’s residual population (ranging from between 0 and

10); further details are given in the text. The labels on the axes shows relative position in tens

of kilometers. The inset (top-left corner) shows a weather station that is situated outside of the

domain.

19

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1038

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1069

1070

1071

1072

1073

1074

1075

1076

1077

1078

1079

1080

1081

1082

1083

Figure 9: Top: The arrow at a given field indicates the average direction to fields with residual

populations that are significantly correlated to the resident residual population. Bottom: The

dotted lines connect fields with significantly correlated residual populations. Further details are

given in the text. Scales indicate relative position in tens of kilometers. The inset (top-left

corner) shows a weather station that is situated outside of the domain.

20

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1101

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1111

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see Fig. 5, left. The existence of a directional aspect (to the time-delayed relationships)

was revealed by plotting a histogram of (weighted) correlation coefficients on a circle,

i.e. as a function of the bearing (Fig. 6). However, these cumulative properties obscure

the role of individual fields. Meanwhile, revealing the contribution of individual fields

may be important for better understanding the process behind the observed pattern. The

contribution from different pairs of fields to the “synchronization vs distance” pattern, as

given by Fig. 5, varies significantly regardless of the distance between them, and hence

one might wish to understand why. Besides, the terrain in the study region is highly

heterogeneous. It includes hills, valleys, plains, urban areas, as well as some considerable

stretches of water (e.g. Firth of Clyde). Different local populations are therefore exposed

to quite diverse environments that vary both in terrain and in weather conditions.

In order to analyse the impact of precise positional relationships between fields, we

now consider all fields individually. In this section, we consider the correlations calcu-

lated without time delay (see Eq. 5). The results are summarized in Fig. 8. For each

given field X, we count the number of the fields where the residual population dynamics

of T. paludosa is significantly correlated (at the 5% significance level) to the residual pop-

ulation dynamics in X. Note that, in this case, all the significant correlation coefficients

had positive sign; see Fig. 5 (bottom, left). The result is shown by the length of a bar

based at the location of each field. The shortest bar (with the length 0.5 unit on the map

scale) means that one other population is significantly correlated to this population. The

largest number of fields significantly correlated to a given one is found to be 10 (with the

length of the corresponding bar thus being set to 5 in map units). For the three fields that

do not have a significant correlation to any other field in the array, the bar has length 0,

and so their position is shown by a dot.

It is readily seen that the map shows a clear divide between parts of the region above

and below the dashed line (which has been included for ease of comparison). In the North-

East region, above the dashed line, most of the populations are correlated with a large

number of other populations (4.27 on average). In contrast, in the South-West region,

below the dashed line, populations are correlated to significantly fewer populations (1.75

on average).

The information shown in Fig. 8 is, however, incomplete until the position of the mu-

tually correlated fields is known. Consider a hypothetical case that field X is significantly

correlated to fields A, B and C; one then might wish to know where fields A, B and

21

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C are situated with respect to X. Figure 8 is therefore complemented with Fig. 9. In

the top plot in Fig. 9, the arrows (now normalized to a unit length) indicate the average

direction to all fields (say, A, B and C) whose populations are significantly correlated to

that in the given one (say, X). If a given field is correlated to just one other field, then the

corresponding arrows point towards each other. In the lower plot in Fig. 9, any significant

relationship between two populations is indicated by a dotted line. Whereas the upper

plot can be considered a summary of the relationships between a field and the remaining

populations, the lower plot provides an overview of the networks of relationships formed

between these populations. Since the weather conditions are expected to be important,

in order to give a visual idea about the impact of the wind velocity, the position of each

weather station is indicated by the intersection of two dashed lines. The solid lines orig-

inating from this intersection replicate the histograms of wind direction given in Fig. 7.

Interestingly, apart from a few exceptions, the strongly correlated fields in the North East

region appears to be mostly correlated between themselves forming a dense network which

largely exlcudes fields in the South West. The fields in the South-West region form a sim-

ilar, if more sparse, network, with limited interconnections with the North-East network.

This provides further evidence of the existence of a division between the North-East and

South-West regions.

The results shown in Fig. 9 confirm that geographical proximity does not appear to

be a factor controlling what fields are correlated. It is readily seen that there are many

situations when a field is not correlated to its immediate neighbour(s) but is significantly

correlated to fields much further away.

3.5 Spatial cross-correlations with a time delay

In the previous sections, we have shown that the standard correlation coefficient (5) is

not able to describe all possible relationships between two fields; e.g. see Figs. 5 and 6. A

dispersal mechanism is likely only to be detected when a time delay is introduced between

two populations. An analysis of the relationships between influencing populations and

influenced residual populations with a generational delay (of a single year in this case)

can be performed in a similar way to that presented in the last section. Again we limit

our attention to those relationships which are significant at the 5% level. The plots below,

see Figs. 10 and 11, are therefore analogous in concept to Figs. 8 and 9.

As we previously observed, the time delay introduced in (7) breaks the symmetry

22

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1254

implicit in the standard correlation coefficient (5). Consequently it is possible to consider

two classes of relationships between a given population and the other populations. The

first class is the set of residual populations which, when delayed by one year, are correlated

to a given population, i.e. the set of populations influenced by a given population. The

second class is the set of populations which are correlated to a given residual population

when it is delayed by one year, i.e. the set of populations influencing a given population.

In Fig. 10, the number of populations influenced by and influencing a given field pop-

ulation is shown by, respectively, the length of a bar above and below the field position

(indicated by a plus sign). The scale of these bars is as described in the previous section.

Note that, in contrast to the relationships presented in the previous section, all of the

statistically significant time lagged relationships now have negative coefficients; see Fig. 5

(bottom, right).

Figure 10: The length of the bar descending (rising) from a given field position indicates the

number of fields with populations significantly influencing (influenced by) that field. Further

details are given in the text. Scales indicate relative position in tens of kilometers. The inset

(top-left corner) shows a weather station that is situated outside of the domain.

23

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Figure 11: Top: The red arrow at a given field indicates the average direction to all fields whose

residual populations are significantly influenced by that field. The blue arrow corresponds to

the average direction to all fields whose populations have a significant influence on that field’s

population. Bottom: The dotted lines connect fields between which significant relationships

exist. The red section of each line emanates from the influencing field, the blue section terminates

at the influenced field. Further details are given in the text. Scales indicate relative position in

tens of kilometers. The inset (top-left corner) shows a weather station that is situated outside

of the domain.

24

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Figure 12: Distribution of bearings from significantly influencing to influenced populations

represented as a histogram. The solid black line indicates the density of significant bearings in

the range shown. The solid gray line shows the overall distribution of bearings within the study

area.

As in the previous section we add reference lines to divide the study area into three

regions, referred to as South-West, Central, and North-East. The average number of fields

influenced by a population in any given region is (approximately) constant across the three

regions, with values of 1.38, 1.94, and 1.54 in the South-West, Central and North-East

regions respectively. In contrast the average number of populations influencing a given

population varies significantly across these regions taking values of 0.125, 1.12 and 3.38

in the South-West, Central and North-East regions respectively. Thus populations in the

South-West are more likely to influence other populations than to be influenced themselves

while populations in the North-East show the opposite trend. Populations in the Central

region influence other populations and are influenced themselves at roughly equal levels.

Similarly to the previous section, we complement these data with plots showing the

directions between fields and the populations that they influence or are influenced by,

Fig. 11. In the top plot of this figure red arrows indicate the average direction to residual

populations which a given population significantly influences. Blue arrows indicate the

average direction to populations which significantly influence a given residual population.

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In the lower plot all significant relationships are represented by a two colour line connecting

the fields between which the relationship exists. The red section of this line connects to

the influencing field while the blue section of the line connects to the influenced field.

Including the positional information in our analysis of this pattern again produces

further understanding. The strong correlations within regional groupings observed in the

undelayed case are, for the most part, absent. Instead South-West populations appear

to influence Central populations which in turn influence North-East populations in an

apparent cascade; see, for example, the relationships between fields along the southern

edge of the study area. Similar interactions can be seen between the populations of the

northern corner and those to their south-east although it is less pronounced.

The distribution of bearings from significantly influencing to influenced is presented as

a histogram in Fig. 12 (black line). It is clear that this distribution deviates noticeably

from the underlying distribution of bearings between fields in the study area (shown by the

grey line); therefore, the observed directional asymmetry cannot be reduced to the effect

of the system geometry. The most significant deviations from this underlying distribution

lie between south east and north. Since all of these relationships have negative correlation

coefficients this is in good agreement with the trend observed in Fig. 6. Furthermore, it

corresponds well to the pattern shown in Fig. 11.

4 Discussion and Conclusions

In this paper, we have considered the population dynamics of an insect pest, T. paludosa,

on a habitat consisting of 38 agricultural fields in South-Western Scotland. The annual

data on population abundance collected in a survey accomplished during 1980-94 were

analysed. Our goal is threefold. Firstly, we want to reveal whether there is a correlation

between the fluctuations in local populations, the phenomenon known as synchronization

(Lundberg et al. 2000; Liebhold et al. 2004). Secondly, we want to reveal the corresponding

spatial pattern, i.e. how the degree of synchronization between fields changes in space, in

particular, with inter-field distance. And thirdly, we want to understand the process(es)

resulting in the observed synchronization pattern, i.e. to relate the observed properties of

the T. paludosa metapopulation to a specific mechanism or factor.

The first goal is relatively simple to reach. We have calculated all cross-correlation

coefficients as given by r0, see Eq. (5), and found that only a small fraction of fields is

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uncorrelated or weakly correlated (say, |r0| < 0.1). A majority of population pairs are

positively correlated, in many cases r0 being as large as 0.5 or even higher (see Fig. 5,

top, left). There is also a considerable number of pairs that are negatively correlated with

typical values of r0 between −0.1 and −0.3.

With regard to the spatial pattern, no matter whether synchronization is due to dis-

persal or the Moran effect, it is reasonable to suppose that the cross-correlation coefficient

should decrease with the distance (e.g. Liebhold et al. 2004). We have shown, however,

that this expectation is rather over-simplified. While the best-fitting line obtained by us-

ing the linear regression analysis indeed has a negative slope (see Fig. 5), a closer look at

the correlation strength versus distance immediately reveals that this ‘prediction’ about

the decay in synchronization is rather superficial. Instead, synchronization exhibits an

intermittent behaviour: For any range of the inter-field distances, from very small (a few

kms) to very large (up to 200 kms), there are fields that can be strongly positively cor-

related, negatively correlated or virtually uncorrelated. In case the analysis is restricted

to the statistically significant correlations only (Fig. 5, bottom, left), the decrease in the

correlation strength with distance is not seen at all, at least up to the scale of 200 kms.

Beside the usual cross-correlation coefficient r0, we also calculated a delay-based cross-

correlation coefficient r1 as given by Eq. (7). Such a delay (assumed to be one year,

i.e. one generation for T. paludosa) can arise if synchronization is induced by dispersal.

Recall that the population data were collected in mid-winter, i.e. when the species is in its

larval stage. Dispersal is however associated with the flying stage that normally happens

in late August/early September. Therefore, the effect of dispersal will not be seen in the

census until the next year. The effect of delay is likely to be felt more strongly if dispersal

is asymmetric, i.e. field X delegates a fraction of its population to field Y but not vice

versa. In contrast, synchronization due to the impact of stochasticity is unlikely to be

subject to delay.

Interestingly, r1 exhibits properties significantly different from r0; see Fig. 5, right.

The values of r1 are predominantly negative, especially if the analysis is restricted to

statistically significant values (Fig. 5, bottom, right) There is no decay with distance at

all as the best-fitting line has a slope very close to zero. Contrary to r0, the time-delayed

coefficient r1 show a clear directional aspect (Fig. 6, right) so that the relationship between

fields appear to be stronger in the East-North-East and South-East directions than on

other bearings (see Fig. 12). This is in good agreement with meteorological data on wind

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direction, thus we suggest that dispersal is wind-assisted. Note that we are not able to

provide a more quantitative proof of the impact of the wind on population synchronization,

e.g. by calculating a correlation coefficient between the bearing of the pairwise correlations

and the wind velocity. Such calculation would require high-resolution data on the wind

direction across the whole study area; unfortunately, such data do not exist.

Note that from the whole range of environmental factors we only consider wind ex-

plicitly; regarding other factors as environmental stochasticity. Another relevant factor

can be precipitation. However, rainfall only has a significant impact on the population

abundance when autumn is exceptionally dry (Milne et al. 1965; see also Blackshaw and

Petrovskii 2007) and there is no evidence of any abnormal precipitation level in South-

Western Scotland during the period of study.

In this work, we have investigated the synchronization pattern obtained for a time

lag of one year. This choice seems to be suggested by T. paludosa life traits. However,

we have also considered longer time delays of two and three years. The corresponding

cross-correlation coefficients r2 and r3 show the properties generally similar to those of

r1; in particular, the networks of inter-field connections (not shown here for the sake of

brevity) have shapes which are only slightly different from that shown in Fig. 11, with a

few links having disappeared and a few new links having emerged. A general tendency

seen with an increase in time lag is a gradual decrease in the average correlation strength.

The essential features of the time-lagged correlations are therefore encompassed by the

coefficient r1.

The differences between r0 and r1 can be used to distinguish between the contribution

from dispersal and the Moran effect, which is the third goal of our study. We associate

the no-delay coefficient r0 with the effect of environmental stochasticity and the delay-

based coefficient r1 with dispersal. Since synchronization is seen both with and without

time-delay, we conclude that both mechanisms are involved.

A counter-intuitive finding is that both mechanisms operate on approximately the same

spatial scale of about 200 kms, as given by the size of the study area. With regard to

dispersal coupling, it seems to be a surprising result as T. paludosa females are known to

be poor flyers with typical dispersal distances thought to be below one hundred meters.

For species with poor dispersal abilities, synchronization due to dispersal is thought to

occur on a spatial scale much smaller than that of synchronization due to the Moran effect

(Sutcliffe et al. 1996; Peltonen et al. 2002). However, this obviously does not account for

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the potential impact of wind, which would not only provide a directional effect but also

uncouple any dispersal-distance relationship at smaller scales. What can be true for forest

insects, may not necessarily be true for insects dwelling on bare plains and hills of South-

Western Scotland. With a wind speed of several meters per second (which is typical for the

study area, see Fig. 7), the air flow is strongly turbulent. Turbulence creates ascending

currents that can keep individual insects in the air5 for many hours (taking also into

account the complicated body shape and the relatively large wing-span of the cranefly),

i.e. the time that is quite sufficient for them to reach another breeding ground situated

a long distance away from their natal field. We note that, although direct evidence of

wind-assisted dispersal for T. paludosa is not available, “sailing with the wind” is a typical

dispersal strategy for many other insect species, with distances covered being dozens and

even hundreds of kilometers (Gatehouse 1997; Compton 2002). We also note that the

number of successfully travelling females does not necessarily need to be large. There

is growing evidence that dispersal coupling by even a small fraction of the population

may bring population fluctuations into synchrony (Haydon and Steen 1997; Kendall et al.

2000; Ripa 2000).

We also mention that there are some other mechanisms that may, in principle, synchro-

nize the population fluctuations. Firstly, synchronization can emerge through interaction

with another species that is itself synchronized (Liebhold et al. 2004). However, we are not

aware of any species that exerts a consistent, regulatory effect on T. paludosa populations

and so we consider this possibility the least likely explanation. Secondly, since T. paludosa

is a pest, its abundance is controlled by pesticides. Should the application of pesticides

be synchronized across the region, it could possibly synchronize the dynamics of the pest.

However, the existing agricultural legislation in the UK does not impose on farmers any

obligatory response to pest infestation. In fact, not only are pest controlling measures

purely voluntary, but so too is participation in monitoring programs. The probability of

a synchronized pesticides application is therefore rather unlikely.

Further evidence of the coupling by the wind-assisted dispersal may also be obtained

by developing a more detailed theoretical framework operating across the whole range of

spatial scales involved. Indeed, a comprehensive model of the dynamics of an individual

population must take all the local populations into account. Correspondingly, it should

5T. paludosa have been caught in suction traps at 14 meters above ground in samples collected as part

of the Rothamsted Insect Survey (Blackshaw, unpublished data).

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take into account environmental processes acting on a regional scale along with those going

on locally. In particular, such a model should make use of the patterns in directionality

observed (e.g. see Fig. 12) in order to estimate the probability that a given population

will be influenced by any other population in the study area. A functional relationship

describing the effect of one population on another could then be derived in a similar way

to that used to obtain Eq. (1) but weighted with the probabilities of given interactions

with respect to distance and bearing. In conjunction with Eq. (1), this would provide

a more complete description of a given population’s dynamics region-wide. However,

parametrization and verification of such a model can hardly be possible until the impact

of wind and terrain are incorporated explicitly into these functional relationships. This

is a complex task which clearly lies beyond the scope of this paper; in particular, more

detailed weather data currently do not exist.

Given the evidence presented, the dynamics of apparently isolated populations of

T. paludosa cannot be completely described by internal mechanisms (e.g. by density de-

pendence). Instead, these dynamics are noticeably influenced by the dynamics of popu-

lations of this species at other locations. Results of our analysis indicate that the wind

is likely to be a factor responsible for the inter-habitat coupling on the spatial scale up

to 200 kms. This is rather counter-intuitive as T. paludosa are usually regarded as poor

flyers. A study to look for genetic similarities between different populations across the

whole area could confirm the existence of inter-habitat coupling by direct transport. We

are considering undertaking such an investigation in the future.

Although in this paper we have focused on the dynamics of a particular species, we

believe that our approach and findings may be useful in a much broader context. Syn-

chronization of population fluctuations often occurs due to a combination of the effects

of environmental stochasticity and dispersal. Discriminating between these two mecha-

nisms is a considerable challenge, especially where they act on the same spatial scale. By

studying the coupling between local populations with a time lag of one generation, we

demonstrate a general method for separating them. Indeed, it is hard to see how spatially-

correlated stochastic fluctuations in weather conditions (as required by the Moran theo-

rem) can possibly deliver a time-lagged coupling. The general message is therefore that

within-generation synchrony can be attributed to the environment whilst that with a shift

between generations, i.e. time-lagged, is due to dispersal.

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Appendix: Statistical analysis of the population data

As described in the main text, in order to quantify the degree of synchronization between

the populations of two fields (say, X and Y ), we calculate the correlation coefficient of

their respective residual time series, RX and RY :

r0(RX , RY ) =

∑n

i=2(RX

i − µXR )(R

Yi − µY

R)√

(∑n

i=2(RX

i − µXR )

2) (∑n

i=2(RY

i − µYR)

2),

where RXi and RY

i are the residual population densities at year i in fields X and Y ,

respectively, and µXR and µY

R are the sample means of the two time series.

It is not immediately clear, however, what constitutes a statistically significant correla-

tion coefficient. Since we work with time series of finite length, and the data are affected

by stochastic factors, any given value of the correlation coefficient (5) may appear by

chance. One must therefore distinguish between the cases when high absolute values of

r0(RX , RY ) are superficial and the cases when these values are the result of actual syn-

chronization. In order to do so, the following approach can be used. Suppose that the two

time series used to calculate the correlation coefficient, r0, are unsynchronised. Then, if

one of these series is replaced with random residuals drawn from LNR and the correlation

coefficient is recalculated, the resulting distribution of correlation coefficients should be

centred on r0. Furthermore it is possible to estimate the probability of a higher abso-

lute value of the correlation coefficient being obtained from a purely random time series.

This approach constitutes a (two-tailed) Monte Carlo test as was originally described by

Professor Barnard in (Bartlett 1963) and later refined by Hope (1968).

In particular, we start from the assumption (or null hypothesis) that two populations

are not synchronised, that is that any correlation between them occurs by chance. The

probability of obtaining the observed correlation coefficient given this assumption, called

the p-value, is calculated as described below. If this probability falls below a certain

significance level, denoted α, then our initial assumption is rejected and instead the pop-

ulations are considered synchronised. Hence, for a single test, the significance level is the

probability of incorrectly rejecting the null hypothesis, i.e. a false positive or Type I error;

e.g. see Gotelli and Ellison (2004).

Monte Carlo test. Random variates from the distribution LNR were used to construct

time series of residuals, R∗. A correlation coefficient, r̄0, for each (RX , R∗) pair was

calculated and compared with r0, the value obtained for (RX , RY ). The number of pairs

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for which r̄0 > r0 was then divided by the total number of trials to determine the estimated

p-value of r0. An initial run of 500 permutations were used to obtain a crude p-value,

pc. If pc > 0.2 then the pair were considered not significantly correlated without further

calculation. If pc < 0.2 a further run of 50000 permutations were used to obtain a refined

p-value, pr.

Note that the number of random permutations used in the refinement run are high

relative to those proposed in (Bartlett 1963) and (Hope 1968). It was determined heuris-

tically that this number was required to ensure consistent results at the lowest significance

level used α = 0.05. The initial crude assessment allows computational run time to be

reduced if the majority of time series assessed are not strongly correlated.

Multiple comparisons and significance level. In assessing the synchronization of

each distinct pair of populations included in the study a large number of statistical tests

must be carried out. It is intuitively clear that undertaking more tests increases the

number of false positives obtained. In order to maintain the desired significance level over

the entire family of inferences undertaken it is necessary to account for this in some way

without excessively compromising the power of the test (Perneger 1998). This is achieved

by defining an acceptable false positive rate (Benjamini and Hochberg 1995).

Note first that we have a finite set of N distinct events, {Ai : 1 ≤ i ≤ N}, representing

each instance where the null hypothesis might be falsely rejected. The union of these

events corresponds to obtaining at least one false positive. A limit on the probability of

this union is given by Boole’s inequality:

P (∪iAi) ≤∑

i

P (Ai), (9)

in terms of the probability of each individual false positive.

Thus to control the false positive rate we first place the correlation values obtained in

ascending order according to their p-value. A given null hypothesis is then rejected only

if the sum of its p-value with those of all tests ranked below it is less than the desired

family significance level. Once the cumulative p-value exceeds this level all remaining

null hypotheses are accepted, that is all remaining populations are considered to be un-

synchronised. Thus the probability of at least one false positive is restricted by Boole’s

inequality to less than the family significance level.

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