• Chrysomelidae;
  • colonization;
  • invasion;
  • metapopulation dynamics;
  • seed dispersal


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
  • 1
    Dispersal is a fundamental ecological process, so spatial models require realistic dispersal kernels. We compare five different forms for the dispersal kernel of the tansy beetle Chrysolina graminis moving between patches of its host-plant (tansy Tanacetum vulgare) in a riparian landscape.
  • 2
    Multi-patch mark–recapture data were collected every 2 weeks over 2 years within a large network of patches and from 2226 beetles. Dispersal was common (28·4% of 880 recaptures after a fortnight) and was more likely over longer intervals, out of small patches, for females and during flooding. Interpatch movement rates did not differ between years and exhibited no density dependence. Dispersal distances were similar for males and females, in both years and over all intervals, with a median dispersal distance of just 9·8 m, although a maximum of 856 m was recorded.
  • 3
    A model of dispersal, where patches competed for dispersers based on their size and distance from the beetle's source patch (scaled by the dispersal kernel) was fitted to the field data with a maximum likelihood procedure and each of five alternative kernels. The best fitting had relatively extended tails of long-distance dispersal, while Gaussian and negative exponential kernels performed worst.
  • 4
    The model suggests that females disperse more commonly than males and that both are strongly attracted to large patches but do not differ between years, which are consistent with the empirical results. Model-predicted emigration and immigration rates and dispersal phenologies match those observed, suggesting that the model captured the major drivers of tansy beetle dispersal.
  • 5
    Although negative exponential and Gaussian kernels are widely used for their simplicity, we suggest that these should not be the models of automatic choice, and that fat-tailed kernels with relatively higher proportions of long-distance dispersal may be more realistic.


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

Ecology is becoming an increasingly spatial discipline. Some of the most important issues in the field, such as the spread of invasive species, responses to climate change and habitat fragmentation, are inherently spatial and as such, dispersal is now viewed as a fundamental process affecting individual fitness, population dynamics, species’ distributions, community structure and evolution (Clobert, Ims & Rousset 2004; Bowler & Benton 2005; Cottenie 2005). Realistic models of dispersal are therefore required to address many key areas of ecological research.

Although highly desirable, it is difficult to develop easily parameterized models of movement behaviour, such as random walks, that generate realistic distributions of dispersal distances (dispersal kernels). This is partly because such models predict Gaussian kernels (Turchin 1998; Okubo & Levin 2002) while measured ones are typically leptokurtic, with ‘fatter’ tails of long-distance dispersal (Kot, Lewis & van den Driessche 1996). This can be caused by heterogeneity in movement behaviour (Skalski & Gilliam 2003) or by responses to landscape features such as habitat boundaries (Morales 2002), which may be hard to quantify and incorporate into models of movement at large spatial scales.

An alternative approach is to fit general functional forms that characterize the kernel to dispersal data (Turchin 1998). An example of this is the Virtual Migration Model for movement between habitat patches within metapopulations (Hanski, Alho & Moilanen 2000), where a negative exponential kernel is parameterized from mark–recapture data. However, negative exponentials may give a poor fit to empirical data, both in plants (Skarpaas et al. 2004) and animals (Hill, Thomas & Lewis 1996; Kot et al. 1996; Thomas & Hanski 1997; Baguette 2003). The precise form of the dispersal function can affect the outcome of models, so it is important to choose an appropriate function for the species under consideration (Kot et al. 1996).

There are many potential proximate drivers of dispersal, which also require consideration (Clobert et al. 2004; Bowler & Benton 2005). For example, individuals may be more likely to emigrate at high population densities because of resource depletion or conspecific interactions, as is the case for the chrysomelid beetle Trirhabda virgata (Herzig 1995). Patch size may also play a role, whereby emigration is typically high and immigration low in small patches (Kareiva 1985; Turchin 1986; Hill et al. 1996). High emigration from small patches may result from increased encounters with the patch edge (Englund & Hambäck 2004) while reduced immigration may be due to a lower chance of finding or settling in a small patch (Byers 1996). There may also be sex differences in dispersal, as females aim to locate multiple oviposition sites, while males act to gain access to mates. For example, female Parpxyna plantaginis flies exhibit density-dependent dispersal, selected by competition for flower heads to lay eggs on, while males sit and wait for females, rendering their dispersal density-independent (Albrechtson & Nachman 2001).

The aim of this study was to investigate and model dispersal of the tansy beetle Chrysolina graminis (Coleoptera: Chrysomelidae) moving between patches of its host-plant (tansy Tanacetum vulgare). As in other insect studies (Hanski et al. 2000), we consider all interpatch movement to be dispersal, which we recorded with mark–recapture in large patch networks. The effects of time, patch area, sex, flooding and beetle density on the likelihood and magnitude of dispersal were analysed. In order to contrast different dispersal kernels, we develop a general model of interpatch dispersal, fit the model to our data with each kernel and compare their performance.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

the species

Chrysolina graminis L. is an 8–10 mm long, flightless and iridescent green chrysomelid beetle that is, in the UK, restricted to the banks of the River Ouse around York, northern England (Chapman et al. 2006). On the Ouse, C. graminis is a specialist herbivore of tansy Tanacetum vulgare L. completes its entire life cycle on and around the plant (Oxford et al. 2003). Tansy is a perennial herb that pioneers disturbed ground along riverbanks (Stace 1991) and grows in dense clumps that form habitat patches for the beetle.

Adult beetles emerge from underground hibernation in the spring to feed and mate around the tops of tansy patches, such that eggs and larvae can be seen in May and June (Oxford et al. 2003). Final instar larvae pupate underground and emerge as adults from mid-August to September, before re-entering the soil to overwinter soon after. During the year, the beetle is relatively sedentary and has no obvious variation in morphological characteristics affecting dispersal (e.g. leg length), although females are generally larger-bodied than males. Sivell (2003) marked beetles on one tansy patch and monitored their dispersal over 80 days, finding a maximum net displacement of just over 100 m.

dispersal data

Data on dispersal were collected in a multi-patch mark–recapture study performed in 2004 and 2005 on roughly 2 km of riverbank at Clifton Ings, York (53°58′13″N, 1°6′47″W). Here, tansy grows along the unmanaged riverbank and on an annually mown flood embankment bounding the bank. Downstream, the bank is wooded and contains no tansy. Upstream, conditions are similar to Clifton, so to minimize edge effects tansy in the upstream half of the site were excluded from the study.

At the start of each field season, the site was thoroughly searched for tansy, defining patches as clumps less than 0·5 m apart. Their locations were mapped with a hand-held GPS (Garmin GPS 12) and their basal lengths and widths were measured, allowing basal areas to be estimated as those of ellipses with these dimensions. In 2004, 158 tansy patches were found while there were 126 in 2005. Twenty-five of these were selected for study in 2004 and 32 were chosen in 2005. In general, clusters of patches containing high numbers of beetles were selected as it was thought that most dispersal events would be over short distances. In 2004, beetles found on all visited patches were marked, while in 2005 this was the case for 24 of the 32 study patches. The remaining eight patches were monitored for the immigration of marked beetles but no marking was carried out on them, allowing a greater number of patches to be visited.

Beetles were permanently marked by lightly abrading the surface of their elytra using a cordless engraving drill (Dremel Multi 7·2 V) to leave an individual code of dark dots. Each elytron was divided into three horizontal rows (top, middle and bottom) and each row was divided into three marking positions (inner, central and outer). At each row, beetles were either given a single dot at one position or the row was left clear, resulting in 4095 unique permutations. A similar marking system has previously been used and is not thought to cause lasting injury (Sivell 2003).

The study patches were visited every 14 days (± 1–2 days three times, twice due to flooding) from 21 April to 6 October 2004 and 12 April to 11 October 2005, encompassing almost all of the species’ active period. Two weeks is approximately half the life expectancy of an active beetle (Sivell 2003) and as beetles must feed over this period, all successful dispersal events are likely to be within this time. During each visit beetles were collected by hand, unmarked beetles were sexed and marked (if on an appropriate patch) and the identities of recaptured beetles were recorded, before release back into the patch centre. Following this, the areas of the study patches were measured to monitor their growth and to calculate relative beetle density from the number of adults sampled. Patch coordinates were also re-recorded so that by taking their means throughout the season, error in the GPS readings was minimized.

Beetle capture histories were processed by extracting pairs of successive captures (without intervening captures) from the same year and recording whether interpatch movement had occurred. If so, the beetle's net displacement was calculated as the straight-line distance between patch centres. For simplicity, analyses of dispersal rates and distances considered the intervals between visits to be whole fortnights and different pairs of recaptures from the same beetle to be independent.

a model for interpatch dispersal

A simple model of the dispersal of insect herbivores was developed and fitted to the mark–recapture data. We assume that individuals on host-plants continually make small exploratory movements, which means that there is a temporal scale above which all individuals in even the largest patches are likely to leave their patch. Work on the movement of the tansy beetle at short time-scales (D. Chapman, unpublished data) suggests that this will be the case over 2 weeks, which is the resolution of our mark–recapture data. Thus emigration did not require explicit consideration in the model.

Once an individual has wandered out of its original patch it will move around in the interpatch matrix in search of host-plants and will not discriminate between the patch it has left and a novel one. Even without specific adaptations for orientation towards host-plants, larger patches will present larger targets and retain immigrants for longer, so observed dispersal should be biased towards large patches (Byers 1996). Dispersal should also be biased towards patches that are closer to the source. As a consequence, an individual leaving a patch with large and close neighbours will be less likely to return to that patch than an individual starting in an isolated one.

These patterns are framed mathematically in a formula giving the probability that at time t, an individual in patch j will move into patch k rather than any of the other available patches, ψjk,t, as follows,

  • image(eqn 1)

where Q is the total number of patches at the site, Ak,t is the area of patch k at time t and the dispersal kernel f(djk) is a function of the distance between j and k, djk, describing dispersal ability. The parameter b > 0 scales the relative bias in movement to large patches. Five alternative forms of f(djk) are shown in Table 1, all of which give the proportion of individuals capable of moving djk and can take any form where f(0) = 1 (as all individuals can move 0 m) and then declines towards zero for greater distances. Different kernels give different shaped curves so it is important to select the most appropriate for describing the movement capacity of the focal species.

Table 1.  Alternative dispersal kernels f(djk), whose performance in explaining the tansy beetle data were compared. All models give the proportion of individuals able to move as far as the distance between patches j and k, djk in terms of parameters σ > 0, αi > 0 and βi > 6. G is a Gaussian kernel, with a mean of 0 and a variance of σ, normalized so that f(0) = 1, as would be the expectation of a simple diffusion model (Turchin 1998; Okubo & Levin 2002). The negative exponential (NE) is frequently used in metapopulation models (Hanski et al. 2000; Moilanen 2004) and is attractive because 1/α1 gives the mean dispersal distance. The other kernels have longer tails (i.e. increased long-distance dispersal). The inverse power function (IP) is commonly fitted to empirically measured distributions of dispersal distances (Hill et al. 1996). The extended negative exponential (ENE) is similar to NE but has an extended tail and is of the form used by Taylor (1978). The fat-tailed kernel (FT) is used in the incidence function metapopulation model (Moilanen 2004)
Ginline image
IPinline image
ENEinline image
FTinline image

The model works by giving the movement from j to k a weighting based on the species’ ability to travel the distance between the patches and the area of k. The realized probability of movement is their weighting divided by the sum of weightings for all the patches at the site (the denominator in eqn 1). Landscape structure in terms of the distribution of patches is thus accounted for in determining movement probabilities. The Virtual Migration Model (Hanski et al. 2000) and the patch accessibility model of Heinz et al. (2005) use similar equations whereby patches ‘compete’ for dispersers on the basis of their size and distance, but here all individuals rather than those that have ‘decided’ to disperse are competed for and the source patch participates in the competition, resulting in a simpler model. Small patches will achieve higher emigration rates as they have weaker competitive abilities.

If patch areas were larger or the timestep smaller, so that individuals did not necessarily leave their patch between captures, emigration could be explicitly included as follows,

  • image(eqn 2)

where parameters η > 0 and c > 0 scale emigration rates from the source patch j to Aj (Englund & Hambäck 2004). If the source and target patches are different, ψjk,t is the probability of emigrating from j multiplied by that of moving to k. Where they are the same, ψjk,t is the probability of not emigrating plus that of emigrating then moving back into the source.

fitting the model to the data

A maximum likelihood approach was taken to fit the model in eqn 1 to the tansy beetle data with each dispersal kernel from Table 1 (preliminary investigation with eqn 2 suggested total emigration, so the simpler model was preferred). This involved calculating the log-likelihood of the data given the model and some set of parameters and then finding the parameters that maximize the likelihood of the data set – the maximum likelihood parameter estimates (MLEs). It is possible to calculate the likelihood of dispersal in each capture history from the first to the last capture as the product of the likelihoods of all the possible paths from the initial to the final patch, via every combination of patches that the individual could have been in when not captured. However, given the number of patches and duration of this study, the computational power required to evaluate the huge number of potential paths is unfeasibly high. Therefore, ‘capture history fragments’ of successive captures over one and two fortnights were extracted from the data.

Given some parameters, the log-likelihood of capture history fragment x, l(x) where a beetle started in patch j and ended up in patch k is,

  • image(eqn 3)

where pi is the capture probability in the intermediate patch i in which the beetle was not captured. In x, known locations of the beetle are represented by j and k and 0 represents a noncapture. If x includes a miss, l(x) is the sum of the log-likelihoods of all possible routes between j and k, accounting for the fact that the beetle was looked for but not found on one occasion.

Capture probabilities in the study patches were not themselves of interest but required estimation. A complicating factor is that C. graminis goes through periods of underground diapause, both for overwintering and occasionally during the summer. For example, some adults enter the soil after mating and reappear in late summer or the following spring, while others delay emergence from overwintering or hibernate early. As underground individuals are unobservable, this violates the ‘equal catchability’ assumption of standard mark–recapture analyses, which is complex to relax. Therefore, we made simple capture probability estimates as the proportion of individuals captured both immediately before and after each visit that were captured at that visit (Hanski et al. 2000). This assumes constancy across time and patches, but was probably based on active individuals only as beetles are unlikely to enter and emerge from diapause in the space of 4 weeks. Capture probabilities in the unvisited patches were zero. As capture probability is related to movement through the time spent inside patches, separate estimates were made from each data set that the model was being fitted to (e.g. male and female capture probabilities were estimated separately when the model was fitted separately but not otherwise).

The overall log-likelihood of the data set, L, given some set of parameters, is the sum of the log-likelihoods of individual capture history fragments (Edwards 1992). The maximum log-likelihood, L* and its MLE parameter values were found by numerical optimization through 2500 iterations of simulated annealing (Kirkpatrick et al. 1983), repeated three times with random seeds to check for convergence. For each parameter, two-unit support intervals were calculated around the MLE as the range of parameter space where the log-likelihood exceeded L* − 2 when the other parameters were held at their MLE values. Parameter 95% confidence intervals were not estimated because of computational demands, but two-unit support intervals are of similar magnitudes (Edwards 1992).

Patch areas between visits t and t + 1 were taken as the mean of their areas at t and t + 1. The areas of the unmarked patches were estimated from their sizes at the start of the season and the mean growth rates of the marked patches thereafter. Error introduced by this should have had limited effect since most of the unmarked patches were far from the clusters of marked ones. For intervals in which flooding occurred, the areas of the bankside patches were halved as the tansy available during these intervals was reduced.

The model was fitted for each alternative dispersal function (Table 1), and the best fitting was selected from their Akaike Information Criteria (AICc) (Burnham & Anderson 2002). Using the optimal kernel, models were fitted for the entire data set, with dispersal being: (1) the same for both sexes and years; (2) different between years but not sexes; (3) different between sexes but not years; and (4) different between both sexes and years. The AICc values of these scenarios were compared to find the best overall model of beetle dispersal.

To assess the degree to which this model explained the data, the observed and expected numbers of emigrants and immigrants associated with each patch were compared. Observed and expected numbers of dispersers in the data throughout the study were also compared to see if there were any phenological patterns in beetle dispersal that were not accounted for by the model. Small sample sizes and nonindependence of the data meant that comparison with chi-squared tests was inappropriate. Therefore, Pearson's correlation coefficients between observed and expected values were calculated (excluding cases without data). Significances were assessed by calculating correlation coefficients for 1000 randomizations of the expected values and seeing whether the true correlation coefficient lay outside the 2·5th or 97·5th percentile of these.

The expected number of emigrants from patch j over the T visits in the season is

  • image(eqn 4)

where Nj,t is the number of individuals caught in j at visit t who were caught again in any patch at t + 1. The expected number of immigrants to j, is

  • image(eqn 5)

The expected number of dispersers in the data between t and t + 1, µt is

  • image(eqn 6)


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

factors affecting dispersal

In total, 2226 beetles were marked, of which we captured 1622 in 2004 and 794 in 2005. Marked beetles were recaptured an average of 0·69 times each. The data contained 1032 successive captures from 2004 and 304 from 2005, separated by up to 10 fortnights. When dispersal (interpatch movement) occurred, beetles moved up and downstream in equal numbers (n = 476, inline image =  1·016, P = 0·313). Forward stepwise logistic regression on the binary variable of whether or not dispersal occurred showed that it was more common over longer intervals (Fig. 1a), if the beetle was female, originated in a smaller tansy patch and if flooding, which happened twice in 2004, occurred between the captures (Table 2). There were no significant differences between years and beetle density did not affect dispersal rate.


Figure 1. (a) Mean interpatch movement rates (with 95% confidence intervals estimated from the percentiles of 1000 bootstraps) against the time between recaptures. (b) The net displacements of dispersing beetles (those moving between tansy patches) (n = 446) in the mark–recapture study. Note the log-scale on the x-axis. Symbols above a bar indicate that there were pairs of study patches within that distance class (and hence the possibility of observing dispersal), with the number of pairs denoted by the symbol (circles = 1–5, triangles = 6–20, diamonds = 21–100, stars  100).

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Table 2.  Logistic regression model for the probability of moving between tansy patches (n = 1336, inline image = 0·093), where positive coefficients indicate an increased probability of dispersal with an increase in that variable. Variables were selected with a (Wald) forward stepwise procedure such that the year of the study (Wald1 = 1·548, P = 0·213) and the density of adult beetles on the source patch (Wald1 = 2·581, P = 0·108) were omitted
VariableCoefficient (SE)Wald1P
Constant−0·720 (0·160)20·33< 0·001
Time between captures (fortnights) 0·384 (0·056)46·82< 0·001
Patch basal area (m2)−0·181 (0·050)13·19< 0·001
Sex (coded males = 1, females = 0)−0·365 (0·119) 9·387  0·002
Flooding (coded flood = 1, none = 0) 0·510 (0·184) 7·661  0·006

Net displacements during dispersal (centre-to-centre interpatch distances) were not normally distributed (Kolmogorov–Smirnov test, n = 476, Z = 7·334, P < 0·001), being highly leptokurtic (kurtosis of 21·17) and right-skewed (skewness of 4·228) (Fig. 1b). The effects of year, sex and interval between recaptures on dispersal distance were therefore assessed with a Scheirer–Ray–Hare (SRH) test, the nonparametric equivalent of multiway anova (Sokal & Rohlf 1995). Neither year (H1 = 0·241, P = 0·624), sex (H1 = 0·151, P = 0·698), nor interval (H9 = 2·664, P = 0·976) affected dispersal distance and there were no significant two- or three–way interaction terms (P > 0·6 in all cases).

comparison of dispersal kernels and scenarios

When the model in eqn 1 was fitted to the data for each sex and year separately, there was almost overwhelming support for the fat-tailed dispersal kernel, with the exception of males in 2004 for whom there was also some support for the inverse power function (Table 3). The Gaussian kernel performed worst overall, with the negative exponential also giving a relatively poor fit. The extended negative exponential and the inverse power function generally performed similarly.

Table 3.  The selection of dispersal kernel (see Table 1) based on maximized log-likelihoods, L*. Each model's Akaike Information Criterion (AICc) was calculated based on Land its number of parameters k (which includes kernel parameters and capture probabilities). ΔAICc (the difference between each model's AICc and the minimum in that candidate set) values were calculated. Lower values indicate better support for that model, with values over 2·5 indicating very low support (Burnham & Anderson 2002). Akaike weights, w, can be interpreted as the probability that each model is the best of the candidate set. For all data sets, FT performed the best, although there was some support for IP among males in 2004
2004 females (n = 454)G−1196·332398·6 892·5< 0·001
NE −919·931845·9 339·8< 0·001
IP −763·331532·7  26·56< 0·001
ENE −757·641523·3  17·23< 0·001
FT −749·041506·1   0> 0·999
2004 males (n = 427)G−1110·532227·01129·3< 0·001
NE −798·731603·4 505·7< 0·001
IP −546·331098·7   1·001  0·377
ENE −555·941120·0  22·25< 0·001
FT −544·841097·7   0  0·623
2005 females (n = 145)G −380·03 766·2 394·8< 0·001
NE −249·73 505·6 134·3< 0·001
IP −185·93 378·0   6·642  0·034
ENE −184·84 377·9   6·543  0·035
FT −181·54 371·3   0  0·931
2005 males (n = 110)G −282·43 570·9 298·7< 0·001
NE −172·73 351·6  79·36< 0·001
IP −138·53 283·2  10·94  0·004
ENE −137·44 283·1  10·84  0·004
FT −131·94 272·3   0  0·991

Fitting various dispersal scenarios with the fat-tailed kernel, we conclude that dispersal ability did not change over the two years of study but differed between sexes, although this was not definitive (Table 4). Parameter estimates (Fig. 2a) suggested that females were slightly more mobile but the male and female kernels were very similar (Fig. 2b) and may not have been significantly different. Beetles were unlikely to disperse further than 100 m in 2 weeks and had strong bias towards larger patches, particularly in females (Fig. 2c).

Table 4.  Comparison of different scenarios for tansy beetle dispersal, using the fat-tailed kernel. Although there was some support for all versions, the most likely result was that dispersal and capture probabilities were different between the sexes but were the same in both years
Dispersal dependent onLkAICcΔAICcw
Neither sex nor year−1619·0 43246·01·7100·181
Sex but not year−1614·1 83244·300·425
Year but not sex−1614·5 83245·10·8060·284
Both sex and year−1607·3163247·02·7120·110

Figure 2. (a) Parameter estimates for the fat-tailed dispersal kernel model for males and females, with two-unit support intervals. Parameter estimates were obtained by fitting the model to both years’ data simultaneously, as was suggested by the results in Table 4. (b) The resultant dispersal kernels and (c) relationship between a patch's size and its relative attractiveness. The kernel estimates for males and females overlapped in their two-unit support intervals (not shown), suggesting that they may not be significantly different.

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model-predicted patterns of dispersal

Under the dispersal scenario selected above, observed and model-predicted numbers of emigrant males and females at each patch were strongly correlated (n = 87, r = 0·821, P < 0·001), as were the numbers of immigrants (n = 128, r = 0·589, P < 0·001) (Fig. 3). Observed and expected numbers of dispersing males and females between each visit were also highly correlated (n = 47, r = 0·960, P < 0·001) indicating that the apparent phenology of dispersal was an artefact of changes in the landscape such as mowing, flooding and patch growth (Fig. 4).


Figure 3. Observed and expected numbers of (a) emigrant and (b) immigrant beetles at each patch throughout each year of the study. Observed and expected data were positively correlated (see main text). Lines of equivalence between observed and expected are shown.

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Figure 4. Observed (filled circles) and expected (open circles) numbers of dispersing (a) female and (b) male beetles in the data set between every visit in both years. The close match between observed and expected values suggests that the apparent phenology was caused by changes in landscape structure rather than behaviour.

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  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

The patterns of dispersal uncovered in the mark–recapture study suggest an organism adopting simple behavioural strategies. Dispersal (interpatch movement) was not biased up or downstream and beetles dispersing on multiple occasions often moved back into previously visited patches. Dispersal rates did not differ between years but were higher in females, over longer time periods and when flooding occurred. Emigration was more common in small patches but unaffected by beetle density. Net displacements were not influenced by the year, interval between recaptures or sex of the disperser. Most dispersal distances were under 100 m but the distribution was highly leptokurtic and right-skewed, up to a maximum of 856 m.

These patterns may be those expected for an organism whose dispersal is driven by day-to-day foraging movements causing it to stray from its host-plant, which would explain the lack of inherent dispersal seasonality (Fig. 4) compared with the extreme seasonality in the beetle's life cycle. Increased emigration from smaller patches would result from increased perimeter-to-area ratios (Englund & Hambäck 2004) and is well documented for beetle–host-plant systems (Kareiva 1985; Turchin 1986). A possible cause of female-biased dispersal rate is their larger body size, which may have affected behaviour inside patches or at their edges, although males and females achieved similar dispersal distances so any such differences may be smaller in the interpatch matrix. Here, their inability to detect tansy (Sivell 2003) means they will wander until they starve, suffer predation or reach a patch. Most forays where the beetle survives will result in the individual ending up at the patch in which it originated, although a minority will lead to successful dispersal.

Our model was based on this concept of dispersal resulting from foraging movement and, as with other insect studies (Hanski et al. 2000), assumed dispersal ability to remain constant throughout the season rather than being a specific natal or breeding event. All host-plants at the site ‘competed’ for individuals on the basis of their distance from the source patch (scaled by a dispersal kernel) and their area (as larger patches are larger targets) so that landscape structure determined achieved dispersal distances. This is preferable to simply regressing functions on to empirical distributions of dispersal distances (e.g. Hill et al. 1996; Baguette 2003), which makes little accounting for the environment encountered by the disperser. Emigration was considered implicitly rather than explicitly because all beetles were likely to leave their patches at some point over the 2 weeks between mark–recapture visits. Source patches competed to retain their beetles and smaller ones had weaker competitive ability leading to higher achieved emigration rates.

In common with other such models (Hanski et al. 2000), we used centre-to-centre interpatch distances and gave intrapatch movements distances of 0 m. We feel this to be reasonable because we do not consider movement within a patch to be dispersal and intrapatch movements could only be small relative to interpatch ones (just nine of 796 interpatch distances were less than the maximum intrapatch distance). Intrapatch movements were not incorporated in the model because the handling of recaptured beetles meant they were disturbed and moved. This will be a feature of the majority of insect mark–recapture studies so treating patches as points may generally be necessary.

When the model was fitted to the data with different forms for the dispersal kernel (Table 1), the Gaussian function, which is expected from diffusion theory (Turchin 1998), and the negative exponential, which is commonly used in spatial ecological models (e.g. Biederman 2004; Moilanen 2004), performed poorly because of their inability to account for long-distance dispersal. Of the remaining candidates, the fat-tailed kernel gave the best fit in all cases. The inverse power function, commonly fitted to empirical data sets (Hill et al. 1996; Baguette 2003) and an extended version of the negative exponential performed similarly overall, although the inverse power function gave a somewhat better fit for males in 2004. Using the fat-tailed kernel, the model suggested that females were slightly more dispersive than males but that dispersal ability did not differ between years and gave good matches to the observed patterns of dispersal phenology, emigration and immigration. However, the fitted kernels for the two sexes were similar, indeed their two-unit support intervals overlapped, which could explain why observed male and female dispersal distances did not differ significantly.

Two causes of fat-tailed dispersal kernels have previously been identified – heterogeneous movement behaviour where the more mobile individuals are responsible for the long-distance dispersal (Skalski & Gilliam 2003) and semipermeable habitat boundaries where individuals that cross the boundary quickly may travel far (Morales 2002). The behavioural heterogeneity explanation predicts that individuals will have Gaussian kernels, but that these will vary among the population and sum to a fat-tailed kernel overall. In the habitat boundary model, individuals behaving according to the same rules in any one patch will have the same fat-tailed kernel, while individuals in different sized and shaped patches will have different kernels dependent on the nature of the boundary and how often it is encountered. However, if individuals vary in their ability to cross boundaries, they will all have unique patch-dependent fat-tailed kernels. Unfortunately, our data do not permit an analysis of heterogeneity in tansy beetle movement behaviour because of a limited numbers of captures per individual (of 2226 marked beetles, just 338 had ≥ 3 captures). Correlation between successive movements of the same individual may not indicate heterogeneity anyway, because dispersal is highly localized and strongly dependent on the landscape. After one movement, an individual will generally be in a similar landscape to the one it left, so may make a similar second movement for this reason. One possible way to address heterogeneity would be to adapt the current model by treating the parameters of each individual as random variables from a specified distribution and fitting the distribution parameters to the data, whereby its estimated variance would indicate the level of heterogeneity. However, this would require multiple observations of the same individual moving in many locations, as are perhaps more often obtained in radio-tracking studies than mark–recapture.

Our model did not consider mortality during dispersal, so in fact we fitted ‘successful dispersal’ kernels. Mortality is likely to increase with distance (Clobert et al. 2004) so the tail of long-distance dispersal of a true kernel would be higher than that observed making our finding that fat-tailed kernels outperformed less fat-tailed ones robust. The Virtual Migration Model (Hanski et al. 2000) includes mortality as a function of patch connectivity, as dispersal from isolated patches will generally be further and thus riskier. This could potentially be used to estimate dispersal mortality from our results. A multistate mark–recapture model where study patches are states (Hestbeck, Nichols & Malecki 1991) could be set up with interpatch transition probabilities between each visit fixed to those predicted by the current model. Assuming dispersal mortality is related to patch connectivity, which can be calculated from the dispersal kernel (Moilanen & Nieminen 2002), patch-specific survival rates could be estimated as a function of connectivity using a constrained linear mark–recapture model (Lebreton et al. 2003).

The structural assumptions of the dispersal model can be tailored to the biology of other species. Emigration could be included overtly as a simple function of patch area, as was shown in eqn 2. Further extensions could replace the patch area terms with alternative measures of size, quality, perimeter-to-area ratios or the density of conspecifics. If the species was polyphagous, modifiers for plant species could be included. Dispersal probabilities could be made conditional on previous locations, although this would require multiple recaptures and marking from eclosion. In this case, it seems unlikely that C. graminis, which cannot detect tansy at distance (D. Chapman, unpublished data), would be capable of employing such a strategy. Data permitting, model performance with and without these modifications would be compared to indicate whether the extra complications were necessary. Given the range of adaptations that could be implemented relatively easily, we suggest that our model provides a flexible tool applicable to other systems.

Although mechanistic models are desirable, pattern-orientated phenomenological approaches are able to reveal important influences on dispersal, in this case sex, patch area and interpatch distance. Also, they are more efficient to implement in spatial models and so will play an important part in the study of dispersal. Many models use Gaussian or negative exponential kernels, presumably because of their simplicity and relationship to theories of animal movement. We add to the body of evidence that ‘fatter-tailed’ kernels are more realistic (Hill et al. 1996; Kot et al. 1996; Thomas & Hanski 1997; Baguette 2003; Skarpaas et al. 2004) and suggest that modellers take account of this.


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References

DSC is supported by NERC Case studentship with English Nature, from whom we thank Roger Key. Additional financial assistance was provided by City of York Council, as part of their Biodiversity Action Plan. We are also indebted to two anonymous reviewers, whose comments greatly improved the manuscript.


  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Acknowledgements
  8. References
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