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- Materials and methods
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.
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.
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- Materials and methods
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.