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Keywords:

  • Bayesian statistics;
  • habitat fragmentation;
  • informative priors;
  • precision

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

1. Informative Bayesian priors can improve the precision of estimates in ecological studies or estimate parameters for which little or no information is available. While Bayesian analyses are becoming more popular in ecology, the use of strongly informative priors remains rare, perhaps because examples of informative priors are not readily available in the published literature.

2. Dispersal distance is an important ecological parameter, but is difficult to measure and estimates are scarce. General models that provide informative prior estimates of dispersal distances will therefore be valuable.

3. Using a world-wide data set on birds, we develop a predictive model of median natal dispersal distance that includes body mass, wingspan, sex and feeding guild. This model predicts median dispersal distance well when using the fitted data and an independent test data set, explaining up to 53% of the variation.

4. Using this model, we predict a priori estimates of median dispersal distance for 57 woodland-dependent bird species in northern Victoria, Australia. These estimates are then used to investigate the relationship between dispersal ability and vulnerability to landscape-scale changes in habitat cover and fragmentation.

5. We find evidence that woodland bird species with poor predicted dispersal ability are more vulnerable to habitat fragmentation than those species with longer predicted dispersal distances, thus improving the understanding of this important phenomenon.

6. The value of constructing informative priors from existing information is also demonstrated. When used as informative priors for four example species, predicted dispersal distances reduced the 95% credible intervals of posterior estimates of dispersal distance by 8–19%. Further, should we have wished to collect information on avian dispersal distances and relate it to species’ responses to habitat loss and fragmentation, data from 221 individuals across 57 species would have been required to obtain estimates with the same precision as those provided by the general model.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

The use of Bayesian statistical methods is becoming more prevalent in ecology (Clark 2005; McCarthy 2007). A key feature of these methods is that they can use prior information when modelling systems and making predictions. Prior information is usually incorporated into the modelling process in the form of a probability density function, which may be estimated from existing data or through expert elicitation (McCarthy 2007; Michielsens et al. 2008). Informative Bayesian priors have been used to improve the precision of parameter estimates (McCarthy & Masters 2005; Michielsens et al. 2008; Lukacs et al. 2009) and to provide informative a priori estimates of ecological parameters in the absence of species-specific information (Martin et al. 2005; McCarthy, Citroen & McCall 2008). Nonetheless, the use of strongly informative priors remains rare in ecology. One reason for this is that there are a few examples of syntheses performed with the explicit aim of determining Bayesian prior estimates for relevant ecological parameters and evaluating their utility. The synthesis of existing data will aid the development of estimates that may serve as informative priors in ecological studies.

Dispersal distance is a critically important ecological parameter, enabling understanding of metapopulation dynamics, colonization and evolution (Greenwood & Harvey 1982; Paradis et al. 1998). Importantly, knowledge of dispersal ability may help us to understand how species will respond to landscape changes induced by climate change and human activities (Dawideit et al. 2009; Tittler, Villard & Fahrig 2009). Unfortunately, dispersal distances are challenging to measure directly and estimates of species’ dispersal ability are therefore rare (Greenwood & Harvey 1982). While the dispersal distance of some species has been thoroughly investigated (for example, the great reed warbler, Acrocephalus arundinaceous (Hansson, Bensch & Hasselquist 2002a; Hansson et al. 2002b)), a few studies have presented dispersal information for multiple species (Paradis et al. 1998; Sutherland et al. 2000). Further, estimates of dispersal distance in these studies are restricted to those species identified in the study. Models that allow us to generalize to other species would be valuable.

The loss and fragmentation of habitat caused by anthropogenic activities are central problems for ecological management and biodiversity conservation (Mac Nally, Bennett & Horrocks 2000; Kindlmann & Burel 2008). In particular, habitat loss and fragmentation are major factors contributing to declines in forest bird species (Schmiegelow & Monkkonen 2002; Radford & Bennett 2007). In the agricultural zone of southern Australia, habitat loss can be as high as 90%, and many bird species of eucalypt woodlands and forests have seriously declined or become locally extinct in this region (Ford et al. 2001). Species’ responses to habitat loss and fragmentation will likely be determined by the severity of fragmentation (in terms of total habitat loss, and loss of connectivity in the landscape: Ferraz et al. 2007; Van Houtan et al. 2007; Sigel, Robinson & Sherry 2010), the characteristics of the surrounding environment (Watson, Whittaker & Freudenberger 2005; Haslem & Bennett 2008a) and the ability of the species to disperse within or between fragments, or patches, of remaining habitat (Fahrig 1998; Kindlmann & Burel 2008; Robertson & Radford 2009). As habitat is increasingly fragmented, the distance between patches will, on average, become greater. Species that have limited ability to colonize new areas of habitat should be more vulnerable to these effects than species with greater dispersal ability (Kindlmann & Burel 2008). Knowledge of a species’ ability to disperse across a landscape may therefore be important for understanding how that species will be affected by habitat fragmentation.

Given the difficulties associated with measuring dispersal, general models that synthesize existing information to provide a priori estimates of dispersal distances will be valuable. Scaling functions that relate body mass to variation in biological functions such as metabolic rate and population dynamics (including home range) are common in ecological studies (Schmidt-Nielsen 1984; Savage et al. 2004; McCarthy, Citroen & McCall 2008), and a number of studies have already demonstrated that dispersal distance is positively correlated with body mass in birds and mammals (Paradis et al. 1998; Sutherland et al. 2000; Tittler, Villard & Fahrig 2009). In addition, Dawideit et al. (2009) have recently demonstrated that wing, tail and bill length parameters may provide useful surrogates for dispersal distance in British bird species. Predictive models of dispersal distance based on ecologically meaningful variables that are easily accessible will be valuable additions to this field.

In this study, we develop a predictive model of avian natal dispersal distance based on data on wingspan and body mass collated from published studies world-wide. The model predicts median dispersal distance well for individual species across feeding guilds, indicating that larger birds with a higher wingspan to mass ratio will have longer median dispersal distances. We use this model to generate informative priors for natal dispersal distance and demonstrate their influence on posterior estimates of dispersal. We also model the contribution of predicted median dispersal distance to the response to habitat fragmentation for 57 woodland-dependent species of Australian birds for which no dispersal data are available, showing that species with shorter predicted natal dispersal distances appear more vulnerable to the effects of habitat fragmentation. To further demonstrate the value of the a priori information that can be obtained from general models, we calculate the effective sample size of the predicted dispersal distances for the birds in the fragmentation study.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

Estimating dispersal distance

Data were compiled from the literature for median natal dispersal distance and body mass for a range of bird species. We chose to use natal dispersal rather than breeding dispersal because natal dispersal distances tend to be longer and are therefore more likely to be of interest in metapopulation and fragmentation studies (Greenwood & Harvey 1982; Dawideit et al. 2009). In total, data were collected for 84 bird species across 12 orders from five studies (See Table S1, Supporting Information). Where multiple estimates were available for a species (including sex-specific estimates), these were recorded individually. Body mass estimates were taken from the study by Dunning (1993) or, where quoted, Sutherland et al. (2000).

In addition to body mass, it is likely that dispersal ability will be influenced by wingspan or length (Lurz et al. 2002; Dawideit et al. 2009). It is assumed that, on average, birds with a larger wingspan will be able to travel further. However, wingspan, S, is likely to scale with body mass, m, such that for every unit increase in body mass, wingspan should increase to the power of three if varying isometrically (Schmidt-Nielsen 1984). As such, we incorporated wingspan into a ‘shape’ parameter (inline image) to investigate the influence of variation in wingspan not accounted for by changes in body mass. Wingspan estimates were collated from published data sources (Table S1, Supporting Information). Where wing length (L) but not wingspan estimates were available, wingspan was estimated from a regression of wingspan as a function of wing length for species for which both were provided (r= 0·98):

  • image

We expected dispersal distance to be greater for females than males (Greenwood & Harvey 1982), so sex (male, female or combined when the data were pooled across sexes) was included as an explanatory variable. Where possible, sex-specific and combined-sex natal dispersal estimates were matched to corresponding male, female or adult-combined information on body mass and wingspan.

Feeding guild is known to influence territory size and may therefore affect dispersal distance (Schoener 1968). The assignment of feeding guild can be subjective and, as such, random effects about a central mean for feeding guild were considered for the intercept as well as slope interactions with body mass and shape. Where the classification of feeding guild is uncertain, the common mean can be used. Each species in the modelling data set was classified as a vertivore (carnivore that eats vertebrates), insectivore (carnivore that eats invertebrates), herbivore or omnivore.

We considered a subset of possible models, focussing on the three regression parameters (sex, mass and shape) and potential variation in the latter two because of feeding guild. To account for further variation in natal dispersal distances, we examined random effects on the intercept for species and taxonomic order (Table 1).

Table 1.   Candidate models of the relationship between median natal dispersal distance, D, and explanatory variables sex (s), body mass (m) and wing shape (sh). ln(D) is the natural logarithm of the median natal dispersal distance, α is the intercept of the linear predictor, βS, βM and βW are the coefficients for the effect of sex, body mass and wing shape, respectively, and εsp, εord and εg are random effects for species, taxonomic order and feeding guild. Square brackets indicate a categorical variable, and parentheses indicate a continuous variable. Evaluation statistics after 200 000 iterations are also presented. The best model is highlighted in bold. DIC is the deviance information criterion (pD refers to the effective number of parameters estimated by each model), r2 is the proportion of variation in D explained by each model using the model-building data set, and r is the Pearson correlation between the predicted and observed dispersal distances using an independent test data set
ModelDIC (pD)Fitted Relationship (Model-building data) Slope, Intercept, r2Predicted Relationship (Test data) r,
1. ln(D) = α  + βS[s] + βM ln(m) + βW ln(sh) + εsp729·3 (59·7)1·08, 0·18, 0·300·58
2. ln(D) = αg] + βS[s] + βMln(m) + βWln(sh) + εsp693·4 (20·2)1·00, −0·02, 0·430·70
3. ln(D) = α + βS[s] + βMg] ln(m) + βWg] ln(sh) + εsp712·3 (48·6)1·06, −0·14, 0·400·54
4. ln(D) = α + βS[s] + βM ln(m) + βW ln(sh) + εsp + εord731·4 (57·2)1·52, 1·31, 0·270·60

The dispersal model

Median natal dispersal distances were assumed to be log-normally distributed and related to the explanatory variables according to a linear regression function with the general form:

  • image

where Di is the ith recorded median natal dispersal distance, α is the intercept, βk are the regression coefficients for K explanatory variables xk, and εn are N random effects. The full set of candidate models is presented in Table 1. The general form of the model was modified to allow for a random effect of guild on the slopes (Table 1, Model 3). The explanatory variables body mass and shape were transformed logarithmically to improve linearity.

Models were run in OpenBUGS version 3.1.0, a freely available statistical software package for conducting Bayesian analyses using Markov chain Monte Carlo (MCMC) methods (Lunn et al. 2009). We used vague prior distributions for α, βk and εn to ensure that the posterior distributions for these parameters were dominated by the data. Prior distributions for parameters α and βk were specified as normal with a mean of zero and a standard deviation of 1000. Prior distributions for species and taxonomic order random effects were specified as normal with a mean of zero and a standard deviation to be estimated from the data. The prior distributions for the standard deviation of the species and order random effects were uniform with a minimum of zero and maximum of 100.

Guild random effects were assumed to vary around a common mean drawn from a normal distribution with a mean of zero and a standard deviation to be estimated from the data. The U(0,100) priors specified for the standard deviation amongst species and taxonomic orders can lead to heavy right tails – and therefore overestimates – in the posterior distribution for the standard deviation where the number of groups is small. As such, we used a weakly informative half-Cauchy prior for the standard deviation amongst feeding guilds, of which there were only four (Gelman 2006). The full model description and code can be found in the Supporting Information. To ensure convergence, we sampled from multiple (two) MCMC chains. In all cases, models had converged within 10 000 MCMC samples, and posterior estimates were taken from 200 000 MCMC samples after discarding the first 10 000.

Candidate models were assessed using the deviance information criterion (DIC: Spiegelhalter et al. 2002), by comparing the fitted and observed dispersal distances, and by comparing the predictions with median dispersal distances from an independent test data set, compiled from a second search for dispersal information after the initial modelling had taken place. This included data that were excluded from the original data set because wingspan information was missing but where we subsequently found these data. Body mass estimates for the test set were taken from the study by Dunning (1993), and wingspan and diet information were collected in an internet search (see Table S2, Supporting Information). The test set comprised 22 observations across 15 species and included representatives from the four feeding guilds identified above. We compared the observed natal dispersal distances with those predicted by each of the candidate models.

Demonstrating the utility of prior information obtained from the general model

Estimates of median natal dispersal obtained from the general model may be used as a priori information in a number of ways. They can be formally incorporated into a Bayesian analysis in the form of an informative prior; in this case as a prior distribution for the median dispersal distance of a species, the parameters of which are estimated by the general model. Alternatively, the general dispersal model can be used to produce a priori estimates of median natal dispersal distance for species for which no other dispersal data are available, for use as a covariate for modelling species occupancy or responses to environmental changes. This is equivalent to a Bayesian analysis in which we have no covariate data, only prior information. We demonstrate both uses here.

To demonstrate the use of the predicted dispersal distances as informative Bayesian priors, information (raw data or probability distribution parameters) on median natal dispersal distances is required. We used the data in Paradis et al. (1998) to represent the range of body masses, wingspans, dispersal distance estimates, variances and sample sizes that would exist in typical data sets. These were used to demonstrate the influence of informative priors on natal dispersal distances. We chose four species (European turtle dove Streptopelia turtur, northern goshawk Accipiter gentilis, willow tit Parus montanus and grey wagtail Motacilla cinerea) whose original dispersal estimates were obtained from a range of sample sizes (n = 4, 9, 14 and 20, respectively). For each species, we had information on the arithmetic mean (AM) and standard deviation (SD) of the natal dispersal distances. When the data and the prior are distributed normally, the posterior will also have a normal distribution. In this case, it is relatively straightforward to calculate the mean and the variance of the posterior distribution based on estimates of the mean and variance of the data and prior (McCarthy 2007). We assumed that natal dispersal distances (d) were distributed log-normally, so ln(d) had a normal distribution. The mean, μdata, and standard deviation, σdata, of the corresponding normal distribution are equal to ln(AM) – 0·5ln(c) and ln(c), respectively, where c is equal to AM2/SD+ 1. We used WinBUGS to estimate μprior and σprior, the mean and standard deviation of the corresponding normal distribution of the predicted natural logarithm of median natal dispersal distance, for each of the four species. We then estimated the mean and variance of the posterior distribution (μpost and σ2post) according to McCarthy (2007):

  • image
  • image

The posterior distributions were then back-transformed to be expressed in units of km. The posterior distribution is a weighted average of the data and the prior, and so including predictions of dispersal distance from the general model as informative priors will reduce the variance (increase the precision) of the posterior distribution. The degree by which precision increases depends on how informative the prior is relative to the information content of the data.

To demonstrate the use of predicted dispersal distances as a priori estimates where no other dispersal information is available, we investigated the relationship between predicted dispersal distance and response to habitat fragmentation in woodland bird species in northern Victoria, Australia. Radford & Bennett (2007) investigated the effect of landscape change on the incidence of woodland bird species in 24 agricultural landscapes. Their study provides a good opportunity to evaluate the application of predicted priors for investigating relationships between dispersal distance and response to habitat fragmentation. In a study area covering 20 500 km2 of agricultural–woodland mosaic, they selected 24 landscapes, each 10 × 10 km, to represent a gradient in remnant tree cover and to contrast landscapes in which tree cover was ‘aggregated’ with those in which tree cover was ‘dispersed’(Radford & Bennett 2007). Pairs of landscapes were chosen that had similar tree cover but contrasting aggregation. In each landscape, 10 survey sites were established in remnant wooded vegetation. Three sites were allocated to riparian vegetation and the remaining seven distributed amongst large (>40 ha) remnants, small (<40 ha) remnants, roadside vegetation and scattered farmland trees according to the proportional representation of each category in the landscape (Radford & Bennett 2007). Species presence was recorded during four 30-min bird surveys conducted along a 400 m line-transect at every site. All species heard or seen during the allocated survey time were recorded as present. Each of the 240 sites was surveyed twice in the breeding season and twice in the nonbreeding season (Radford & Bennett 2007).

We constructed prevalence models for 57 bird species considered to be woodland dependent. As in the original study, each landscape represents a single sampling unit (= 24) and the incidence of each species in each landscape is the response variable. In this case, the incidence of each species – or the number of surveys in which the species was present – is the realization of 40 Bernoulli trials, each with a probability, p, the proportion of sites in the landscape where the species is observed, which we refer to as prevalence. Our aim was not to build the best possible prevalence model; rather, we wanted to build a model whose outcome would allow us to assess any relationship between predicted dispersal ability and response to fragmentation. Habitat aggregation is more likely to capture the difference in distance between patches than habitat cover per se and was therefore the best choice of variables available to us. We constructed a model that relates pij, the prevalence of species i in landscape j, to the aggregation (aggj) of tree cover using the logit link (Agresti, 1996):

  • image
  • image

where κi and γi are the intercept and regression coefficient for species i, and Yij is the observed number of presences of species i in landscape j, and ηj and φij are random effects; ηj represents additional variation between landscapes and φij extra-binomial variation between species and landscapes. The parameters ηj, φij and κi were each assumed to have been drawn from a normal distribution with a mean and standard deviation to be estimated. The prior distribution for each mean was specified as normal with a mean of zero and a standard deviation of 1000. Prior distributions for the standard deviations were specified as uniform, ranging between 0 and 100.

Estimates of γi can be used to infer the strength of the influence of tree cover aggregation on the presence of each species in the study area. Values of greater magnitude indicate a stronger influence than those values close to zero. Comparison of values of γi and predicted dispersal distance for each species allows inference about the relationship between predicted dispersal ability and sensitivity to fragmentation of tree cover for woodland-dependent bird species in northern Victoria. We constructed a hierarchical model in which the value of γi depends on the predicted dispersal distance of species i:

  • image

where θ is the intercept, inline image is the predicted median dispersal distance for species i, ζi is a random effect term describing variation in the response of species i to aggregation, γi, not explained by inline image, and δ is the slope of the relationship between median dispersal distance and response to increasing aggregation of tree cover in the landscape. Uninformative priors (mean = 0 and standard deviation = 1000) were specified for δ and θ. The prior for ζi was specified as normal with a mean of zero and standard deviation to be estimated from the data. Dispersal distances were predicted using the best dispersal model, parameterized on a combined data set including the initial modelling data as well as the test data. The standard deviation of posterior estimates of median dispersal distance from the dispersal model was constant across species (average = 0·79). To include uncertainty in dispersal estimates in this analysis, inline image was drawn from a normal distribution with a mean equal to the predicted median dispersal distance for species i and a common standard deviation of 0·79. Sex was unspecified in dispersal predictions.

When compared with the global data set, the precision of these median dispersal distance predictions can be expressed in terms of the effective sample size, n. If the coefficient of variation in the data is expressed as:

  • image

and the coefficient of variation of the predicted median dispersal distances is expressed as

  • image

then, assuming a common mean and standard deviation,

  • image

and

  • image

In our study, the coefficient of variation in the data and predictions was greater for shorter dispersal distances. The relationship was such that the natural log of the coefficient of variation and the natural log of the median dispersal distance were negatively, linearly correlated. This means that the effective sample size of the priors is highest for short dispersal species and lower for long distance dispersers. The slope of this relationship was different for our predictions and the published data. Allowing for the different relationships between CV and dispersal distance, we modelled the effective sample size provided by the Bayesian priors for a range of dispersal distances in OpenBUGS and estimated the effective number of observations across all 57 species. Detailed workings and code are provided in the Supporting Information.

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

Predicting median natal dispersal distance

The best candidate model was the one that included sex, body mass and shape, with variable intercepts for feeding guilds (Model 2, Table 1). This model performed best in all measures of model evaluation (DIC, model fit and correlation with test data) in this study (Table 1). Comparing the observed dispersal distances with those fitted by this model for an average species suggests that the best model is relatively free from bias and explains around 43% of the variation in the data (Fig. 1a: r= 0·43). When using this model to predict the test data set, the correlation between predicted ln(dispersal distance) and observed ln(dispersal distance) is positive with a correlation coefficient of 0·70 (Fig. 1b, Table 1).

image

Figure 1.  Plots of predicted vs observed median dispersal distances for the DIC best model (Model 2, Table 1). (a) Fitted responses using the initial modelling data set; (b) predicted dispersal distances for species included in an independent test data set; (c) fitted responses using a combined data set comprised of the initial modelling data set and the test data set. Observations in plots (a) and (c) are categorized by feeding guild: vertivores (⋄); insectivores (•); herbivores (bsl00066); and omnivores (crosses). Fitted relationships for plots (a) and (c) are = 1·00x−0·02 (r= 0·43) and = 1·07x−0·15 (r= 0·53), respectively. Dispersal distances predicted to the test data set are positively correlated with observed dispersal distances (on the natural log scale) with a correlation coefficient, r, of 0·70 (plot b).

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Posterior parameter estimates (produced using the combined data set) for the best dispersal distance model suggest that dispersal distances are shorter for males than females and greatest for vertivores compared with other feeding guilds (Table 2). In addition, dispersal distance increases with body mass and wing shape such that, for any given body mass, birds with a longer wingspan will disperse further on average. Alternatively, as wingspan is held constant, birds with a larger body mass will travel shorter distances. The standard deviation for the species effect is clearly above zero, indicating that there is extra variation in dispersal distance between species that is not captured by sex, guild, body mass or wingspan.

Table 2.   Posterior parameter estimates (mean and 95% credible intervals) for the best model of median natal dispersal distance (Model 2, Table 1) using the combined (model-building + test) data set. Estimates are taken from 180 000 MCMC samples after discarding 10 000 samples as a burn-in
ParameteraPosterior Mean95% Credible Interval
  1. aLetters in square brackets denote relevant category for categorical variables: U = sex unspecified; M = male; F = female; V = vertivore; I = insectivore; H = herbivore; and O = omnivore. βS[U] is specified as a reference class, and the coefficient for this parameter is therefore set to zero.

αmean1·95[0·27, 3·64]
α[V]2·90[2·37, 3·42]
α[I]2·06[1·45, 2·67]
α[H]1·79[1·22, 2·36]
α[O]1·10[0·51, 1·70]
βS [U]0 
βS [M]−0·91[−1·38, −0·45]
βS [F]−0·52[−0·98, −0·07]
βM0·12[−0·02, 0·27]
βW0·55[0·10, 1·00]
Std Dev (εsp)0·74[0·49, 0·97]
Std Dev (εg)1·44[0·43, 4·51]

When used as informative priors, predicted dispersal distances from the general model reduced the variance of posterior estimates of median natal dispersal distance. For the four species for which it was tested, including the informative prior reduced the 95% credible interval of the posterior estimate by between 8 and 19% (Fig. 2). The influence of the prior was greatest where sample size was low and where estimated dispersal distance was small.

image

Figure 2.  Distributions for the median natal dispersal distance for Streptopelia turtur, Accipiter gentilis, Parus montanus and Motacilla cinerea based on the prior (•), data (○) and posterior (•). Error bars show 95% credible intervals (CIs), and n is the sample size of the data. The percentage reduction in the credible intervals of the posterior distribution for each species is also given.

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Predictions of median natal dispersal distances for woodland-dependent bird species in Radford & Bennett’s (2007) study ranged from 1·4 km for the white-browed scrubwren (Sericornis frontalis) to 7·8 km for the tree martin (Petrochelidon nigricans) (See Table S3, Supporting Information).

The effective sample size of a priori estimates ranged from 2·9 [95% C. I.: 2·5, 3·3] individuals for a species with a predicted median dispersal distance of 6 km to 7·5 [5·6, 10·0] individuals for a species with a predicted dispersal distance of 1 km. Summed across the 57 species for which dispersal distance was estimated, the total effective sample size was 221 [186, 262] individuals.

We found a clear positive effect of habitat aggregation on the incidence of woodland bird species in the landscape (Fig. 3). This is demonstrated by all the symbols reflecting the coefficients for the effect of aggregation being above the x-axis. Standard deviations of the parameters κi, ηj and φij were all clearly above zero (Table 3), indicating that there is extra variation in species’ incidence attributable to species, landscape, and interactions between species and landscapes.

image

Figure 3.  Coefficient estimates for the response to habitat fragmentation plotted against the natural logarithm of the predicted median dispersal distance for 57 woodland-dependent bird species in northern Victoria. Estimated natal dispersal distances are as predicted by Model 2. Error bars show standard deviations of the estimates of aggregation coefficients. When uncertainty around dispersal estimates is included, the slope of the fitted relationship between dispersal ability and response to fragmentation is −1·33.

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Table 3.   Posterior estimates for parameters included in the hierarchical model of the response of woodland bird species to fragmentation of tree cover in northern Victoria. κi, ηj and φij are factors in the equation describing the relationship between pij, the incidence of species i in landscape j, and the aggregation of tree cover in the landscape: κi is the regression coefficient for species i; ηj is a random effect representing additional variation between j landscapes; and φij is a random effect representing extra-binomial variation between species and landscapes. θ, δ and ζi are factors in the equation describing the relationship between γi, the effect of tree cover aggregation on the prevalence of species i, and the predicted median natal dispersal distance for species i: θ and δ are the intercept and regression coefficients, respectively; and ζi is a random effect representing additional variation in the species response to habitat aggregation not explained by estimated natal dispersal distance. Estimates are taken from 900 000 samples. Posterior estimates of γ are presented in Fig. 3
ParameterPosterior Mean95% Credible Interval
Std Dev (κi)2·05[1·46, 3·01]
Std Dev (ηj)0·41[0·26, 0·60]
Std Dev (φij)1·11[1·02, 1·21]
θ5·60[3·85, 7·53]
δ−1·33[−2·94, 0·23]
Std Dev (ζi)1·26[0·08, 2·81]

We also found evidence for a relationship between the size of the effect of aggregation experienced by a species and the dispersal ability of the species (Table 3, Fig. 3: r= 0·19). The effect of habitat fragmentation (as measured by the slope parameter estimates for aggregation, γi) tends to be experienced most strongly by species with shorter predicted natal dispersal distances on the natural log scale. δ, the slope of the relationship between γi and predicted dispersal ability, inline image, is −1·33 [−2·94, 0·23]. The posterior probability that the slope is negative is 0·94. Thus, woodland-dependent bird species with longer predicted natal dispersal distances appear less affected by tree cover fragmentation than species with limited dispersal ability, although this relationship is still uncertain.

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

We have demonstrated how predictions of dispersal distance from the general model may be used as a priori information to improve our understanding of species’ response to habitat fragmentation. We have developed a model for predicting avian natal dispersal distance from sex, body mass and wingspan. This relatively simple model predicts median dispersal distance remarkably well, explaining 53% of the variation in the fitted data on the combined data set (model-building plus test data). We have used predictions of median natal dispersal distance to help explain responses to habitat fragmentation in 57 woodland-dependent bird species in northern Victoria. Our best estimate suggests that there is evidence of a negative relationship between dispersal ability and vulnerability to the effects of habitat fragmentation for these species; however, there is considerable uncertainty (i.e. the 95% credible interval for δ includes zero). The effective sample size of the dispersal distance predictions in this study is equivalent to between 3 and 8 observations per species depending on dispersal ability, and a total of 221 observations across the 57 species for which we estimated dispersal distance. This represents a moderate yet significant sample size, particularly where existing data are rare, and difficult and costly to collect, and is evidence of the value of synthesizing existing information for use as informative Bayesian priors and a priori estimates where specific information does not exist.

Predictive model of avian natal dispersal

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

Using a world-wide (but predominantly northern hemispherical) data set, we have developed a model for predicting median natal dispersal distance of bird species. This model is an improvement over previous models of dispersal distance in terms of inter-specific variance explained. Furthermore, because it was built on an international data set, including 100 species across 12 orders, and uses explanatory variables that are both ecologically sensible and easily accessible, our model offers improvements in generality and ease of use.

The influences of sex, body mass and wingspan on dispersal distance found here concur with previous studies of avian dispersal. Female-biased dispersal is prevalent in most bird species for which dispersal has been estimated (Greenwood & Harvey 1982). Allometric scaling theory suggests that dispersal distance should increase with increasing body size (Sutherland et al. 2000), and a positive correlation between avian dispersal distance and body mass has been demonstrated previously (Paradis et al. 1998; Tittler, Villard & Fahrig 2009). Finally, a number of recent studies have shown that dispersal distance increases with wing length (Skjelseth et al. 2007).

We also found that feeding guild influences dispersal distance, with carnivorous species (vertivores and insectivores) exhibiting the longest natal dispersal distances. Diet is recognized as an important factor in determining the home range size of birds and mammals (Grant, Chapman & Richardson 1992). Because of the limitations associated with food availability, carnivores generally have larger home range sizes than omnivores or herbivores of a similar body mass (Schoener 1968). It is reasonable to expect that home range size would be positively correlated with natal dispersal distance, and this would explain the differences we found in dispersal distance across feeding guilds. In a recent study of British passerine birds, Dawideit et al. (2009) found that dispersal distance tended to be longer for species with shallower bills. They suggested that this could be indicative of a correlation between diet and dispersal, but were unable to confirm this mechanism. Our study has clearly identified a relationship between dispersal distance and feeding guild and strengthens the hypothesis of diet as the mechanism for the role of bill depth as an explanatory factor in Dawideit et al.’s (2009) study. Future exploration of the correlation between ecomorphological factors, such as bill depth, and diet may help to minimize the uncertainty associated with assigning classifications of diet.

When predicting to an independent test data set, there was a high correlation between observed median natal dispersal distances and those predicted by the model; however, the model appears to underestimate long natal dispersal distances and overestimate shorter dispersal (Fig. 1b). Dispersal distances are difficult to measure (Greenwood & Harvey 1982; Tittler, Villard & Fahrig 2009), and there is often significant variation in estimates of median dispersal distance for a single species (see, for example, Hansson et al. (2002b)). Banding and recovery and other mark–recapture methods used to estimate dispersal are often restricted to a finite study area, thereby potentially underestimating dispersal distance (Dawideit et al. 2009; Tittler, Villard & Fahrig 2009). Further, estimates of dispersal distance have been derived from landscapes which themselves vary greatly in the extent of modification experienced by the species studied. The relatively simple model presented in this study accounts for 53% of the variation in estimates of median natal dispersal distance from a world-wide data set. Given the uncertainty and possible biases in data on natal dispersal distance that were used in our study, and the range of species covered, this model predicts natal dispersal distance remarkably well.

Dispersal ability and vulnerability to habitat fragmentation

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

In the past, the categorical classification of mobility has impeded the identification of correlations between vulnerability to habitat changes and dispersal ability (Ferraz et al. 2007; Mac Nally et al. 2009). By using predictions of median natal dispersal distance for 57 woodland bird species, we found compelling evidence of a negative relationship between predicted dispersal distance and the size of the response to tree cover fragmentation in northern Victoria. Given the modest effective sample size, uncertainty in the relationship is to be expected. While future validation of the predictive capacity of the models presented in this study is warranted, prior estimates of dispersal ability might be used in the future to help identify species at increased risk from landscape-scale changes in habitat aggregation.

A number of factors may complicate the relationship between dispersal distance and response to habitat fragmentation in this study. The predictive model of dispersal distance is built predominantly on bird species of the northern hemisphere and may not accurately predict natal dispersal distance for Australian birds. For example, cooperative breeders are more common in Australia than other regions (Cockburn 2003). Blackmore, Peakall & Heinsohn (2011) recently reported estimates of dispersal for the cooperatively breeding grey-crowned babbler that were significantly shorter than our model estimates. Juveniles of cooperative breeders may improve breeding success by waiting to fill breeding vacancies within their social group. Dispersal in these species therefore tends to be delayed and occurs less frequently than in other species. The predictive power of the natal dispersal distance model for Australian birds will be improved through the expansion of the data set to include natal dispersal estimates for Australian bird species as they become available. More broadly, as more data are incorporated and knowledge becomes more refined, contextual modifications of the general model can be made to predict particular faunas or biogeographic regions.

The fitted relationship between dispersal ability and response to habitat fragmentation has an r2 of 0·19. This is typical of the amount of variance explained in complex ecological studies, where many factors may influence the response variable being investigated (Møller & Jennions 2002). A species’ response to habitat fragmentation is influenced by a complex interaction of many factors, such as sensitivity to surrounding land uses and propensity to use modified habitats, metapopulation characteristics, resource availability, interactions with other species and habitat quality (Watson, Whittaker & Freudenberger 2005; Van Houtan et al. 2007; Haslem & Bennett 2008b; Mac Nally et al. 2009; Coulon et al. 2010; Sigel, Robinson & Sherry 2010). It is unreasonable to expect a single factor to predict well. Nonetheless, our analysis suggests that predicted median natal dispersal distance can explain some (∼20%) of the variation in a species’ response to fragmentation. This finding is consistent with the long-held but seldom demonstrated theory that disrupted dispersal is one of the main mechanisms by which habitat fragmentation adversely impacts on avian populations, leading to local extinctions (Ferraz et al. 2007; Sunnucks 2011).

Utility and value of prior information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

We have demonstrated here the use of informative Bayesian priors for improving the precision of demographic estimates and estimating unknown demographic parameters and responses to disturbance in ecological studies, as in previous work (Martin et al. 2005; McCarthy & Masters 2005; McCarthy, Citroen & McCall 2008). We have also demonstrated the value of prior information by estimating the amount of information – expressed as an effective sample size – contained in the predictions of dispersal distance. Over the range of median natal dispersal distances predicted in our study, the information contained in the predictions is equivalent to sample sizes of between around 3 and 8 individuals for each species. Because priors become less influential as the sample size of the data increases above the effective sample size of the prior (McCarthy 2007), the estimates of median dispersal distance presented in this study will be most valuable as informative priors when actual data on dispersal distances is available for relatively a few individuals. Still, the investment required to collect data on natal dispersal distances for even small numbers of birds is not trivial, and it is not uncommon for published dispersal distances to be estimated from less than 10 observations (see Paradis et al. 1998; Sutherland et al. 2000).

Caveats and limitations

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

It is possible that the apparent relationship between predicted dispersal distance and response to fragmentation may be at least partially attributed to another correlate of body mass, rather than dispersal ability. For example, larger birds tend to be longer-lived than smaller species (Ricklefs 2000; McCarthy, Citroen & McCall 2008). While larger animals are thought to be more vulnerable to the effects of habitat fragmentation over the long-term, they may take longer to show responses such as changes in species prevalence than smaller animals simply because they are longer-lived (Ewers & Didham 2006). Without comprehensive and long-term data sets, it is difficult to tease apart causal factors in fragmentation studies: future investigations, particularly demographic studies (e.g. Major, Christie & Ivison 1999; Holland & Bennett 2010), will continue to improve our understanding of these processes.

Our study is subject to a number of assumptions and limitations, some of which are common to fragmentation studies. First, tree cover is used as a surrogate for habitat cover in determining aggregation. While we restricted the analyses to woodland species that depend on wooded habitat, not all species will necessarily be equally affected by the fragmentation of tree cover in the landscape. Some, such as those associated with the shrub and ground layer, may be more vulnerable to the quality of the wooded habitat rather than presence or absence of tree cover per se. Similarly, we have assumed that the surrounding environment is consistently inhospitable to all species. Different species respond to the increasing proportion of matrix in fragmented landscapes in different ways (Watson, Whittaker & Freudenberger 2005; Van Houtan et al. 2007) and have differing abilities to cross ‘gaps’ between suitable habitat (Sigel, Robinson & Sherry 2010). Other species are able to take advantage of new ‘habitat’ provided by intervening land uses, such as scattered trees in farmland (Andren 1994).

Second, we used the prevalence of a species in a landscape as a measure of the ‘health’ of the species. This means that we are not detecting demographic changes within each landscape other than prevalence. Declines in abundance that are not reflected in patterns of prevalence, for example, are not accounted for in this study. It is also critical to note that sampling effort was fixed per landscape and only wooded habitats were sampled. Sampling was representative of the different types of wooded habitats remaining but not proportional to tree cover (Radford & Bennett 2007). Thus, any decline in incidence represents a decrease in occurrence for a fixed area sampled as a function of landscape tree cover. That is, it is a disproportionate decline, over and above that because of habitat loss alone. Finally, in this study, we have used the aggregation of habitat in each landscape (as calculated by Radford & Bennett (2007)) as an indicator of landscape fragmentation. Other studies have used indicators such as patch size, distance to nearest neighbouring patch, and length or proportion of edges (Fahrig 2003). The variety of metrics used impedes direct comparison with other studies.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

This research was supported by an Australian Research Council Discovery Grant DP0985600. Much of the morphological and dispersal data were collated from the literature by Rebecca Citroen. We are grateful to three anonymous referees who offered advice on an earlier version of this manuscript.

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Predictive model of avian natal dispersal
  8. Dispersal ability and vulnerability to habitat fragmentation
  9. Utility and value of prior information
  10. Caveats and limitations
  11. Acknowledgements
  12. References
  13. Supporting Information

Table S1. Median dispersal distance and species’ attribute data used to develop a predictive model of dispersal ability.

Table S2. Species attribute and dispersal information and sources for 17 bird species used to investigate the predictive capacity of the natal dispersal distance model.

Table S3. Species attribute information and predicted ln[median Fdispersal distance] (mean and 95% credible intervals] – as estimated by the DIC-best model - for 57 woodland bird species of northern Victoria.

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