The resultant breeding design is complex, and we are interested in assessing two questions: (i) how heritable is dispersal and (ii) does dispersal ability show a genetic shift associated with invasion history (i.e. distance through the transect across tropical Australia)? We assessed both of these questions using a single statistical model that accounts for these possible sources of variance and simultaneously estimates (i) the mean within-population additive genetic variance across the transect and (ii) the effect of transect distance on additive genetic values (breeding values). Our model was instituted in WinBUGS using code and model (an ‘animal model’) modified from that of Waldmann (2009). We used standardized log-transformed mean daily displacement as our measure of dispersal for each individual (*d*_{i}, which was approximately normally distributed). Thus, for parents,

where *S* is body size (also standardized), *T* is start date for radiotracking (incorporated to account for weather and time effects on field movement), *B* is an additive genetic value (a factor, the breeding value), and *ε* is the error in fit. Subscript i refers to the individual under consideration. *T* is a random factor (mean of zero), and *B* is a random factor with a mean proportional to distance from the transect midpoint (set to zero), where a negative distance represents populations close to the invasion source and a positive distance represents populations close to the invasion front, i.e.

where *K*_{i} is transect distance and *A*_{i} is the remaining additive genetic effect (random factor, mean of zero and coefficient fixed at one). Thus, *B*_{i} estimates the individual’s breeding value but partitions that value into within- and between-population effects. The between-population relationship is assumed linear (with the caveat that an actual nonlinear relationship will result in biased parameter estimates). Offspring had a similar model, but with five additional factors,

where *C* is the breeding cohort, *X* is whether or not the individual came from a population cross (which is confounded with paternity order in our design), *N* is the large tadpole-rearing container, *R* is basket within container, and *D* is the effect of dam (the maternal effect). *X* is a fixed factor, whereas *C*, *N*, *R* and *D* are random factors. In the case of the offspring, *B*_{i} is considered a random factor with mean equal to the mean of the breeding values of each parent, and a variance equal to half that around parental additive genetic variance (Lynch & Walsh, 1998; Waldmann, 2009):

where *B*_{D} and *B*_{S} are the breeding values of an individual’s dam and sire, respectively (estimated in first equation). This error structure should exclude all but heritable variance from our estimate of *B*.

We fitted this model using minimally informative priors in WinBUGS (Lunn *et al.*, 2000; Waldmann, 2009). Convergence was assessed using three chains with randomly generated initial values. Convergence was reliably achieved within 50 000 iterations, after which we took an additional 200 000 samples from each chain to estimate the posterior distributions of our parameters.