Field methods for photo-identification of Hector’s dolphins have been described in detail elsewhere (Slooten, Dawson & Lad 1992; Bräger et al. 2002). Briefly, standardized along-shore transects around Banks Peninsula were followed in 4–6-m outboard-powered boats. When a dolphin sighting was made, all distinctive dolphins in the group were photographed before continuing the transect. Photographs were judged usable if the dorsal fin was in focus, completely visible and perpendicular to the photographer, thereby ensuring that any identifying marks would be clearly visible if present. Individual dolphins were judged to be either unmarked or marked and subsequently assigned to one of the three categories of mark quality defined by Slooten, Dawson & Lad (1992: see this paper for examples). Only dorsal fins with permanent, unambiguous marks were used for further analysis, minimizing the possibility of mark changes and subsequent misidentification of individuals on recapture (Du Fresne 2005). A catalogue of identifiable individuals was maintained, along with a data base containing each individual’s sighting history. Because of involvement in line-transect surveys (e.g. Dawson et al. 2004), no photo-ID fieldwork was conducted in 1998 or 1999.
Data were restricted to captures during November to February inclusive, to maximize the available data whilst attempting to satisfy the assumption of population closure within each sampling period (Pollock et al. 1990) for the capture-recapture model (see below). The data were summarized into a capture history for each individual, where xi,t = 1 indicates that individual i was observed at least once during sampling period t, and 0 otherwise, where t ranges from 1986 to 2006.
We used a modified form of the Cormack–Jolly–Seber (CJS) model to estimate annual survival (Cormack 1964; Jolly 1965; Seber 1965). The CJS model allows for imperfect detection, that is, individuals that are alive may be missed during some sampling periods. We implemented the CJS model using a state-space modelling approach that includes a process model that describes the true state (alive or dead) of each individual at each time period, and an observation model that describes whether an individual was captured at each time period, conditional on it being alive.
The alive state of each individual was modelled as a random variate from a Bernoulli distribution, where Ai,t = 1 indicates individual i is alive at time t, and 0 otherwise:
where φt is the probability of survival from t to t + 1, n is the total number of individuals, ai is the period individual i was first observed, and T is the total number of sampling periods. Survival was modelled on the logit scale allowing for different mean survival before and after the sanctuary was established, with annual variation treated as a random effect:
where and denote the mean survival rate on the logit scale before and after the sanctuary was established, and εt is the random effect on survival over time, normally distributed , where is the annual process variation in survival.
Heterogeneous capture probabilities were included by using the number of times an individual was observed in a sampling period as a covariate for capture during the next period (Fletcher 1994). The probability of observing each individual was modelled as:
where pi,t is the capture probability of individual i at time t, and:
where zi,t is the number of times individual i was observed in period t, and αt and βt are the regression intercept and slope coefficients, respectively. These were modelled as normally distributed random effects:
(eqn 5) (eqn 6)
where and are the random effects on α and β over time, respectively, normally distributed and .
The difference between post- and pre-sanctuary survival (on the logit scale) was calculated as .
The model was fitted in a Bayesian framework using WinBUGS (Spiegelhalter et al. 2004). Vague prior distributions were specified for all parameters: logistic (0, 1) for the mean survival parameters ( and ), normal (0, 100) for the mean coefficient parameters (μ and μ) and uniform(0, 100) for all variance parameters (, and ). The capture probability covariate zi,t was standardized to improve convergence. Three Markov-chains were started from different initial values and run for 10 000 iterations to tune the algorithm. These ‘burn-in’ samples were discarded and the algorithm run for a further 100 000 samples. Each chain was thinned by taking every fifth value and then combined to give a posterior sample of 60 000 for each model parameter. Convergence was assessed visually (Fig. S1, Supporting information) and by confirming that the Brooks–Gelman–Rubin statistic (Brooks & Gelman 1998) had converged to one (Fig. S2, Supporting information). Goodness-of-fit was assessed using a posterior predictive checking procedure (Link & Barker 2010) for a general CJS model (Appendix S2, Supporting information).
Survival rates on the probability scale, which include both sampling and process variation, were obtained by taking 10 000 samples from the posterior distributions of , and :
(eqn 7) (eqn 8)
Our capture-recapture analysis differs from the classical approach in that a single model was specified in which all the parameters are modelled as random effects, that is, the parameters are acknowledged to vary in a random manner with time about some mean, but not in a way that can be explained by covariates and trends (Royle & Link 2002; Schofield, Barker & MacKenzie 2009). This approach is flexible in that it can adequately model constant or time-varying survival without needing to specify different competing models. Furthermore, it naturally separates out the process and sampling variation, enabling us to more easily make inference about parameters of interest such as changes in mean survival. Our code is provided in Appendix S1 (Supporting information).
A fixed-duration, stage-structured matrix model was specified to carry out population projections (Caswell 2001). We used a stage-structured model with three stages: calf (from birth to age 1), juveniles (age 1 to age of first reproduction) and adults (age of first reproduction to maximum age).
The projection model requires a small number of life-history parameters: stage-specific survival (SC, SJ and SA for calves, juveniles and adults, respectively) and fecundity (m), as well as the conditional probability of moving from one stage to the next given survival (γC, γJ and γA). The fecundity rate is defined as the average number of female offspring, per mature female, per year, and therefore, the fecundity rate for juveniles and calves is by definition equal to zero. Estimates of the transition parameters γ are not available directly; however, they can be estimated recursively as functions of the other parameters and asymptotic population growth λ (Caswell 2001). The recursive method for estimating the conditional growth parameters is not guaranteed to converge for all combinations of parameter values; however, they did for the range of parameter values we considered. For our model:
where α and ω are age at first reproduction and maximum age, respectively. Note that γC = 1 by definition (i.e. all surviving calves become juveniles at age one).
Adult survival was estimated from the capture-recapture data using the estimation model described previously (SA = φ). We also assume juvenile survival was equal to adult survival. Field observations of Hector’s dolphins show that like most cetaceans, new-born calves are dependent on their mother for at least the first year of their life. We, therefore, assume that the survival of a calf is dependent on both the survival of that calf, and also of its mother; hence, calf survival is defined as φ2.
Fecundity (m = 0·025) and age at first reproduction (α = 7·55) were obtained from Gormley (2009), who used observations of calf–mother pairs, accounting for adult survival, calf and adult detection probabilities, as well as information on age and reproductive status from the teeth and ovaries of dead animals (see also Slooten 1991; Slooten & Lad 1991).
Maximum age (ω) was inferred from the mark-recapture data set using the number of years that individuals are in the catalogue as a minimum estimate. Four individuals have been seen over a span of 20 years corresponding to an age of at least 22 (Hector’s dolphins do not enter the catalogue until they are at least 2 years old as 1-year-old juveniles typically do not yet have identifying features). This estimate of maximum age is likely to be low: we did not know the age of the four individuals above when first seen, their age was likely to be greater than two, and they may subsequently survive for some years to come. For the purpose of model simulations, we represent uncertainty in maximum age by a simple triangular distribution, with a mode of 26 and minimum and maximum values of 22 and 30, respectively.
The number of parameters required to specify the matrix reduces to annual survival rate (φ), age at first reproduction (α), maximum age (ω) and fecundity (m; Table 2).
The population projection model was run under two different estimates of survival reflecting the pre-sanctuary and post-sanctuary periods. Fecundity, age at first reproduction and maximum age were assumed to be constant for both scenarios.
Simulations were initialized with a starting population of 500 animals (i.e. approximately, the female population size assuming a sex ratio of 0·5). A total of 10 000 projections were each run for 50 years. For each iteration, single values for fecundity and age at first reproduction were sampled from their respective posterior distributions, along with a value for maximum age from the triangular distribution specified earlier. Values for μ and σ were sampled for each iteration, and a value for annual survival rate for each year ϕt was sampled from eqns 7 and 8 for the pre- and post-sanctuary periods, respectively. Each year, the matrix projection model was specified and values for the transition parameters derived. The population was projected with demographic stochasticity included on all parameters. The distribution of population growth rate was obtained by:
where NT is the population size after T = 50 years, and N0 is the initial population size. Population growth is considered to be positive when λ > 1 and negative otherwise. The proportion of projections where λ > 1 is interpreted as the probability of positive population growth.
We calculated the sensitivity of λ to each of the projection model parameters, that is, the relative change in λ for a small absolute change in any parameter θ (Caswell 2001):
Uncertainty in λ was decomposed into contributions because of uncertainty in each of the estimates of the projection model parameters to estimate the amount of variation in λ explained by the variance associated with each parameter. This was carried out for each parameter in the projection model using the sensitivity and variance of that parameter and then scaled to sum to 1:
Because we are interested in the variation because of uncertainty in our parameter estimates (sampling variation), we excluded environmental stochasticity (process variation) in survival and used the inverse-logit of and .