#### study site and data collection

The dispersal of a cohort of YOY brook charr along the lake shoreline was studied in Mykiss Lake (2480 m perimeter, 23·5 ha), Algonquin Provincial Park, Ontario, Canada (45°40′ N 78°13′ W) in the spring of 2003. Mykiss Lake has an unexploited, self-sustaining population of brook charr with high densities of YOY. A single littoral spawning site spans approximately 60 m of shoreline on the west end of Mykiss Lake (Fig. 1). Two types of groundwater habitat along the north shore of the lake act as thermal refugia during the summer: three seasonally flowing inlet streams, fed primarily by shallow groundwater, and 14 zones of groundwater discharge (Borwick, Buttle & Ridgway 2006; see Fig. 1). None of the inlet streams hold resident populations of brook charr, but during the spring one stream drains a smaller lake holding a small population. A permanent block net was set across this stream at the outlet of the smaller lake to prevent downstream migration into Mykiss Lake. Temperatures in groundwater habitats and at a single shoreline location without groundwater discharge (260 m from the centre of the spawning site, north shore), were recorded at 8-min intervals (StowAway TidbiT loggers, Onset Computer, Bourne, MA, USA).

The lake shoreline was delimited into 248 sections, each 10 m in length, which collectively covered the entire lake perimeter (Fig. 1). Field observations began on 26 April, which closely coincided with the start of the emergence period (as determined by field observations of embryonic YOY with residual yolk sac), and continued until 21 June. Over this period, the spatial distribution of YOY was determined 12 times, in surveys conducted 3–5 days apart, by visually counting YOY brook charr while snorkelling the lake perimeter (Table 1). Counts were made between 08.00 and 18.30 h by a single observer (MFC) that moved across 10-m sections, staying within 1 m from the edge of the inundated shoreline vegetation and woody debris, and called counts through the snorkel to a second observer floating offshore in a canoe. YOY in Mykiss Lake are consistently found close to the surface and to structural cover such as fallen trees projecting offshore over deeper water, and do not appear to be disturbed by the presence of an observer (Biro & Ridgway 1995). Fallen trees were carefully surveyed by swimming out from shore on one side and returning to shore on the other. YOY in shallow, inaccessible areas were counted from above the surface while the observer kneeled on the substrate. Sampling time per section was 5·45 ± 2·16 min (mean ± SD for 24 sections without offshore trees). All YOY counted within a given section were assigned a common dispersal distance, measured from the midpoint of the section to the midpoint of the spawning site.

Table 1. Date, time since beginning of emergence, shoreline temperature in nongroundwater habitat (daily mean), total counts of fish over all sections sampled, fraction of dispersal range sampled (*F*), and estimated cohort size (total counts × *F*^{−1}) for 12 lake-wide surveys of YOY brook charr Survey | Date | Time (days) | Temperature (°C) | Total counts | Fraction sampled | Cohort size |
---|

1 | 26, 27 April | 3·5 | 6·7 | 64 | 1 | 64 |

2 | 30 April; 2, 3 May | 8·5 | 8·6 | 218 | 1 | 218 |

3 | 6, 7 May | 13·5 | 10·9 | 365 | 1/3 | 1095 |

4 | 10, 11 May | 17·5 | 14·0 | 1064 | 1/2 | 2128 |

5 | 14, 15 May | 21·5 | 12·3 | 1045 | 1/2 | 2090 |

6 | 19, 20 May | 26·5 | 15·9 | 737 | 1/3 | 2211 |

7 | 23, 24 May | 30·5 | 14·8 | 703 | 1/3 | 2109 |

8 | 27, 28 May | 34·5 | 15·3 | 673 | 1/3 | 2019 |

9 | 2, 3 June | 40·5 | 15·5 | 359 | 1/3 | 1077 |

10 | 8, 9 June | 46·5 | 18·0 | 372 | 1/3 | 1116 |

11 | 14, 15 June | 52·5 | 18·3 | 332 | 1/3 | 996 |

12 | 20, 21 June | 58·5 | 21·2 | 250 | 1/3 | 750 |

Counts were made in all sections along the full range of the spatial distribution of YOY in the first two surveys. Once the distribution of YOY covered the entire shoreline, it was not feasible to inspect carefully all 248 sections within a short period of time. Therefore, for surveys 3–12, counts were made only for a subsample of sections from strata of equal length distributed uniformly over the entire lake perimeter. In surveys 4 and 5, two sections were randomly sampled from individual strata comprising four sections each, and in the remaining surveys one section was randomly sampled from strata comprising three sections each, so that either one-half or one-third of the lake perimeter was sampled in each of these surveys (Table 1). Sampling began at a randomly selected section and continued around the lake perimeter in a randomly determined direction. Although YOY did migrate up the three inlet streams, only YOY observed within the boundaries of the lake shoreline were considered in this study. A permanent net set across the Mykiss Lake outlet was checked regularly and used to monitor emigration. When the 10-m section encompassing the outlet was sampled in one of the 12 surveys, captured YOY were assigned the dispersal distance corresponding to that section (735 m) and released to the stream.

To assess the accuracy of the counting technique, 52 10-m sections distributed around the lake were sampled twice, first at normal search speed and again, 5 min later, using as much time as necessary to ensure detection of all YOY present. Sections sampled twice had representative levels of cover and first sample counts covering the full range of counts recorded in this study. Because accuracy of counts declined with increasing YOY abundance, a fitted power function was used to correct all counts from the 12 surveys: corrected count = 0·85 (original count)^{1·20}; *r*^{2} = 0·97; *n* = 52; *P* < 0·0001.

To confirm that YOY dispersal was primarily along the lake shoreline rather than in deeper water, fine-meshed gillnets (12·5 mm stretched mesh, 0·1 mm filament diameter; Lundgrens Fiskredskap, Stockholm, Sweden) were set at the west end of Mykiss Lake, where YOY densities were highest. Six gillnets (10 m × 1·5 m) were set along the bottom of the lake and perpendicular to the shoreline (depth range: 0·5–4 m), from 16.00 h on 22 May to 10.00 h on 23 May. This procedure sampled the area beyond the edge of the inundated shoreline vegetation, where YOY brook charr were observed most frequently in a study using nets of the same specifications (Venne & Magnan 1995). A similar survey was conducted on 19 and 20 July to determine whether disappearance of YOY from most of the lake shoreline by the last survey in late June could be due to movement to deeper, cooler waters. Thirteen nets were set on the bottom throughout the lake in offshore regions (depth range: 3–6 m). No YOY brook charr were captured in netting surveys in May or July, although cyprinids of similar size as YOY (May only) and numerous 1- and 2-year-old brook charr were caught (May and July). These observations, together with those of Biro & Ridgway (1995) and Biro *et al*. (1997) strongly suggest that the YOY cohort was restricted to the shallow shoreline of Mykiss Lake throughout the 12 surveys.

#### dispersal models

The models evaluated here were based on two types of dispersal kernel, *f*(*x*,*t*), specifying the probability density function of dispersers along a single dimension of space, *x* (distance from the point of origin), at different times, *t*. The first type of kernel is the normal distribution arising from a simple diffusion process:

where *D* is a diffusion coefficient (m^{2} day^{−1}) quantifying the rate of spread from the point of origin. The second type of kernel, depicting exponential decay of the frequency of dispersers to either side of the point of origin, is a Laplace distribution modified to allow for temporal variation in δ, the mean lateral displacement of dispersers (m):

specifically, δ_{t} = δ_{0}*t*^{r}, where δ_{t} is the mean lateral displacement up to time *t* (d), δ_{0} is the mean lateral displacement 1 day after the onset of dispersal, and the exponent *r* determines how the lateral displacement changes through time. For both types of kernel, the distributions are symmetric about the origin.

The kernels described above are one-group models assuming that the population comprises a single component that is homogeneous with regard to movement behaviour, i.e. that all individuals share a common displacement parameter. To account for potential heterogeneity arising from the presence of population components differing in movement behaviour, dispersal kernels were also combined as discrete mixtures of two distributions, one for sedentary individuals and the other for mobile individuals, to form two-group models (Skalski & Gilliam 2000; Rodríguez 2002) of the general form:

*f*(*x*,*t*) = *p g*_{s}(*x*,*t*) + (1 − *p*) *g*_{m}(*x*,*t*)

where *f*(*x*,*t*) is the dispersal kernel for the entire population, *p* is the proportion of sedentary individuals in the population (assumed constant through time), *g*_{s}(*x*,*t*) is the dispersal kernel for sedentary individuals, and *g*_{m}(*x*,*t*) is the dispersal kernel for mobile individuals. In all, four dispersal models were assessed: the one-group diffusion (D1S) and exponential (E1S) kernels, and the two-group diffusion (D2S) and exponential (E2S) kernels. For the D2S model, the diffusion kernels *g*_{s} and *g*_{m} had different diffusion coefficients, whereas for the E2S model the exponential kernels *g*_{s} and *g*_{m} had different mean lateral displacements but a common exponent *r*. Although both diffusion and exponential models can account for asymmetrical dispersal about the origin (e.g. by incorporating a term for directional drift in advection–diffusion models; Skalski & Gilliam 2000), only symmetric models were considered here because there was little evidence for directed dispersal of YOY brook charr in Mykiss Lake (see Results, below).

Because the lake perimeter forms a closed loop, discrepancies between the spatial distribution predicted assuming an infinite domain of dispersal and that observed in the lake can result from ‘looping’ movement of fish beyond ± 1240 m from the spawning site, the maximum displacement observed (Fig. 1). Looping movements appeared to be negligible only for the first two surveys (see Results, below). Therefore, prior to calculation of all predicted values, the dispersal kernels were adjusted to account for truncation of the spatial range of observation at 1240 m, i.e. for the fact that displacements > 1240 m would not have been recorded correctly. Compound probability densities over the truncated range, *h*(*x*,*t*), were obtained for all models as the sum of densities for displacements that do not traverse the boundary 1240 m from the spawning site, *f*(*x*), those that traverse exactly once, *f*(*x* ± *L*), and those that traverse exactly twice, *f*(*x* ± 2*L*):

where *L* is the lake perimeter and index *i* takes integer values (Fig. 2). For all models, the area under the density function *h*(*x*,*t*) was always > 0·999 when *n* = 2. The compound densities *h*(*x*,*t*) model the spatial distribution of the whole cohort along the truncated range of observation and are used for model comparisons, but truncation is an arbitrary observational constraint the importance of which varies from lake to lake as a function of perimeter length. Therefore, substantive ecological interpretation of dispersal behaviour should be based on the dispersal kernels *f*(*x*,*t*), which better reflect how the fish would actually move along an unbounded domain of dispersal, independently of truncation.

#### model fitting and cross-validation

A two-step approach based on cross-validation was used to assess model predictions. The lake-wide dispersal distributions from the 12 surveys were split into two subsets, a calibration subset consisting of the distributions from the first two surveys only, and a validation subset consisting of the remaining 10 distributions. In the calibration step, parameter estimates for the four models were obtained by minimizing the sum of negative log-likelihoods over all of the observed displacement distances simultaneously for the first two surveys. Then, keeping these parameter values fixed, the models were used to predict the dispersal distributions for the 10 remaining surveys, i.e. predictions for surveys 3–12 were projections based on model fits to data from only the first two surveys. A similar approach was used in a previous study of fish dispersal, in which models calibrated to data from the first two of four surveys were validated against data from the last two surveys (Skalski & Gilliam 2000).

In the validation step, model predictions were compared with observed values for five properties of the dispersal distributions: (1) counts of YOY brook charr in individual shoreline sections; (2) mean lateral displacement (the mean, or first moment, of distances from the point of origin, in absolute value, i.e. ignoring dispersal direction); (3) variance of displacements (the second moment of distances from the point of origin, which measured spatial spread of the cohort from the spawning site); (4) kurtosis of displacements (*g*_{2}, the fourth moment of distances from the point of origin, standardized for variance, a measure of the shape of the ‘shoulders’ of the distribution); and (5) the percentage of YOY brook charr in the ‘tail’ region of the dispersal distributions, defined arbitrarily as the 12 shoreline sections most distant from the spawning site in either direction around the lake (24 sections in all, covering 10% of the lake perimeter; −1120 to −1240 m and 1120 to 1240 m; Fig. 1).

For the first property of the dispersal distributions, counts in individual shoreline sections, model performance was assessed by comparing the relationship between observed and predicted counts for each of the 12 surveys with that expected if model predictions were unbiased (corresponding to a 1 : 1 relation). Predicted counts for a section *x* metres away from the spawning site at time *t*, *C*_{pred}(*x*,*t*), were obtained from the relation:

where *H*(*x,t*) is the proportion obtained from the compound density function *h*(*x*,*t*), and *C*_{obs}(*x*,*t*) the observed count, at position *x* and time *t*. The summation index *j* runs through all sections examined by the snorkeller at time *t*. This formula apportions the total counts spatially so that the proportion of counts at a given position matches that derived from the compound density function. For the four remaining properties of the dispersal distributions, models were evaluated by comparing the temporal trajectories of observed and predicted values and assessing the mean error of prediction over surveys 3–12. Because the variance of the observed values and the deviation between observed and predicted values tended to increase with the magnitude of the observed values, two measures of error incorporating different weighting functions for the deviations were used; weighted squared error: *w* (*x*_{obs} − *x*_{pred})^{2}, where *w* is inversely proportional to the variance of *x*_{obs}, and relative error: |* x*_{obs} − *x*_{pred} |/*x*_{obs}.

Variances and confidence intervals for observed values were obtained by assuming an underlying binomial distribution for proportions, and by bootstrapping for statistical moments (percentile method; 10 000 draws). Confidence intervals for model parameters and predicted trajectories were obtained by bootstrapping based on individual draws that randomly sampled 64 observations from survey 1 and 218 from survey 2 (cf. Table 1), to mimic the actual sampling scheme. For all models, the predicted statistical moments and proportion of YOY under the tails of the compound density function *h*(*x*,*t*) were obtained by numerical integration. Calculations were done in the R environment (version 2·1·0; R Development Core Team 2005).

Dispersal of a hypothetical cohort was simulated to compare the predictions of the four models for movement along an unbounded spatial domain. The bootstrap fits of the dispersal kernels, *f*(*x*,*t*), were used to obtain predicted values for the mean lateral displacement and the percentage of long-distance displacements expected after 58·5 days. Long-distance displacements were defined arbitrarily as those > 1240 m away from the spawning site, to provide an estimate of the percentage of the cohort that moved beyond that boundary. However other boundaries for long-distance displacement could have been used as well, because the ecologically interesting question is how predictions of the four models differ when fish displacements can be observed precisely, i.e. in the absence of looping.