Figure 1.
Scheme of an asymptotic adult-recruit relationship.
Reproductive output of the adult stock is proportional to the stock size, but through density-dependent mortality during the pre-recruit phase, a stock-recruitment relationship is lacking (after [12]).
Table 1.
Sampling time scheme with corer areas (cm2).
Figure 2.
Illustration of the steps within the analysis of density dependence.
A: regression through observed data of Macoma balthica for the time interval May to June 1997. B: preset slope forced through observed data (in this example slope = 1 for density independence). C: deterministic data calculated from observed data to lie on hypothetical regression line from previous step. D: regression after measurement error, estimated by bootstrapping, was added to deterministic data, one example of 10000 simulations. E: additionally, before adding measurement error, process error was added to logDt+1, to arrive at observed residual deviance from first step, one example of 10000 simulations. F: modelled slope values with prediction intervals for various preset slopes, simulated with observed measurement error alone (white line for slope values and grey bootstrap interval) and observed process error additionally (black dots and whiskers). The dashed line gives the observed slope from step A. In this case the observed slope falls within the modelled prediction interval, whatever the preset slope (i.e. assumed true slope) is.
Figure 3.
Simulation results for Macoma balthica in the time intervals of density decline.
Dashed line: slope value of the original regression through the observed data. White line: average slope values resulting when measurement error is added to deterministic data on preset regression lines. Grey area: corresponding 95% confidence intervals. Black dots: average slope values resulting when process error and measurement error are added to deterministic data on preset regression lines. Whiskers: corresponding 95% confidence intervals. When the lower end of the confidence interval for the preset slope 1 ends above the observed slope value, the observed slope differs significantly from 1 and density dependence is concluded (arrows).
Figure 4.
Simulation results for Cerastoderma edule in the time intervals of density decline.
Dashed line: slope value of the original regression through the observed data. White line: average slope values resulting when measurement error is added to deterministic data on preset regression lines. grey area: corresponding 95% confidence intervals. Black dots: average slope values resulting when process error and measurement error are added to deterministic data on preset regression lines. Whiskers: corresponding 95% confidence intervals. If the lower end of the confidence interval for the preset slope 1 would end above the observed slope value, the observed slope would differ significantly from 1 and density dependence would be concluded.
Figure 5.
Mortality in relation to initial density.
Range of log densities per m2 at the beginning of a time interval, and total mortality (mean logDt - mean logDt+1) during the time interval. Solid line: Macoma balthica (bold: density dependence detected). Broken line: Cerastoderma edule.