Appendix A. Robustness analysis. In addition to testing the effects of different minimum length limits and harvest probabilities as described in the main text, we also tested the robustness of model results to changes in several other functions and parameters.
The growth-survival trade-off (main text Eq. 2d) was expected to influence growth evolution: raising gmax reduces the severity of the trade-off and causes a lower cost of faster growth. Therefore, not surprisingly, raising gmax elevated the genetic growth capacity after 100 years of harvest (Fig. A1a).
For the other robustness tests, we focused on effects on the PMRN intercept, because this trait consistently showed a large evolutionary response to harvest. Nevertheless, detailed results for all traits are presented in Table A1. The magnitude of evolution was positively correlated with changes in the parameter s0, which determines the density-dependent mortality of newborn offspring (main text Eq. 3a); however, the overall difference in the position of the PMRN after 100 years of harvest was small (Fig. A1b). Higher values of s0 produce larger population abundances, which, in turn, produce smaller phenotypic length increments; possibly, there is then compensation in the evolution of the PMRN in response to this slower growth.
We also tested the effect of changing the relationship describing density-dependent growth by varying the parameter c11 (main text Eq. 3f) between 0 (density-independent growth) and 21.5 × 10-8 g–1(causing a strong reduction in growth with increasing population biomass). We observed an increase in the evolutionary response in the PMRN intercept when c11 was increased (Fig. A1c), so that the slowest evolutionary response occurred when the density dependence in growth was switched off entirely.
For the final component of our robustness analysis, we tested the effect of changing the assumed genetic coefficient of variance CVG in the initial population on evolution of the PMRN: as expected, increasing CVG between 0 and 12% accelerated the evolutionary reduction of the PMRN intercept (Fig. A1d).
TABLE A1. Detailed results of the robustness analysis. Results shown are the mean genetic trait values after 100 years of fishing with a harvest probability of 0.5 and a minimum size limit of 60 cm. Intercept and slope are the probabilistic maturation reaction norm (PMRN) intercept and slope, respectively. GSI is the gonado-somatic index.
|Parameter name and value||
|Maximum length increment gmax|
|Density-dependent mortality constant s0|
|1.0 × 103|
|2.0 × 103|
|3.0 × 103|
|4.0 × 103|
|5.0 × 103|
|6.0 × 103|
|7.0 × 103|
|8.0 × 103|
|9.0 × 103|
|10.0 × 103|
|Density-dependent growth constant c11|
|0 10-8 g-1|
|0.01 10-8 g-1|
|0.1 10-8 g-1|
|0.4 10-8 g-1|
|1.0 10-8 g-1|
|2.1 10-8 g-1|
|3.4 10-8 g-1|
|6.6 10-8 g-1|
|10.2 10-8 g-1|
|15.2 10-8 g-1|
|21.5 10-8 g-1|
|Genetic coefficient of variation CVG in the initial population|
|FIG. A1. Robustness analysis for the eco-genetic model of Atlantic cod. Panels show the (a) genetic growth capacity and the (b-d) probabilistic maturation reaction norm (PMRN) intercept after 100 years of harvesting. Parameters varied are the (a) maximal length increment gmax (Eq. 2d), the (b) density-dependent mortality constant s0 (Eq. 3a), the (c) density-dependent growth constant c11 (Eq. 3f), and the (d) genetic coefficient of variation CVG in the initial population. Harvest probability was 0.5, minimum length limit was 60 cm, and results show the means and standard deviations for 30 independent model runs. Triangles indicate the default parameter values from Table 2 and error bars show the standard deviations.|