Appendix A. The derivation of fitness.
To calculate fitness, we use the expected lifetime reproductive output, . Generally, the expected lifetime reproductive output in an age-based model is the fecundity at age , , multiplied by the survivorship to age , , integrated over all ages:
After maturation, the fecundity depends on size according to constants and ,
TABLE A1. Fecundity parameters for Eq. A.3. is the only calibrated parameter (see text), where is from length-weight relationships; all other parameter values are available from the references cited in the Methods: Parameterization section.
The survivorship is
Finally, if size represents length, we assume individuals have a piecewise growth function, with a constant growth rate before maturation and a slower, asymptotic growth rate toward maximum at rate after maturation:
Combining Eqs. A.2–A.8
indicates how size-at-maturation phenotype and location in
a protected or harvested area determine the fitness of an individual. This fitness definition uses a size-structured
approach to derive fitness for a model without size structure. Underlying this
approach is the simplifying assumption that size at maturation evolves slowly
enough such that it is approximately constant across size classes within a given
Heino, M. and Kaitala, V. 1997a. Evolutionary consequences of density dependence on optimal maturity in animals with indeterminate growth. Journal of Biological Systems 5:181–190.
Heino, M. and Kaitala, V. 1997b. Should ecological factors affect the evolution of age at maturity in freshwater clams? Evolutionary Ecology 11:67–81.
Perrin, N. and Rubin, J. F. 1990. On dome-shaped norms of reaction for size-to-age at maturity in fishes. Functional Ecology 4:53–57.
Roff, D. A. 1983. An allocation model of growth and reproduction in fish. Canadian Journal of Fisheries and Aquatic Sciences 40:1395–1404.