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:

(A.1) |

Using the growth function to substitute size for age simplifies the mathematics of describing as a function of size at maturation and incorporating size-selective harvesting. In a size-based population, the expected lifetime reproductive output of an individual with size at maturation in location (in a protected or harvested area) is

where is the initial size and is the asymptotic maximum size.

After maturation, the fecundity depends on size according to constants and ,

where incorporates elevated levels of mortality during the pelagic larval stage; i.e., represents the number of settling offspring before density-dependence. We base the fecundity parameters and on reported values of length-weight conversion and eggs produced per unit weight, with the proportion of eggs produced that become settling juveniles calibrated to produce realistic values of (Table A1).

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.

Species | ||

Atlantic cod | ||

Bocaccio | ||

Yelloweye rockfish | ||

Red snapper |

The survivorship is

where the mortality depends on natural mortality and the size relative to minimum harvest size

and the harvest mortality depends on the location

In a slot fishery, with both a minimum harvest size and a maximum harvest size , the mortality is

Here, after the maximum size limit individuals experience catch-and-release mortality at a rate .

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:

Roff (1983) bases an analogous discrete-time approach on the assumptions that (1) the observed decrease in growth after maturity is due to investment of resources in reproduction rather than in growth and (2) the linear growth function observed in juveniles would continue in the absence of maturation (see also Heino and Kaitala 1997

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
generation.

Heino, M.
and Kaitala, V. 1997*a*. 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. 1997*b*. 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.