Ecological Archives A015-024-A1

Marissa L. Baskett, Simon A. Levin, Steven D. Gaines, and Jonathan Dushoff. 2005. Marine reserve design and the evolution of size at maturation in harvested fish. Ecological Applications 15:882–901.

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 $ a$, $ m(a)$, multiplied by the survivorship to age $ a$, $ l(a)$, integrated over all ages:

$\displaystyle R_0 = \int_0^{a_{max}}l(a)m(a)da.$ (A.1)

Using the growth function $ g(s) = ds/da$ to substitute size $ s$ for age simplifies the mathematics of describing $ R_0$ 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 $ f$ in location $ x$ (in a protected or harvested area) is
$\displaystyle R_0(x,f) = \int_{S_0}^{S_{max}}\frac{l(s,f,x)m(s,f)}{g(s,f)}ds,$ (A.2)

where $ S_0$ is the initial size and $ S_{max}$ is the asymptotic maximum size.

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

$\displaystyle m(s,f) = \left\{ \begin{array}{ll} 0 & s < f \\ A s^B & s \geq f \end{array}\right.,$ (A.3)

where $ A$ incorporates elevated levels of mortality during the pelagic larval stage; i.e., $ m(s,f)$ represents the number of settling offspring before density-dependence. We base the fecundity parameters $ A$ and $ B$ 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 $ R_0$ (Table A1).

TABLE A1. Fecundity parameters for Eq. A.3. $ A$ is the only calibrated parameter (see text), where $ B$ is from length-weight relationships; all other parameter values are available from the references cited in the Methods: Parameterization section.

Species $ A$ $ B$
Atlantic cod $ 3.6273\times10^{-6}$ $ 3.0$
Bocaccio $ 3.7682\times10^{-6}$ $ 3.061$
Yelloweye rockfish $ 2.6563\times10^{-6}$ $ 3.222$
Red snapper $ 2.6286\times10^{-6}$ $ 2.953$







The survivorship is

$\displaystyle l(s, f, x) = \exp\left[-\int_{S_0}^s \frac{u(S, x)}{g(S, f)}dS\right],$ (A.4)

where the mortality $ u(s,x)$ depends on natural mortality $ d$ and the size relative to minimum harvest size $ S_h$
$\displaystyle u(s, x) = \left\{ \begin{array}{ll} d & s < S_h \\ d+h(x) & s \geq S_h\end{array}\right.,$ (A.5)

and the harvest mortality $ h(x)$ depends on the location
$\displaystyle h(x) = \left\{\begin{array}{ll} h_R & x \in \mbox{reserve} \\ h_H & x \notin \mbox{reserve}\end{array}\right..$ (A.6)

In a slot fishery, with both a minimum harvest size $ S_h$ and a maximum harvest size $ S_m$, the mortality is
$\displaystyle u(s,x) = \left\{ \begin{array}{ll} d & s < S_h \\ d+h(x) & S_h \leq s \leq S_m \\ d+h_m(x) & s > S_m \end{array}\right..$ (A.7)

Here, after the maximum size limit individuals experience catch-and-release mortality at a rate $ h_m(x)<h(x)$.

Finally, if size $ s$ represents length, we assume individuals have a piecewise growth function, with a constant growth rate $ k_j$ before maturation and a slower, asymptotic growth rate toward maximum $ S_{max}$ at rate $ k_a$ after maturation:

$\displaystyle g(s, f) = \left\{ \begin{array}{ll} k_j & s < f \\ k_a(S_{max}-s) & s \geq f \end{array}\right..$ (A.8)

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 1997a,b; Perrin and Ruben 1990). Our growth parameters ($ S_{max}$, $ k_a$, and $ k_j$ ) depend on the available von Bertalanffy growth parameters ( $ L_{\infty}$ and $ k$ ) according to $ S_{max}=L_{\infty}$, $ k_a=k$ , and $ k_j=L_{\infty}(1/(1-k)-1)$, analogous to Roff (1983).

Combining Eqs. A.2A.8 indicates how size-at-maturation phenotype $ f$ and location in a protected or harvested area $ x$ determine the fitness $ R_0$ 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. 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.

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