Here we assume reduced fitness resulted from selection on a *single* quantitative trait. We used a quantitative genetic model (Fig. 2) to evaluate the parameter space in which selection could result in a >30% genetic fitness decline in naturally spawning offspring from first-generation hatchery-born parents. In other words, here we are asking: just how extreme would values of key parameters have to be for selection on a single trait to cause 30% declines after a single generation? Domestication selection is considered to work during captive rearing, in which a quantitative trait (e.g. growth rate) is selected differently from in the wild (and so the optimal trait value is shifted from that in the natural environment). We assume truncation selection for this step, in which only fish with trait values above a threshold are viable (Fig. 2). Such traits under selection in a hatchery are not well understood yet, although some candidate traits exist (see below). After hatchery fish are released into the wild, they are allowed to produce natural-born offspring. At this step, viability selection will work against the trait selected in captivity. To measure the strength of natural selection in this step, we used a Gaussian model of stabilizing selection, following Lande (1976) and Ford (2002). So to summarize, we begin with individuals adapted at equilibrium under stabilizing selection to a natural environment. They experience strong truncation selection in the novel (hatchery) environment, which has a different fitness optimum from the natural environment. Their offspring are then exposed to the original, natural environment, and experience strong, directional selection back towards the original optimum phenotype. We ask under what conditions this offspring generation will suffer mortality >30%.

Before selection, we assume that the wild population is at an equilibrium state at which fitness in the wild is maximized for individuals with phenotypic value (*z*) equal to zero (Fig. 2). For simplicity, the standard normal phenotypic distribution, *N*(0, 1), is considered at this state. Therefore the mean trait value is zero and the phenotypic variance is one in the wild population. This assumption should not restrict our results because any phenotypic variables that are distributed normally can be easily standardized. In this model, we also assume that the selection function acting on this trait is also Gaussian with mean zero and variance *ω*^{2 }+^{ }1, (Ford 2002; Estes and Arnold 2007). Because *ω*^{2} determines how tightly phenotypic variation is restricted by stabilizing selection around the optimal value, it represents the strength of natural selection in the wild (smaller *ω*^{2} represents stronger stabilizing selection).

We assume that broodstock are collected at random from the wild population and that the number of broodstock is large enough to represent the phenotypic distribution in the wild. After hatchery fish are created, they are subject to truncation selection favoring the trait values adaptive to the captive environment. For comparison, we consider four different levels of truncation selection (*L*_{T} = 0.15, 0.50, 0.90, and 0.99), which determine the truncation points (*T*) in the phenotypic distribution (Fig. 2). The selection differential (*S*) between the natural-born offspring from the hatchery fish and those from the natural-born fish can be obtained from the breeder’s equation (Lynch and Walsh 1998) as

- (1)

where *h*^{2} is the realized heritability and is the mean trait value (shifted from zero) after truncation selection and before reproduction. is calculated as (Lynch and Walsh 1998)

- (2)

where *p*(*x*) is the probability density function of *N*(0, 1) in this case. When *ω*^{2} >> *h*^{2}, we can also obtain the relative fitness *(*RF) of the offspring from hatchery-born fish to those from natural-born fish (from Lande 1976; Eqn (3) in Ford (2002)) as

- (3)

Thus, in the simplest model, *RF* is determined by only three parameters, the level of truncation in a hatchery (*T*), the realized heritability (*h*^{2}), and strength of stabilizing selection in the wild (*ω*^{2}). In Fig. 3 we show the relationship between *RF* and *ω*^{2} with three different levels of heritability (*h*^{2 }=^{ }0.2, 0.5, 0.8). It is intuitively obvious that *RF* is low when the strength of natural selection is strong, the level of truncation in a hatchery is high, and the heritability is high. However, Fig. 3 shows that all three conditions are required to explain a >30% fitness decline following a single generation of captive rearing if selection acts only on a single trait. For example, low levels of truncation (*L*_{T} ≤ 0.5) cannot explain the >30% fitness decline, and neither can *ω*^{2 }>^{ }5 or *h*^{2 }≤^{ }0.2. Thus, the conditions necessary to explain the >30% fitness decline per generation due to selection on a single trait are fairly extreme (e.g. *h*^{2} of various traits in salmonid species are generally lower than 0.5. Carlson and Seamons 2008). On the other hand, if all these conditions are met, then Fig. 3 illustrates that the fitness decline can be very severe (even >50%) within a generation. In the following sections we discuss how realistic each condition is for hatchery fish.