Hatchery-induced transition of the effective size in a Pareto population

It seems paradoxical to have observed the absence of reduced effective population sizes $N_{\mathrm{e}}$ under marine hatchery practices. This paper studies the Ryman-Laikre, or two-demographic-component, model of the hatchery impact related to inbreeding in a population with power-law family-size distribution, where hatchery inputs are represented by a Dirac delta function. By examining the asymptotic (i.e. large-population limit) behavior of the normalized sizes (or weights) of families of the mixture population, I derive the distribution properties of the average weight of families (i.e. the sum of the squared weights, $Y$) over the population existing at any given time. The reciprocal of the average weight $Y$ gives the effective number of families (or reproducing lineages) in the population, $N_{\mathrm{e}}=1/Y$. When the specific production in the hatchery (i.e. the number of offspring per broodstock) is low, the most probable value of $N_{\mathrm{e}}$ is close to the lower bound of the $N_{\mathrm{e}}$-distribution. When the specific hatchery-production is increased to a critical value with fixed mixing proportion of hatchery fish, a discontinuous transition takes place, so that the most probable $N_{\mathrm{e}}$ jumps to the upper extreme of the distribution. This hatchery-induced transition is attributed to the breaking of reciprocal symmetry, i.e. the fact that the typical value of $Y$ and its reciprocal (the typical $N_{\mathrm{e}}$) do not vary with the population size in opposite ways. At a high specific hatchery-production, the symmetry breaking disappears.