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

TL;DR

This paper models how hatchery practices can cause abrupt changes in the effective population size in populations with Pareto-distributed family sizes, revealing a transition driven by hatchery input levels.

Contribution

It introduces a mathematical framework analyzing the impact of hatchery inputs on effective population size using a two-component model with power-law family-size distribution.

Findings

Effective population size can undergo a discontinuous jump due to hatchery input levels.

High hatchery production can break symmetry, stabilizing the effective size.

The model explains paradoxical observations of stable effective sizes under hatchery influence.

Abstract

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.