Single gene dynamics under controlled mating

TL;DR

This paper develops a stochastic model for genotype dynamics in controlled livestock populations, incorporating herd management strategies and using jump diffusion processes for simulation, to better understand genetic evolution under artificial constraints.

Contribution

It introduces a novel stochastic differential equation model for genotype evolution in controlled herds, including herd management strategies and a discrete-time version reflecting real-world practices.

Findings

Herd size stabilizes below the upper control limit, indicating partial extinction.

The model effectively simulates genotype dynamics under controlled mating and herd restrictions.

Monte Carlo simulations demonstrate the impact of herd management strategies on genetic stability.

Abstract

We seek models for the genotype evolution of agricultural animals, animals involved in primary production processes. Classical models for genotype evolution have tended to be very simple in order that analytic methods may be employed in their study. Unfortunately these models fail to describe processes in artificially controlled populations including agricultural livestock. It is particularly important {to describe such processes} in order to make better use of the massive genotyping data becoming available. We describe an approach to stochastically modeling the dynamics of a biallelic polymorphism herds under conditions of controlled mating {and restriction of herds size from above}. The system of stochastic differential equations that we propose is based on jump diffusion processes to provide an effective platform for Monte Carlo simulation. A feature of the model is the division of the {population} into a {\it main herd} comprised of animals involved in the production process and a {\it replacement herd} of {animals not currently in the production process, typically juvenile animals}. This feature allows for exploring different strategies for adding replacement animals to the main herd without altering the part of the model concerned with the dynamics of the main herd. A discrete-time version of the model has been developed which reflects the typical practice of New Zealand dairy herd management. Our Monte Carlo simulation has demonstrated that an isolated deme whose size is bounded above demonstrate size stabilization at a level less than the control limit, it is looks like partial extinction, the effect being well known in classic models.