  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.

Quantitative Methods;Populations and Evolution