Single Gene Dynamics under Controlled Mating

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 of 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 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 (by imposition of a fixed size control requirement) 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. Another interesting feature of the model with a size control rule is its sensitivity to a form of a control. We have found that even change a rule to different moment of choice of animal substitution ( from replacement herd to a main one) results in observable variation in herds’ temporal characteristics. We demonstrate several simulation results under the condition of Mendelian inheritance and its corresponding rule of summation. We also propose a variant of the model taking into account animal inflows and outflows providing exchange through an external market.