Optimising breeding strategies for sheep populations in response to dynamic price: a case study in Xinjiang

Price plays an important role in maintaining the balance between supply and demand in the market. There exists a mutually restrictive relationship between price, consumer demand and the supply of farm produce. Therefore, the purpose of this paper is to provide the optimal breeding strategy for farmers to maximise profits and adjust the balance between supply and demand. This paper took sheep breeding as an example, establishing a dynamic model based on the market price and the relationship between market supply and demand. We solved the equilibriums of the model and analysed their stabilities. Parameters of the model were estimated from sheep stocking and the price of live sheep in Xinjiang Uygur Autonomous Region. Then, we proposed four optimal breeding strategies based on the dynamic models through the Pontryagin maximum principle. Optimal breeding strategies can be achieved by controlling the sales rate and breeding size of livestock. In order to further analyse profits, we selected different price and breeding scale to solve the optimal breeding strategy. From that, we obtained several conclusions: Government subsidies can provide stable profits for farmers. When prices are on an upward trajectory, it is more profitable to increase sales rate or to expand the breeding size. However, there exists a threshold value for market price. In order to gain more profits, farmers should choose to expand the scale of breeding when the market price is higher than the threshold value, and choose to increase the sales rate when the market price is lower than the threshold value.