The distribution of urine deposited on a pasture from grazing animals

Intensive agricultural production practices are known to cause far-reaching effects on water quality. The current paper addresses and quantifies these effects caused by high stocking rates.A set of stochastic difference equations describing the development of the proportion of a grazed field either unaffected by urine deposition, or affected by multiple (1, 2, …) urine depositions is described. A solution to this set of equations is found for the expected value of multiple (0, 1, 2, …) urine depositions, and the variances of these depositions. It is assumed that an animal voids urine with a Poisson probability distribution, and that each urine deposition covers a random area with a Gaussian probability density. Given these reasonable assumptions, the probability distributions for each multiplicity of patch distribution can be found numerically.The utility of the results obtained is illustrated for a problem in assessing the nitrogen (N) pollution of ground water from different grazing strategies. It is demonstrated quantitatively that mob stocking (typical of winter management regimes in New Zealand) is often caused by rotational grazing. The latter is often used to optimize grass growth and intake, especially in winter. This increases (more than linearly) the level of N pollution in ground water. This is because of the increased frequency of multiple urine depositions, i.e. more than one urine deposition on the same patch of land in a short time.