A PIECEWISE SMOOTH RANGELAND GRAZING MODEL SHOWS THAT SETTING RECOVERY PERIOD OF PLANT IS BENEFICIAL TO PREVENT OVERGRAZING

Abstract

Recently, people pay more attention to overgrazing due to its extensive damaging effects on ecological systems. How to solve the remarkable problem is essentially an important topic. In this paper, setting the recovery period of plant population, we simulate the dynamics of rangeland grazing pattern with a mathematical model. Using the corresponding difference system determined by the model, we obtain the critical value of recovery period which is a ratio of the birth rate of grazer population during the grazing period to the sum of it and the death rate of grazer population during the recovery period. If the recovery period is above the critical value, the grazer population will become extinct; if the recovery period is below the critical value, we obtain the explicit expression of the positive periodic solution which is globally stable, i.e. two kinds of population are all persistent. Further, we find the maximum of grazer population's density when the density of grass population is not below a threshold. Our research demonstrates that properly timed grazing plays a key role in protecting grassland environment, or say, the recovery period of plant population is below the critical value; managed grazing is arguably the only natural process by which grassland can be improved on a sustainable basis. Controlled grazing can regenerate grassland, preserve healthy soil and improve livestock production simultaneously. Our results are valuable for rangeland grazing management in practice.