Abstract

Dynamical systems of forage biomass accumulation under continuous harvesting with and without nutrient cycling are formulated and studied. For one dimensional dynamical system with  being a measure of forage intake relative to forage growth, there is bifurcation at whereas at  the system exhibits discontinuous stability, and at the system collapses. For a two dimensional dynamical system, there is bifurcation at, stable equilibrium at  and unstable equilibrium at. For the three dimensional dynamical system where  is a measure of soil nutrient replenishment, at  the system exhibits two different stable states (a fixed point and a limit cycle) and one unstable state (limit cycle), and at  there is only one stable state (limit cycle), and there is hysteresis. For  and , there is local stability but no global stability. It is shown that the long-term stability of the livestock-forage system primarily depends on the stocking levels and initial forage biomass. Through bifurcation analysis, this study has identified critical points of the system for timely intervention by the farmer. Finally, by using dimensional analysis, it is possible to more readily compare relative effects of biological processes.

Key words:  Livestock, forage growth, nutrient, dynamical system, sustainability.