If the area colonized by the perennials becomes too small relative to that colonized by the annuals, it is possible for the perennials to be completely eliminated through competition. Let denote an area in which perennial grasses and annual grasses are in competition. We consider that grasses and weeds may share a patch of ground, so that both w and g may vary from zero to one. The equations describing their dynamics are The parameters and represent intrinsic growth rates of the grasses and weeds, respectively. The cattle stocking rate is introduced through E .
E decreases linearly with the number of cattle on the plot.
We want to find threshold values of g and w dividing the case in which the grasses are able to compete successfully with the weeds, from the case in which the perennial grasses are so sparse that they can no longer compete, and die off in the area . We are interested in the phase plane only in the unit square. We assume that g is on the horizontal axis, and w is on the vertical axis. Following is a summary of the three equilibria on the axes. One of these points is a stable node , henceforth denoted Y , while the other is a saddle, denoted X .