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where ∆is the growth increment to ungrazed plant standing crop in kg / ha over 3

months, Vis the plant standing crop in kg / ha at the beginning of those 3 months,

and Ris rainfall in mm over those 3 months. The 2.5 coefficient of Rused here dif-

fers from Robertson’s (1987) 3.946 for reasons given by Caughley (1987).

Feeding response

I=86[1 −exp(−0.029V)]

where Iis intake of food in kg dry weight over 3 months, per red kangaroo, assum-

ing a mean body weight of 35 kg and no shrubs in the pasture layer (Short 1987).

Numerical response

r=−0.4 +0.5[1 −exp(−0.007V)]

where ris the exponential rate of increase of red kangaroos on a 3-monthly basis.

The numerical response of the herbivore allows us to calculate the equilibrium

level to which plant biomass will converge in a constant environment under the

influence of an unrestrained population of herbivores, that is, the null isocline. It is

the x-intercept of the regression of rate of increase of the herbivores against plant

biomass or, put another way, the plant biomass at which rate of increase of the

herbivore is zero (see Fig. 12.6). In the absence of seasonality, and of year-to-year

variation in rainfall and temperature, this will be the equilibrium plant biomass imposed

by grazing. The numerical response curve of this example was fitted as:

r=−d+a[1 −exp(−fV)]

and the level of plant biomass Vat which r=0 is solved simply by setting rto zero

and solving for V. Thus:

V=(1/f)loge[a/(a−d)]

which, when loaded with the values of the constants given here on a 3-monthly basis,

yields:

V=(1/0.007)loge[0.5/(0.5 −0.4)] =230 kg / ha dry weight plant biomass

This value is immensely important ecologically. It is the equilibrium level of plant

biomass imposed by grazing in a constant environment. This is of some theoretical

interest but of limited practical importance because environments are not constant.

However, it is also the level of plant biomass above which the herbivore population

will increase and below which it will decrease (thecritical threshold), and that is

true whether the environment is constant or variable and whether the density of her-

bivores is high or low.

Using similar logic, we can calculate the combination of kangaroo and plant

densities at which consumption exactly matches regrowth by plants. This will occur

204 Chapter 12