untitled

when NI=∆. We can rearrange terms to isolate kangaroo density on the left-hand

side of the equation, N=∆/I. Both null isoclines for the kangaroo–plant system are

plotted as broken lines in Fig. 12.7.

We can now reassemble the response functions of the system in their proper rela-

tionships to examine dynamics in the absence of stochastic variability in rainfall. We

shall see what these in combination reveal about the system’s dynamic behavior:

V(t) =∆[V(t), R(t)] −N(t)I[V(t)]

N(t) =N(t)r[V(t)]

Under a constant rainfall regime (in this case 60 mm per 3-month period), the

system converges on the equilibrium, that is, the point of intersection of both null

isoclines (Fig. 12.7). This shows that the equilibrium is stable, as one might have

guessed, based on the negative slope of the plant null isocline. Convergence on

the equilibrium is circuitous, involving a burst of plant growth, followed by plant

decline as the kangaroo population stabilizes.

Rainfall can also be simulated as a sequence of random events from a normal

distribution with a mean and standard deviation identical to the Australian data

(Fig. 12.8), and the consequent changes in plant biomass and kangaroo numbers can

be calculated accordingly.

Figure 12.9 demonstrates a typical time trend for kangaroos as generated by the

equations describing the unpredictable rainfall (Fig. 12.8) and the responses to it of

the plants and herbivores. The only external input other than starting conditions are

the random values from the 3-monthly rainfall distributions whose observed means

and standard deviations are given above. The kangaroo population trajectory is a

mathematical consequence of that rainfall, as its effect feeds through to plant growth,

herbivore population growth, and grazing pressure.

d

dt

d

dt

CONSUMER–RESOURCE DYNAMICS 205

0 100 200 300 400

Plant biomass (kg / ha)

Kangaroo density (individuals / ha)

2.0

1.5

1.0

0.5

0

Fig. 12.7Dynamics of
kangaroos and plants
over time, based on
the Australian model
discussed in the text.