untitled

predators are presumed to increase and level off according to Holling’s (1959) Type

II functional response (see Chapter 10).

Depending on the parameter values that one uses, this model is capable of a

variety of dynamics. In Fig. 12.1, we demonstrate one possible outcome: cyclic fluctu-

ations of both consumers and resources over time.

Rather than plot densities of both resource and consumer populations against time,

ecologists often plot the density of consumers against that of resource. This is known

as a phase-plane diagram. For example, in Fig. 12.2 we re-plot the data shown in

Fig. 12.1 as such a phase-plane diagram.

The phase-plane trajectory shown in Fig. 12.2 displays a pattern spiraling outwards

from the starting point until it converges on a repetitive pattern known as a stable

limit cycle. For most realistic consumer–resource models, this is a common outcome

(Rosenzweig 1971; May 1972, 1973). If we had started at values outside the stable

limit cycle, we would observe a spiral inwards until the trajectory converged once

again on the stable limit cycle.

198 Chapter 12

80

60

40

20

0

0 100 200 300 400

Time, t

Population density

Resource

Consumers

Fig. 12.1Cyclic
dynamics over time for
the general
consumer–resource
model with the
following parameter
values: a=0.1, h=0.2,
c=0.2, d=0.3, rmax=
0.4, and K=120.

15

10

5

(^0) 0 20406080100120
Resources, V
Consumers,
N
Fig. 12.2Data from
Fig. 12.1 re-plotted as a
phase-plane diagram of
consumers versus
resources.