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addition of noise (Markus et al. 1987). The notion is similar to the observation
that whitening can actually enhance a signal (Bezrukov and Vodyanoy 1997).
The consequence of randomness in a predator-prey system depends on the
magnitude of noise, the autocorrelation structure and distribution of the sto-
chastic process, and nonlinearities in the model. To illustrate the consequences
of stochastic variation in a predator-prey model I have plotted the outcome of
adding normally distributed white noise to populations in a stable limit cycle
(figure 8.2). Using Caughley's (1976) two-species model I simulated trajecto-
ries with variable K ( t ) (prey population). With
s
= 500 the population follows
the stable limit cycle of the deterministic model, but with
s
= 5,000 the under-
lying stable limit cycle is difficult to see (figure 8.2, bottom). Still, the popula-
tion persists.
In the predator-prey model I developed for wolf recovery in the Greater
Yellowstone Ecosystem (Boyce 1992b), an interesting pattern emerged with
the addition of stochastic variation. In the deterministic model, we found con-
vergence to equilibrium. With the addition of noise, however, autocorrelated
oscillations emerged in the number of wolves, with the population apparently
fluctuating on an attractor (Boyce 1992b). Tuning parameters in the model
permits detection of such an unrealized attractor, but this attractor is visited
only when stochastic variance causes the system to take excursions away from
the simple equilibrium behavior predicted for the mean parameter values.
AUTOREGRESSIVE MODELS
Our ability to collapse the essential dynamics of a multiple-equation set of dif-
ferential equations into a difference equation has important ramifications.
Even though one might not have data for all components of a complex eco-
logical system, Poincaré's results suggest that we can capture the essential
dynamics in a much simpler model in discrete time (Schaffer 1985). This idea
is fundamental to the use of autoregressive models for characterizing the
dynamic features of the system. Royama (1992) provides a useful introduction
to this statistical mechanics approach to modeling that can easily embrace
predator-prey dynamics.
The general form of the model is
ln[ N ( t + 1)/ N ( t )] = a 0
+ a 1 ln[ N ( t )] + a 2 ln[ N ( t - 1)]
(8.15)
or, equivalently,
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