02-05-2012, 01:19 PM
ESTIMATING CHAOS AND COMPLEX DYNAMICS IN AN INSECT POPULATION
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INTRODUCTION
Nonlinear theory in ecology can be summarized as
a broad hypothesis: that the fluctuation patterns of
abundances in many population systems are explained
largely by relatively low-dimensional nonlinear interactions.
Population fluctuations in this view are the
result of stable points, stable periodic and aperiodic
Manuscript received 24 June 1999; revised 3 February 2000;
cycles, chaos, stable and unstable manifolds of invariant
sets, and multiple attractors, with the addition of
some unexplained noise. Some recent applications of
nonlinear modeling in ecology have met with encouraging
success, due in part to improved statistical methods
and stochastic modeling approaches (Costantino et
al. 1995, 1997, 1998, Dennis et al. 1995, 1997, Ellner
and Turchin 1995, Begon et al. 1996, Stenseth et al.
1996, Higgens et al. 1997, Leirs et al. 1997, Bjornstad
et al. 1998, Cushing et al. 1998a, b, Finkenstadt et al.
1998, Dixon et al. 1999, Henson et al. 1999).
Demographic variability
Different types of stochastic mechanisms produce
different patterns of variability. In particular, two broad
classes of stochastic mechanisms important to populations
have been widely discussed: demographic stochasticity
and environmental stochasticity (May 1974b,
Shaffer 1981). In order to connect a deterministic population
model with time-series data, the population
model has to be converted into a stochastic model (Dennis
et al. 1995). The types of stochastic mechanisms
affecting the population therefore have to be carefully
considered and formulated.
Parameter estimation
The method of conditional least squares (CLS) was
used for estimation of the parameters in the stochastic
demographic LPA model (Eqs. 10–12). CLS methods
relax many distributional assumptions about the noise
variables in the vector Et (Klimko and Nelson 1978,
Tong 1990). CLS estimates are consistent (converge to
the true parameters as sample size increases), even if
Et is non-normal and autocorrelated, provided the stochastic
model (Eq. 9) has a stationary distribution.
CLS estimates are based on the sum of squared differences
between the value of a variable observed at
time t and its expected (or one-step forecast) value,
given the observed state of the system at time t 2 1.
For fitting the LPA model to a single time series, there
are three such conditional sums of squares: