Should all the species of a food chain be counted to investigate the
Letellier C., Aguirre L., Maquet J.,
Labo. Math., Le Havre University
Chaos Sol. Fractals, Vol. 13(5), pp 1099-1113 (2002).
The fairly realistic three species food chain model
given in another paper
(click here), is studied
from a single time series ...
When one uses a single time series, he may be ensured to have obtain the best
quality of reconstruction when a diffeormorphism between the reconstructed
state portrait and the original one is found. Such a quality may be obtained
when the embedding dimension $d_E$ is sufficiently large. According to the
Takens theorem, it must be greater than or equal to $2D_H +1$ where $D_H$
ideally refers to the Haussdorff dimension which could be conveniently
approximated by the counting box dimension estimated by the Grassberger \&
Procaccia's algorithm . Theoretically, i.e. with an infinite
amount of data without any noise, all the dynamical variables are equivalent
and may be used. Unfortunately, when one is facing data from the real world,
necessarily corrupted by noise and discretized in time, such equivalence is
not always observed and we may find that some variables are better than others.
Such an annoyance may be amplified by the non availability of
data with a sufficient quality as often observed in studying ecological
system where an uncontrolled nonlinear superimposition of
fluctuations may be present. It is therefore rather important to
state whether the three species may be equivalently used to investigate the
dynamics of the whole food-chain.
Typical strange attractor exhibited by this system.
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