Should all the species of a food chain be counted to investigate the global dynamics

Letellier C., Aguirre L., Maquet J.,

Aziz-Alaoui M.A.

Labo. Math., Le Havre University

Chaos Sol. Fractals, Vol. 13(5), pp 1099-1113 (2002).

http://www.elsevier.com/gej-ng/10/14/11/63/36/show/toc.htt

Abstract

The fairly realistic three species food chain model given in another paper (click here), is studied from a single time series ...

Introduction ...
... 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. ...
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Typical strange attractor exhibited by this system.

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