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The "curve of autocatalytic growth" (and also the generalized logistic) is essentially the hyperbolic tangent. Use of this fact simplifies the theoretical and practical handling of the equation. Demonstration of a good fit of the autocatalytic curve to a given growth-curve is no proof that the growth is autocatalytic, since the autocatalytic curve can be made to fit curves of similar appearance but very different nature. The variety of forms which the autocatalytic curve can assume is evidence of the futility of such an attempted proof.
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Kavanagh et al. (1934) studied this question.
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