Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

In the field of ecology, a well known relation between the number of species and the size of an island can be approximated by a power function of the form: $S = c A^z$

  • $S$ = number of species
  • $c$ = a fitted constant
  • $A$ = area of the island
  • $z$ = a constant equal to $\log_{10}(S) / \log_{10}(A)$

Variable $z$ is, in fact, the slope of the linear relation between $\log_{10}(S)$ and $\log_{10}(A)$. I’ve never encountered a multivariate equation where one of the independent variables consists of the slope of a linear relation where a non-linear transform has been applied to the major independent and dependent variable.

I’m curious if the previously mentioned equation belongs to a more general class of equations? Is there a special name for these types of equations?

share|improve this question
add comment

3 Answers

In engineering fields we simply call this a power law relationship. Power law representations are almost always empirical.

Your equation isn't really multivariate. Its better to think of it as a parameterized single variable relation, where $z$ is computed from a body of data outside the scope of the studied region; alternatively, $A$ is fixed for the region in question and $z$ is estimated from experiments.

Regardless, the parameters are fixed in any given context and a single variable is fit to data.

Power law relationships often do not have a causal mechanism such that they take that form, and as such only apply over a specific domain. These models are used widely in engineering fields, and substantial effort must usually be done to justify their prognostic abilities.

share|improve this answer
add comment

Are you sure you're not missing something? Because from what you said it can be derived that

$S=cA^z$

but on the other hand you say we have $z=\frac{\lg S}{\lg A}=\log _A \left( S \right) $ so we have $S=c\cdot S$.

share|improve this answer
add comment

Please look at the Wikipedia entry on allometry. Variable $z$ is estimated from the data.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.