I received this question about scaling relationships and I figured that I would share the answer with everyone because it might be kind of useful to others who are confused by what they wrote down in their notes:
"I saw in notes that we need to know a general form of the scaling equation..i cannot find it in my notes or the slides...could youhelp me out with this?"
And here is my response:
The most general form of a scaling equation is simply referring to the relationship between two variables. In the context of what we have talked about, these are usually in logarithmic relationships and refer to things such as body mass, average population density, etc. - however, a scaling relationship doesn't have to fall within these categories.
For a basic logarithmic scaling relationship between any two variables, we can represent it with the generalized equation:
Y = Y0X^(b) [that is "Y equals Yzero times X to the b power"]
For a basic logarithmic scaling relationship between any two variables, we can represent it with the generalized equation:
Y = Y0X^(b) [that is "Y equals Yzero times X to the b power"]
Prof Witman probably wrote it in class with an M instead of the X, because one of the variables is usually body mass. So again, the equation that you most likely should have seen on the board would look like:
Y = Y0M^(b)
which can be written in a logarithmic form by taking the log of everything (or the natural log):
log(Y)=log(Y0) + b*log(M)
[aside from the log bit, this should remind you of algebra class and everyone's favorite equation for a line "y=mx+b": this graph looks like a straight line; whereas the other form is a log graph, which can be harder to interpret.]
Okay. So that's the general equation logarithmic scaling equation. Hopefully you should have some ideas about what it means, but just in case I've confused you, here are some places to start thinking about it.
Y0 is a constant. It's like a starting point. In the log/log scale plot, it will, in fact, be the y-intercept of the graph.
b is also a constant, and it's the interesting part of the equation. It serves to relate the two variables - so ecologists want to find b and try to think about the implications of it,and why it is so. In the log/log plot, it will be the slope.
Therefore:
if b=1, the relationship between the variables is directly proportionate; the slope is 1; if mass increases by a certain amount, the Y also increases by the same amount. This isisometric scaling.
if b does not = 1, the relationship between the variables is not directly proportionate; the slope is not 1; and if the mass increases by a certain amount, the Y will change by a different amount. This is allometric scaling.
I think I will leave off there and hopefully your notes, and the White et al review paper can help you fill in the gaps as far as examples and relevance. Again, body mass is the variable that we talked about most, but this is actually a general concept, so it can be used for other ideas as well.
Let me know if you have more questions.
