# Self-thinning: slope of log (biomass) vs log (density) is -1/2 whereas slope of log (weight) vs log (density) is -3/2, why?

In self-thinning or intraspecific competition (more relevant for trees I guess) why does logarithm of Biomass vs logarithm Density give you a slope of -1/2 whereas logarithm of Weight vs logarithm of Density gives you a slope of -3/2? Biomass is mass (or quantity) of an organism per volume(or area), weight is mass scaled by gravitational acceleration which is a constant, unless the organism you are dealing with is very very tall. So the plots can be shifted accordingly but shouldn't slopes be more or less the same ?

source: Ecology: Concepts and Applications by Manuel Molles www.mheducation.com/highered/product/ecology-concepts-applications-molles/M0077837282.html

• Welcome to Biology.SE. This is a question of statistics, not biology. You should give it a try on Stats.SE. Sep 20 '16 at 17:49
• I would not expect the slope of y~x to be equal to the slope of log(y)~log(x) (to the exception maybe of when the slope is 1?). Consider this R code for example. x=rnorm(100,40,7);y=x/6+rnorm(100,0,1);plot(x=x,y=y);lm(y~x);lm(log(y)~log(x)) Sep 20 '16 at 17:53
• @Remi.b I understand log plots. For example, if you know Stephan's law of radiation; radiated energy E is proportional to fourth power of temp T, in this case, if we plot log(E) and log(T) we will get slope to be 4. But that (value) is not my concern, why are they different is my concern. Sep 20 '16 at 18:42
• @Remi.b I don't understand how is this a Stats question, I think definition of biomass and physics are more relevant here ! Sep 20 '16 at 18:44
• @Remi.b I know that this is not exactly a Biology question, it is more about the understanding of those quantities, physics behind them and the mathematical analysis of it. I am not exactly sure where to post it. I googled definitions of Biomass and weight and crosschecked if there was any misinterpretation, but there wasn't any. And as far as I understand it should only offset ( y = mx + c, I mean c by offset ) the plot and not change the slope. BTW I still don't get it how is it a Stats question XD. Sep 20 '16 at 19:20

Biomass here refers to the total mass of all the organisms in the population (in a unit of area), added together.

Weight here refers to the average weight or mass of an individual.

If biomass decreases, that could be because the individuals got lighter, or it could be because there are fewer of them, or both.

I think I see what's happening.

Let $W$ be the average mass of an individual, $N$ be the number of individuals, and $B$ be their total biomass. Then of course,

$B=WN$

• $W$ is proportional to $N^{-\frac{3}{2}}$
• $N$ is proportional to $N$ (duh). Let's rewrite that though, as:
• $N$ is proportional to $N^1$
$B=WN$ is proportional to $N^{-\frac{3}{2}} \cdot N^{1} =N^{-\frac{3}{2}+1} = N^{-\frac{1}{2}}$