Colimitation indices

Resource limitation is a fundamental structuring force in ecosystems. I've written a lot about how we need better ways to analyze the simple factorial experiments that are used to determine limitation.  Boiling down patterns of limitation to a single number or two is going to be a big help understanding broader patterns of limitation. Below, I introduce a colimitation index, that separates out some key patterns. 

Let’s start with a basic limitation experiment. 4 treatments: Control (0), +N, +P, +NP.

B₀ is biomass of controls

B_N is biomass with +N

B_P is biomass with +P

B_NP is biomass with +N+P

To make matters simple, let’s assume that there is no effect of P added alone and that the biomass of plants with N and P added is greater than unfertilized biomass.

With this, we’re trying to separate three cases.

1) Classic co-limitation (co-limitation by supply) where there is no direct effect of N (or a relatively small one).

2) Primary limitation by N and secondary limitation by P

3) Single resource limitation where there is an N effect, but no effect of P on N-fertilized plants.

There are two ways to calculate a co-limitation index (CI) based on absolute or relative changes in biomass.

First we can compare the absolute changes in biomass relative to controls and compare the N effect to the NP effect. If we calculate the co-limitation index as

(BN-B0)-(BNP-B0)/2

then a plant that had a CI > 0 would be primarily limited by N and secondarily by P. A CI <>

The interpretation of the pattern would be that soils with low P availability are co-limited by N and P. As P availability increases plants are more likely to be primarily limited by N and secondarily limited by P.

That’s a pretty clean story, but the problem with this approach is that you cannot separate if a plant is secondarily limited by P or just limited by N.

To get around this we can calculate a co-limitation index as:

(BN-B0)/(BNP-B0)

With this index, 0≤CI<0.5><1> 

Here are the patterns for the 100 soils data:

Pretty much the same story. Plants start out co-limited by N and P at low P availability. Then as P availability increases P limitation becomes more secondary until ~30 ppm available P at which we’re into strict single limitation by N. The problem with this approach is that the relative index is sensitive to BNP. For the graph at left I had to exclude two points that had |CI|>10.  

With these approaches statistical significance relative to threshold values, e.g. CI = 0 for the first index, are possible. I’m not sure how to extract them from JMP yet.

Note with a factorial resource addition experiment there are something like 9 different basic responses when you include inhibition and responses by individual resources. There will be no way to boil, but we might be able to get the most important patterns down to 2.