I came across a pretty exciting news story. It seems that scientists have found a measure of food web stability that shows increasing stability with increasing complexity; this would solve a long-standing paradox in population ecology.
The basis of the paradox was described in Robert May’s 1973 book Stability and Complexity in Model Ecosystems. Robert May was a Theoretical Physicist who helped create the foundations of population biology. He noted that beyond a critical point, his model ecosystems became less stable as they became more complex. This was odd, as coral reefs and tropical jungles were both complex and long lived; so presumably stable. To explain this he concluded that the stability was external to the structure of the food web, it must come from their environment. He reasoned that if the environment was stable there would be few perturbations to disturb the status quo, so even unstable complex ecosystems could survive.
This mismatch between observation and mathematics has been investigated and criticized for decades but until now no clearly superior algorithms were proposed. Some amazingly fragile theories were proposed resolve the conundrum, and the meanings of words such as resilience and elasticity were twisted to achieve the desired results. Here we define stability as: the ability of an ecosystem to endure over time.
That leads to today. Samuel Johnson and his coauthors propose that the paradox comes from the structure of the food web combined with the stability measure. They propose adjustments to both.
May’s ecosystems were built with many random species connections (everything can be connected) while Johnson imposed a tighter structure on his. In Johnson’s webs, species are placed into neat trophic levels based on the average of their food sources. That is, they are mostly: predators, herbivores, or plants. He called this constraint ‘trophic coherence’.
Now, real ecosystems are not completely coherent, but they are nearly so. We don’t, for example, expect lions to survive eating acacia leaves. However as I noted in a previous post some herbivores will occasionally eat other animals. To account for this, Johnson allowed trophic levels to have fractional values, and he used the standard deviation of the trophic distances in the food web to determine the level of coherence.
Changing the structure of the food webs by itself was not enough to replicate the expected behavior; the stability measure itself needed adjustment.
May represented his model ecosystems with differential equations. He collected the equations into a matrix and solved for the eigenvalues. Then he used the real part of the leading eigenvalue as his representation of stability. The leading eigenvalue measures the degree of self-regulation each species requires in order for the system to be linearly stable. Since this technique is used to model many types of physical structures and systems it is logical to apply it to ecosystems.
Johnson’s team found that the leading eigenvalue of the biomass change matrix was only weakly correlated to expected stability measures. It was most highly correlated to incoherence when self-links were removed. Self-links represent cannibalism, which we can safely assume is an insignificant transfer of energy.
One of the challenges in an analysis of this type is assigning a stability metric to the web. This is typically done my assuming a biomass distribution model and a growth function – for example Lotka-Volterra — and using those to build the matrix. It isn’t hard to see that the model is inextricably confounded with the web and any stability test applies equally to the selected model and the web. In an attempt to reduce the impact of this the team applied different growth models, calculated stability metrics for each, and tested those against the coherence measure.
This isn’t perfect, but rarely is enough data available to do anything else. As a note this is linear stability, and when one see’s a linear approximation to a natural system, it can only hold for a short time period.
Which brings up a profound weakness of food webs; they don’t include measures of size. I didn’t collect and weight all of the pinacate beetles in my study area. I inferred their weight much after the fact.
The weight distribution has an effect on the predicted stability. If it is relatively equally distributed on all trophic levels then it is just a shift in the eigenvalue. But if some species are much more prevalent and better known then it is not possible to predict the results. I don’t mean this as a criticism of Johnson’s methods; it’s just one of those things that make this problem more interesting.
I like this idea. I’m not sure that I can use it on the project I’m currently working, but I will have to write out my model ecosystem in a form to which Johnson’s model can be applied. It would be exciting if, after nearly 40 years, I can finally show that the ecosystem modeled by my High School’s food web was stable. But that will have to wait for bandwidth.
However, I am reminded of May’s words:
“With the contemporary upsurge of interest in these questions, accumulating evidence suggests that the relation between complexity and stability is substantially more complicated than appears at first sight.”
I don’t expect trophic coherence to be the last word on this topic. None the less, it’s an exciting step.