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I use the term topological extinction to refer to a simple approach to secondary extinction in food webs. The concept basically asserts that the extinction of a species occurs when all in-links to that species have been removed. In terms of the CEG model, where species links vary in strength, link strength will vary in order to compensate for lost resources, and species can be assigned extinction thresholds permitting extinction even though one or more in-links might yet exist, topological extinction can be viewed as a special case. It is certainly the simplest, and while biologically unrealistic (in my opinion), can be used to gain insight into some of the macroscopic behaviours of the system. And the behaviours of these systems can be quite complex, unpredictable, and inexplicable.

canonical_ceg_slide

One of the main features of the model that has to be explained is something we call “typical CEG behaviour”. This behaviour is exhibited by food webs that are perturbed bottom-up by the reduction or removal of primary production. We’ve examined this for a wide range of communities, including Early Permian, Late Permian and Early Triassic terrestrial communities, Miocene-Pliocene shallow coastal marine communities, and the modern marine community of San Francisco Bay. Almost all these communities exhibit moderate to strong resistance to secondary extinction over a broad range of perturbation magnitude. There is little to no increase of secondary extinction as perturbation magnitude is increased. A threshold exists, however, at which the level of secondary extinction increases “rapidly” (though time is not involved here) or discontinuously, and above which levels of secondary extinction are always high (eventually reaching 100%). I term this the critical perturbation magnitude \omega_{c} (or \Omega_{c} when the perturbation involves multiple guilds). Here are some interesting observations and questions:

  1. Even though the state space (see this post) of a typical community (i.e. the number of possible variants of the food web) is astronomical and beyond, most species-level variants exhibit very uniform behaviour (and we’ve run tens of thousands of simulations of most of these communities).
  2. The result of a specific perturbation is deterministic. But let’s say that we are interested in perturbation of a particular magnitude, so we remove x species (or amount of primary productivity) from a guild in repeated simulations, not caring if we remove the same x nodes from the guild, only the number. The perturbation is therefore stochastic within the confines of guild membership. The result of course is that there is a variance of secondary extinction response associated with a single species-level network now, but that variance is very low. Low, that is, except in the region of \omega_{c}, where it peaks. There seems to be an “undecidability” of the network response at this point; sometimes secondary extinction is low, and sometimes it jumps to the high regime.
  3. Why is there an \omega_{c}? Combinatoric and differential calculus models that estimate the CEG model are smoothly exponential. They do not exhibit the threshold.
  4. Why does the variance of secondary extinction peak at this threshold? Some of it is certainly associated with the stochasticity of the perturbations, but in some way combined with the existence of the threshold itself.

CEG modeling and simulations of so many different types of communities has certainly been enlightening. We continue to model more communities, and are beginning to ask questions about community stability/resistance and changes over time. But I believe to really make anymore meaningful progress on understanding why the model behaves the way that it does, I almost have to begin to re-build it. An inspirational, enlightening, and depressing quote from Strogatz: “…now that we’re dealing with network systems with millions of nodes, we can’t use geometry either. We need something else. We have graph theory, that’s something. We have simulation but the simulations are often as hard to understand as the reality. So I think we have a real psychological question here as to whether we will ever be able to develop what’s needed to understand this modern world.”

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