Posted by: Jeremy Fox | May 16, 2012

Maybe we need even more stability concepts!

p.s. to the previous post: Commenter Christopher Eliot (indirectly) makes the important point that all those stability concepts related to equilibria and other attractors assume that whatever system you’re studying can be described by a model with unchanging structure and parameter values. It’s only species’ densities (or whatever your “state variables” of interest are) that are allowed to change over time. Of course, in nature it’s probably hardly ever the case that a perturbation just changes species densities, while having no effect on any other aspect of the ecology of the system (e.g., the species’ behaviors, levels of key abiotic factors, etc.).

So maybe we need some new stability concepts! ;-) Just kidding. There are of course theoretical models which allow one or more key parameters to vary over time, often due to extrinsic variation in the environment (that’s sometimes called “external forcing”). And there are models which allow intrinsically-generated temporal variation in parameter values as well, which effectively just makes those parameters into additional state variables (e.g., models of eco-evolutionary dynamics, which allow some parameters to evolve via natural selection). So I don’t think we actually need any new stability concepts. But we probably do need a lot more work on models in which parameter values and even model “structures” can change over time. That’s really difficult and messy of course, which is why many theoreticians understandably hesitate to do those kinds of models.

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  1. I tried to publish a paper a few years ago looking at how changes in specific model parameters affected local stability/persistence in community models. It was even submitted to Oikos! The best way to summarise the referees’ responses was ‘utter confusion’, over what we were doing and why we were doing this. I keep meaning to go back and clarify these points, but it’s clearly challenging and keeps falling down the MS pile. You’ve given me some nice ideas to start from, thanks!

    I think the basic idea is related to “press perturbation” theory though, which has been developed to some extent in population ecology.

    A possible counter argument, is that the sort of models I think you’re talking about are ‘mean-field’ models. An important underlying assumption here is that the mean value of each biological/model parameter is a sufficiently good representation of what’s driving population dynamics. If you don’t think this assumption holds for your study system/question, then mean field models are the wrong tool to use. It’s not that the models are inherently wrong, just that they’re not meant to be applied to that problem. Similarly, changes in behaviour (or other the other features you suggest) following perturbation (a term which can also be defined in various ways) might not have any appreciable effect leading to changes in population dynamics in a changing environment – behavioural changes may just be required to maintain the status quo under the new conditions.

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