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.