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Autonomous versus non-autonomous equations. Lotka-Volterra belongs to the former category, a time-invariant system -- is that a stretch for describing real systems where populations depend on the environment? What good will that behavioral description do when the prey species is susceptible to e.g. drought cycles?
So it then becomes a forced system and the focus on an attractor orbit goes out the window. I actually get annoyed by how much people cling to the notion that internal eigenvalues have to be the solution to everything. In many practical cases, it's the forced response and not the natural response that governs the evolution. For this predator-prey system, it appears that ENSO climate cycles and not the internal L-V dynamics drive the cycles. Moreover, even ENSO isn't an internally natural response system, as that is obviously forced by external tidal cycles. Perhaps that's why the scientists are all mystified by this, as they may be deeply attached to the mathematical idealism of eigenvalue-based solutions. But then even this is odd, because climate change and AGW is well-agreed to be a forced response system, driven by adding CO2 to the atmosphere. So I can't generalize either.
I can discuss this aspect of natural vs forced response all day.