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@WebHubTel wrote:

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.

## Comments

@DavidTanzer wrote:

I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium.

...

But the charge against those who are "deeply attached to the mathematical idealism of eigenvalue-based solutions" has an edge to it which could easily be taken out of context.

Your point is that the 'first order' L-V model, which doesn't take into account forcing, won't give accurate predictions in real-world scenarios. Fine.

But eigenvalue-based solutions and equilibria are absolutely foundational concepts, which apply almost perfectly in cases like chemical reactions, which are not dominated by external forces. So let's not get discouraged from opening up the topic just because it's empirical applicability needs to be evaluated on a case-by-case basis.

I can also imagine that even where it doesn't literally apply, the theory of equilibrium may provide a framework for understanding the disruption of equilibrium by slight to moderate amounts of forcing. For example, with slight forcing, there will no longer be points of equilibrium, but perhaps small fuzzy regions of "quasi-equilibrium" around the theoretical equilibrium points.

`@DavidTanzer wrote: I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium. ... But the charge against those who are "deeply attached to the mathematical idealism of eigenvalue-based solutions" has an edge to it which could easily be taken out of context. Your point is that the 'first order' L-V model, which doesn't take into account forcing, won't give accurate predictions in real-world scenarios. Fine. But eigenvalue-based solutions and equilibria are absolutely foundational concepts, which apply almost perfectly in cases like chemical reactions, which are not dominated by external forces. So let's not get discouraged from opening up the topic just because it's empirical applicability needs to be evaluated on a case-by-case basis. I can also imagine that even where it doesn't literally apply, the theory of equilibrium may provide a framework for understanding the disruption of equilibrium by slight to moderate amounts of forcing. For example, with slight forcing, there will no longer be points of equilibrium, but perhaps small fuzzy regions of "quasi-equilibrium" around the theoretical equilibrium points.`

@WebHubTel wrote:

Distinction perhaps between equilibrium and steady-state. Tidal cycles for a given geographical location will not deviate from predictions over decades. Even though this has the properties of a stationary equilibrium, we are taught that this should be referred to as steady-state behavior since forcing is supplied externally. The only equilibrium behavior observable for this system would be to remove the moon and the sun, and only then will the sea surface achieve a stable "equilibrium" level.

Isn't that wild? The equilibrium calculation is trivially uninteresting. This happens in electrical circuit theory as well:

The "broken circuit" is essentially the one where someone clipped a connection and the circuit doesn't do anything. Totally uninteresting, yet that's the definition for equilibrium.

But then if one reads further across disciplines:

In terms of tidal forces only, the Earth/Moon/Sun (+other planets) system is closed to other flows. So perhaps by definition, the observed tidal cycles actually are the equilibrium solution. Or that whatever caused the initial orbital motions long ago, and in the context of extremely low dissipation of energy, the tides are still part of a natural response to the original (solar-system formation) stimulus.

`@WebHubTel wrote: > "I do get your point, though, which can be boiled down to saying: what's the point of studying equilibrium solutions if the system is being driven by external forces, so there is no equilibrium." Distinction perhaps between equilibrium and steady-state. Tidal cycles for a given geographical location will not deviate from predictions over decades. Even though this has the properties of a stationary equilibrium, we are taught that this should be referred to as steady-state behavior since forcing is supplied externally. The only equilibrium behavior observable for this system would be to remove the moon and the sun, and only then will the sea surface achieve a stable "equilibrium" level. Isn't that wild? The equilibrium calculation is trivially uninteresting. This happens in electrical circuit theory as well: > ["The simplest example is a broken circuit vs a closed circuit consisting of a battery and a wire. The first is in equilibrium, the second is in a steady state."](https://forum.allaboutcircuits.com/threads/difference-between-steady-state-and-equilibrium.70546/) The "broken circuit" is essentially the one where someone clipped a connection and the circuit doesn't do anything. Totally uninteresting, yet that's the definition for equilibrium. But then if [one reads further across disciplines](http://www.projects.bucknell.edu/LearnThermo/pages/Equilibrium%20and%20Steady%20State/equilibrium-and-steady-state.html): > "Equilibrium and Steady State. A state of chemical equilibrium is reached when the concentration of reactants and product are constant over time (Wikipedia). ... In contrast, steady state is when the state variables are constant over time while there is a flow through the system (Wikipedia)." In terms of tidal forces only, the Earth/Moon/Sun (+other planets) system is closed to other flows. So perhaps by definition, the observed tidal cycles actually are the equilibrium solution. Or that whatever caused the initial orbital motions long ago, and in the context of extremely low dissipation of energy, the tides are still part of a natural response to the original (solar-system formation) stimulus.`

@WebHubTel wrote:

Interesting point.

We can get a richer model for Petri nets if we allow the rate coefficients to vary with time. Then we could build a Petri net model with processes like reproduction of prey, death of prey, predation, and death of predators. By allowing the processes for death-of-prey and death-of-predators to have time-dependent rate coefficients, external driving forces like the effects of droughts can be expressed.

In this way, the structure of the network will capture some aspects of the dynamics, but not all. It could be coupled with other models or data for the external factors.

Note this is not a full leap to the generality of non-autonomous equations, as the rate equations are still

derivedfrom a network structure and the assumptions about its dynamics. This will of course lead to non-autonomous rate equations - but only to a specific subset of them.`@WebHubTel wrote: > 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? Interesting point. We can get a richer model for Petri nets if we allow the rate coefficients to vary with time. Then we could build a Petri net model with processes like reproduction of prey, death of prey, predation, and death of predators. By allowing the processes for death-of-prey and death-of-predators to have time-dependent rate coefficients, external driving forces like the effects of droughts can be expressed. In this way, the structure of the network will capture some aspects of the dynamics, but not all. It could be coupled with other models or data for the external factors. Note this is not a full leap to the generality of non-autonomous equations, as the rate equations are still _derived_ from a network structure and the assumptions about its dynamics. This will of course lead to non-autonomous rate equations - but only to a specific subset of them.`