Chaos in Ecology

Question 1

What is Takens theorem?

Answer 1

Takens’ theorem allows that for any dynamical system, we can recover the dynamics of the system from a time series of a single observable, using time lags as surrogate dimensions of the attractor.

This enables a reconstruction of the attractor manifold, such that given the right time steps and the correct number of embedding dimensions, the embedded dynamics in the reconstructed manifold have the same topology as the original state space.

Question 2

What are its main assumptions?

Answer 2

• the system needs to be nonlinear and deterministic
• it cannot be an oscillator or a periodic orbit with high
  stability
• the system must be stationary

Question 3

What is stationarity?

Answer 3

Stationarity refers to a number stream or data set with statistical qualities, such as the mean and variance, which remain constant over time.

Question 4

Explain how violations of stationarity might affect the application of Takens’ theorem

Answer 4

Takens’ theorem uses time lags to reconstruct the system’s manifold, so if the fundamental properties of the time series change between lags or at a different scale than the time lags, the manifold projection will be distorted and therefore unable to represent the topology of the original system dynamics.

Question 5

What is the main conclusion of May’s 1977 paper on thresholds and break points?

Answer 5

Many ecosystems have dynamical regimes with multiple stable states. E.g. continuous change in fisheries harvest can cause fisheries collapse

Question 6

What does this mean for theory application?

Answer 6

That multiple equilibrium points exist for a variety of ecological systems, and that these systems possess a regime of dynamical behavior where continuous variation of a control parameter can produce discontinuous effects.

Question 7

What are the constraints about generalizing the results of May’s 1977 paper?

Answer 7

All the examples he offered are of single populations with their control parameters, but are not multi-species assemblages or communities.

Question 8

Why is it still promising to try to discern these dynamical regimes?

Answer 8

Because many complex communities are actually loosely coupled low-order systems that are discernable.