Petraitis Multiple stable states in natural ecosystems

[MKLau]

Theory Terms State variables = measured value of interest Parameter = value that determines relationships among variables Equilibrium = no change over time in a state variable Linear = equilibrium solution obtained by setting the equation equal to zero Stability = Return to the same equilibrium point after perturbation Perturbation = moving the state variable away from equilibrium Multiple stabile states = same parameters, more than one equilibrium point Aka, Basin, attractor

[MKLau]

For models with more than one state variable, basin shape is determined both by parameters and by state variable paths when perturbed.

See Chapter 8!

[MKLau]

Parameters are abstractions of processes and rates of change.

[MKLau]

Theoretical Support

1. Allee model
2. Consumer models (1 2 4 equations)
3. Competition models

Requirements

1. Multiple basins
2. Boundaries
3. State variables change, not parameters

Peterson's Criteria

1. Single environment
2. Two or more distinct communities
3. Communities are self replicating
4. Pulse perturbations

Priority Effects

One community can not be invaded by another

Hysteresis

1. System is resistant to environmental change
2. Sentinel species undergoes a shift
3. Reversion to original environmental conditions does not revert to original community

[MKLau]

Catastrophe Theory

Systems with 1-2 state variables and four or fewer parameters have seven elementary catastrophes, aka discontinuous jumps.

A catastrophe machine. (E.C. Zeeman)

[MKLau]

Go back over catastrophe theory and where do we go from here