1. NATO ASI, October 2003William Silvert Special Topics Some subjects to think about for the future

2. NATO ASI, October 2003 Modelling Issues  There are some fundamental issues which modellers eventually have to deal with. These include:  Stability  Bifurcation  Fuzzy logic

3. NATO ASI, October 2003 Stability  Models are not always stable, which can be a good or bad thing.  If a model is unstable because it is poorly designed or programmed, that is bad – for example, there is “numerical instability” due to bad mathematical algorithms.  But systems can be unstable, so models of those systems should alson be unstable.

4. NATO ASI, October 2003 Resilience  Stability is often confused with resilience, but they are different.  A stable system is one which returns to its original state if perturbed.  Resilience refers to how much a system can be perturbed before it returns to its original state.

5. NATO ASI, October 2003 Stability vs.Resilience  Stability and resilience are usually inversely related to each other.  An oak tree is stable, but if bent more a few meters it will break.  A willow is far less stable, but it can bend very far before it breaks.  The same analogy applies to stiff and stretchy springs.

6. NATO ASI, October 2003 Types of Instability  There are several standard ways in which instability can arise.  One common pattern is related to instability and chaos.  Some systems follow a “fixed point trajectory” and then break into a chaotic mess.

7. NATO ASI, October 2003 The Ricker Model  Consider the Ricker model of salmon recruitment (which is here simplified).  This relates next year’s stock, x t+1, to this year’s stock, x t, by the equation x t+1 = Ax t exp(-x t )  For low values of A the values of x tend to a limiting value, but for higher values of A the solutions bounce around and ultimately become chaotic for high A.

8. NATO ASI, October 2003 Catastrophe Theory  Catastrophe theory will be discussed later on in this ASI, so I will only mention it briefly.  A catastrophe in the mathematical sense arises when a system becomes increasingly unstable and then collapses into a totally different state.  Ecological applications are plentiful but controversial.

9. NATO ASI, October 2003 Super-cooling  The super-cooling of water is a common example of a catastrophe.  Normally water freezes at 0°C.  Pure water can be cooled below 0°C without freezing, but any dust or vibration makes it freeze.  The colder it gets, the more violent the eventual phase transition.

10. NATO ASI, October 2003 Regime Shifts  Regime shifts in ecosystems are probably symptomatic of catastrophes.  Insect outbreaks are the most widely discussed examples.  Ecosystem collapse, mass extinctions, and successful invasions can be understood in terms of catastrophe theory.

11. NATO ASI, October 2003 Le Châtelier’s Principle  Henri Louis Le Châtelier pronounced what is probably the most important law in science:  If you displace a system from equilibrium, it will fight back and try to return.  This is very general and almost always true.

12. NATO ASI, October 2003 Thermodynamics  When you squeeze a balloon the pressure inside increases.  This is a common example of Le Châtelier’s Principle, since the harder you squeeze, the higher the pressure and the greater the force resisting you.

13. NATO ASI, October 2003 Epidemiology  If there are too many organisms in a fixed space, something will happen to reduce the population.  Every time there is a mass explosion of sea urchins, they end up being wiped out by an epizootic.  The same happens to humans in large over-crowded cities.