Special Topics

1. Special Topics Some subjects to think about for the future

2. Modelling Issues • There are some fundamental issues which modellers eventually have to deal with. These include: • Stability • Bifurcation • Fuzzy logic

3. 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. 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. 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. 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. The Ricker Model • Consider the Ricker model of salmon recruitment (which is here simplified). • This relates next year’s stock, xt+1, to this year’s stock, xt, by the equation xt+1 = Axt exp(-xt) • 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. 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. 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. 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. 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. 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. 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.