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Special Topics - PowerPoint PPT Presentation

Special Topics. Some subjects to think about for the future. Modelling Issues. There are some fundamental issues which modellers eventually have to deal with. These include: Stability Bifurcation Fuzzy logic. Stability. Models are not always stable, which can be a good or bad thing.

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Special Topics

Some subjects to think about for the future

• There are some fundamental issues which modellers eventually have to deal with. These include:

• Stability

• Bifurcation

• Fuzzy logic

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.

• 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.