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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

Special Topics

Some subjects to think about for the future

modelling issues
Modelling Issues
  • 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 vs resilience
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.
types of instability
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.
the ricker model
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.
catastrophe theory
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.
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.
regime shifts
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.
le ch telier s principle
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.
  • 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.