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

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

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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