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NATO ASI, October 2003William Silvert Model Characteristics Types of Models and their Different Features

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NATO ASI, October 2003 Steady-State Models Steady-state models are not the same as equilibrium models. EQUILIBRIUM: all the forces on a system are in balance and there is no change and no fluxes STEADY-STATE: there is a net force on the system and thus there is a flux, but at a constant level.

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NATO ASI, October 2003 Illustration A bowl of water at rest is in equilibrium. There is no net force on the water and it is at rest. If we stir the water at constant speed then there is a constant force on it (in this case a torque) and the water moves around the bowl at constant speed. This is a steady-state situation.

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NATO ASI, October 2003 So What? Is this the sort of detail that only makes sense to theorists? Actually it makes a big difference. For systems near equilibrium we can apply powerful methods that do not apply to systems near steady-state. What is the equilibrium state of any organism?

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NATO ASI, October 2003 What is Steady-State? Are steady-state models constant? Consider a steady-state population. Births and deaths must balance, but they are discrete events, so actually the system suffers a constant series of small discontinuities. Steady-state actually refers to averages, often annual.

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NATO ASI, October 2003 Analytic Models We call a model “analytic” if it can be written down as a system of equations which can be solved by purely mathematical means. Exponential growth is a typical analytic model, described by the equation dx/dt = ax But exponential population growth is actually discrete!

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NATO ASI, October 2003 Numerical Models Some models can only (or can best) be solved by numerical methods, such as computer simulation. If a model like dx/dt = ax has a variable a, for example a(T) is temperature- dependent and T is given by a time series, then we have to solve the equation numerically.

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NATO ASI, October 2003 THE Model One model more than any other shows up in ecology, and certainly in marine ecology and aquaculture impacts. This is the uptake-clearance model, dC/dt = a – bC where C is a concentration, a is an input or uptake rate and b is the clearance rate.

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NATO ASI, October 2003 dC/dt = a – bC If a and b are constants, this can be solved analytically and the concentration C approaches the limiting value a/b, since when C = a/b, dC/dt = 0. If a or b varies (for example, if they are temperature-dependent) then the equation needs to be solved by numerical methods.

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NATO ASI, October 2003 Typical Applications The uptake-clearance equation describes: Kinetics of toxins in shellfish Nutrient dynamics of estuaries Carbon loading of the seabed Hydrocarbon accumulation on beaches The list goes on and on!

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NATO ASI, October 2003 Aggregation When building a model you have to decide how much to aggregate the variables. Do you combine: Males and females? Different stocks? Different cohorts? Separate pens?

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NATO ASI, October 2003 Trade-Offs Too much aggregation loses detail and your model may not be informative. Too little aggregation (too much detail) requires too many parameters and can lead to enormous estimation errors. Figuring out just how to structure the model is a large part of modelling.

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NATO ASI, October 2003 Resolution Resolution refers to the smallest scale in the model in both space and time. Are you concerned with local, regional, national or global impacts? Is your time step an hour, a day, a month or a year? Mixing scales is the best way to create a bad model!

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