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

2. 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.

3. 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.

4. 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?

5. 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.

6. 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!

7. 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.

8. 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.

9. 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.

10. 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!

11. 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?

12. 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.

13. 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!