Foundations of Modeling Models are simplifications of real systems They help us to understand the behavior of these systems by focusing on what (we believe) is important They can be used both to explain the past and to predict the future They are often based on observation

Example Isaac Newton observed the motion of objects in response to force, and found he could describe this behavior with F = ma. This is a mathematical model (though physicists glorify it by calling it a “law”). It correctly describes past behaviors and predicts future ones... …but only under some conditions!

Kinds of models Mathematical –e.g. F = ma Physical –e.g. a stream flow model of board and soil Mental –e.g. our expectation of future behavior based on past experience The first two can be used to simulate reality. The last one helps us navigate life.

How do we approach modeling? 1.Define the system 2.Draw a system diagram 3.Enumerate relationships 4.Determine the units for variables and parameters/coefficients 5.Calibrate 6.Validate 7.Document

1. Define the system System boundaries Flux (flows across boundaries) Time frame under consideration Temporal and spatial resolution Scope (what’s included, what’s excluded) Assumptions

2. Draw a system diagram A diagram will help you understand your conceptual model of the system and explain it to others Doesn’t have to be art – but should be clear

3. Enumerate relationships Mathematical relationships Graphical relationships (e.g. spatial associations) Other relationships, e.g. –Political/power/influence –Economic (could be mathematical)

4. Determine the units Mathematical models will generally consist of variables and coefficients (or parameters). Units of measurement must be determined, and should be analyzed for consistency BEFORE any calculations are performed.

5. Calibrate While the variables will be either manipulated (independent) or recorded (dependent), coefficients/parameters must be set. This calibration process involves using real data from past observations. The coefficients are adjusted so that the model output matches the observed system output.

6. Validate Once the model is calibrated, we validate it by using a different set of real data. This time the model coefficients are not changed, and model output is compared to the observed system output.

7. Document Models need to be documented in order to be given credibility All of the above need to be included in the documentation The most important things to document are your assumptions

Characteristics of Environmental Models Common relationships –Linear –Exponential –Logistic –Overshoot & Collapse –Oscillation Mass balance Difference equations