## Parameters

A A Voinov, University of Vermont, Burlington, VT, USA © 2008 Elsevier B.V. All rights reserved.

Parameters are an important component of any model. In the most general sense, parameters drive the system that we observe. When we model a system we try to reproduce its behavior in terms of certain values, which we observe and which we call state variables, or output variables. These are the outputs from the model. These outputs are determined by inputs that go into the model, and by the model internal organization, the wiring. Generally, any quantity that is used to describe and run a model can be called a parameter.

Parameters are an important component of any model. They may be further classified into following categories:

Boundary conditions. These describe the spatial and temporal boundaries of a system. For a spatially homogeneous system we have only 'initial conditions', that describe the state of the variables at time t = 0, when we start the model. For spatially distributed systems, we may need to define the conditions along the boundary, as well as the geometry of the boundary itself. These are known as 'boundary conditions'.

Constants or parameters in a narrow sense. These are measured or calibrated quantities that describe the rates of processes in a process-based model. In empirical models, these are the constants that are statistically derived based on empirical observations. We may want to distinguish between real constants, such as gravity, g, and, say, half-saturation coefficient, K, in the Michaelis-Menten function. While both of them take on constant values in a particular model run, g will be always the same from one run to another, but K may change quite substantially while we calibrate the model. Even if K comes from observations, it will normally be measured with certain error, so the exact value will not be really known.

Forcing functions. These are parameters that describe the effect of the outside world upon the system. These may change in time or space, but they do not respond to changes within the system. They are external to it, driven by processes at the higher hierarchical levels. Climatic conditions, rainfall, and temperature certainly affect the growth of tomatoes in the garden, but the tomatoes hardly affect the temperature or the rainfall patterns. If we build a model of tomato growth, the temperature will be a forcing function.

Control functions. They are also parameters, except that they are allowed to change to see how their change affects systems dynamics. It is like tuning the knob on a radio set. At every time, the knob is dialed to a certain position, but you know that it may vary and will result in different performance of the system.

Note that in some texts parameters will be assumed only in the narrow sense of constants that may sometimes change, like the growth rate or half-saturation coefficients. However, this may be somewhat confusing, since forcing functions are also such parameters if they are fixed. Suppose we want to run a model with temperature held constant and equal to the mean over a certain period of time, say the 6 months of the growth season for a crop. Then suppose later on, we want to feed into the model the actual data that we have measured for temperature. So now it is no longer a constant but changes every day according to the recorded time series. Does this mean that temperature will no longer be a parameter? For any given moment, it will still be a constant. It will only change from time to time according to the data available. Probably, it would make sense to still treat it as a parameter, except now it will be no longer constant but will change accordingly.

Suppose now that we approximate the course of temperatures by a sine function with some constants that control the amplitude and the period. Now temperature will no longer be a parameter. Instead, temperature will be an intermediate variable, while we will have two new parameters in the sine function that specifies temperature - one parameter (B = 4) will make the period equal to 6 months, the other parameter (A) will define the amplitude and make the temperature change from a minimal value (0) to the maximal value (40, if A = 20) and back over this period of time, like in the function

Here t is time, k is a constant (=3.14), and A and B are parameters. If B = 2, then the period will change from 6 to 12 months.

There may be a number of ways to determine model parameters, including the following:

1. Measurements in situ are probably the best ones since they define the value of exactly what is assumed in the model. However, such measurements are the most labor and cost intensive, and they also come with large margins of error. Besides, in many cases, such measurements may not be possible at all, if a parameter represents some aggregated value or an extreme condition, that may not occur in reality (say, the maximal temperature for a population to tolerate - it may differ from one organism to another, and such conditions may be hard to find in reality).

2. Experiments in lab (in vitro) are usually performed when in situ experiments are impossible. Say we take an organism and expose it to high temperatures to find out the limits of its tolerance. We can create such conditions artificially in a lab, but we cannot change the temperature for the whole ecosystem.

3. Values from previous studies (found from literature, web searches, or personal communications) can be used; it certainly makes sense if data are available for similar systems. However, it should be always kept in mind that there are no two identical ecosystems, so most likely there will be some error in the parameters borrowed from another case study.

4. Calibration (see Sensitivity, Calibration, Validation, Verification) is another way. When we know what the model output should look like, we can always tweak some of the parameters to make the model perform best. For empirical models, this and literature search are the only ways to identify parameter values.

5. Basic laws, say conservation principles and therefore mass and energy balances, can also be employed.

6. Allometric principles, stoichiometry, and other chemical, physical, etc., properties can be taken advantage of. Basic and derived laws may help establish relationships between parameters, and therefore identify at least some of them based on the other ones already measured or estimated.

Note that in all cases there is a considerable level of uncertainty present in the values assigned to various model parameters. Further testing and tedious analysis of the model is the only way to decrease the error margin and deal with this uncertainty.

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(eds.) (1978) Handbook of Environmental Data and Ecological

Parameters, 1162pp. Vaedose, Denmark: ISEM.