Kinds of Models

Written language

Icon models

Analoguous models

Topological models

Arithmetical models

We will be concentrating on what are called "flow - reservior models or "box models".

Combines topological and arithmetic models

Our example of a model will be population growth.

We have all seen discussions of population growth, especially of humans.

What are some of the possible relationships that a population can have with time?

1. Linear Growth

2. Exponetial Growth

3. Logistic Growth

4. Overshoot and collapse growth

5. Cyclical Growth

What kinds of information do we need to examine population growth? (list)

How might we make a graphical map of this information? (draw)

Case 1: Only Births

Case 2: Births and Deaths

Case 3: Births Dependent on Population

Case 4: Births and Deaths Dependent on Population

Case 5 Births and Deaths Dependent on Population

and Birth Rate or Death Rate Dependant on Population

Lets talk about representing these mathematically

We are all familar with the equations of at least some of these kinds of lines.

Linear: y = mx + b or N(t) = rt + NO

where N0 = the initial population size; r = growth rate; t = time

Exponential: y = be^mx or N(t) = NOe^rt

Logistic: y = bk / [b + (k - b)e^mt]
or N(t) = N0K / [ N0 + (K-N0)e^-rt]

where K = carrying capacity of the environment.

But these can also be solved numerically:

The simplest numerical solutions are called difference solutions which solve the function iteratively:

We can do this very easily in EXCEL

BUT for our purposes we whant to use STELLA to solve the eqations numerically for us by representing the logic of the system in its graphical language.

STELLA uses Euler's method to numerically solve the equatiosn, which prevents certain pecularities of discrete numerical solutions.

Lets look at Stella and try the exercise.

We can build a logistic growth model in EXCEL and an exact equivalent in STELLA:

For small values of the growth rate, the EXCEL and STELLA models are equivalent. But for larger ones, the EXCEL model behaves in a classic chaotic way. Its what you might actally see in a population with descrete breeding periods.

STELLA does not behave this way because Euler's method prevents it.

Tommorow we will use a STELLA model of the Bio2 atmosphere.