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Minimal model in Chaos theory

by marellasunny
Tags: chaos theory, minimal models
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marellasunny
#1
Nov26-12, 03:24 AM
P: 234
I am reading a book on chaos theory by Robert M.May.There is a reference to a 'minimal model of a ecosystem' through which the author describes hysteresis and bistable states.
Q.
What does a minimal model mean in mathematical terms,&also intuitively?Is it a concept of topology referring to homotopy between topological spaces or does it take another definitions?
Q.
How does one arrive at a 'minimal model equation' from a system of differential equations?
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Mute
#2
Nov26-12, 12:52 PM
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P: 1,391
As far as I know there is no connection to topology here. Typically, a "minimal model" is a simple model which mimics qualitatively the essential features of the problem you are modeling, without regard to getting every detail correct. For example, for a simple population model you might be interested in the dynamics of predators and prey, and so you want to understand the basics underlying that process by writing down a simple system of differential equations for the population size q(t) of the predator and p(t) of the prey as

$$\dot{p} = p(a-bq) \\ \dot{q} = -q(c-gp)$$
for constants a, b, c, g with have some interpretation as birth rates, death rates, etc.

The point is that this is a very simple model that doesn't take into account things like other predators or prey, spatial variations in the populations, seasonal variations, etc. There are lots and lots of details left out, but that's okay - our goal with the minimal model is to understand the essential qualitative features of the process. In this example, the interactions between predator and prey populations. We can always add more detail later.

To develop minimal models one usually has to have some idea or hypothesis of what are the essential features of the problem one wishes to study, and then write down a differential equation (or system of equations) that describes how things change with time, position, other variables, etc. It's a bit of an art; there is not necessarily a formulaic way to come up with a minimal model.


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