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Pause to Reflect.
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Whether expressed as difference equations or differential equations, the model
I used in the fox and rabbit simulator is too complicated to yield a
nice analytical solution. There are many models that can be solved. In many
texts you will run across a simple model the solutions of which are loops:
rabbits breed; foxes eat better, and they breed; many foxes deplete rabbits;
foxes starve; fewer foxes mean happier, more fecund rabbits; etc. This is all well
and good, but of course we pretty much know that answer before we start
modelling it. Outbreaks, bifurcations and chaos are more interesting
phenomena, and the mathematical modelling of them can give us more interesting
insights. But it often happens that without computer simulations very
little of this complicated mathematical realm would be accessible to us. The real world, of course, is always going to be well beyond our models and simulations. It's a really hard problem. On the next page we'll look at a biologically more sophisticated model that will require the use of matrices and matrix multiplication. |
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