Originally posted by FreakyKBH
So, anyway, where were we? Oh, that's right: we were in the middle of you guys telling me how natural selection (not a force) is a process found in nature like photosynthesis and what not.
Now, math is not my stong suit (just ask Telerion, he'll tell you), so you'll have to bear with me on this part. Is not the preferred means of describing the processes of nature differential equations?
There's nothing fundamental about differential equations, necessarily (in the sense that there not being a DE describing natural selection does not mean that it can't be a natural process). They are a mathematical construct, but are useful in natural sceince because the natural scientist abstracts some property of the natural process and compares its behaviour with that of a solution to an initial value/boundary value problem, where the boundary conditions have been chosen as part of the process of abstracting properties of the natural process.
For example, I could consider the vibrations in the head of a circular drum when I hit it. In this case, I want to find a function h of the radial distance from the centre of the drum, the angular distance around the drum, and the time since hitting -- h is the height of the drum (measured from the height of a nonvibrating drum) at any given point in space and time. I thus start abstracting things from the physical, drum-hitting situation (ie, I abstract things about the natural process of vibration) to build a mathematical model which will predict the shape and size of the drum's vibrations. Maybe it will explain why, harmonically, drums sound like ass.
The first thing I assume is a boundary condition saying the drum stays put together -- ie h = 0 for all radial distances greater than or equal to the radius of the drum. Then I assume that no matter the t, h is always finite. Furthermore, I have to hit it, so perhaps when t = 0, h is some given function of radial distance and angle defined by the hitting. In fact, hitting seems radially symmetrical about whatever point I hit it, so this initial condition needn't depend on the angle. Finally, I might stipulate that h is solution of the wave equation. Given all of the conditions, I can solve this and no everything about the vibrations, given these assumptions.
It's a powerful technique; it tells us, in this instance, why drums are harmonically unpleasant, for instance. However, the understanding of natural processes we gain is only as good as our assumptions. For instance, solving the BVP I set up above will tell us that the vibrations never dampen, because the wave equation includes no dampening term. It's still a good model, for limited amounts of time, but it illustrates the fact that maths in natural science is not like maths alone because it in no way perfectly describes the phenomena in question, and setting up and solving differential equations is only as useful as the assumptions about the natural process which they represent.
Not all natural processes lend themselves to good modelling by DEs. Some natural processes are basically discrete, and this can lead to very complicated behaviour that a DE would miss (compare the logistic differential equation to its discrete counterpart, and their respective uses in population models). Therefore, asking what DE describes natural selection is not necessarily very sensible, especially since DEs (like you just saw with the wave equation) model specific phenomena under narrow ranges of conditions. Natural selection is a very broad and varied process, and requires much more complex modelling (potentially involving DEs or systems of them).
Forces of Nature
31 Mar '06 04:08
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