by Olivia DeLane

I was looking through some of my old scratchings and ran across this little item that I discovered long ago, but had forgotten about.

In a previous column I used a modified version of the Lotka-Volterra predator-prey model to investigate if lions should forbid hunting kid gazelle (the way that humans tend to prohibit taking fish that are smaller than a certain size).

The modification to the equation set was that there were two state variables for the prey (one for juvenile kids and one for the full-grown adults) rather than just a single variable.

The equations looked like this in the case where Lions do not discriminate (but for this exercise it turns out that the specific equation for the number of Lions does not matter):

I was interested in the long-term ratio of Adults (A) to Kids (K) in the case where the r and t parameters are equal, so I would like to find when that ratio is constant. We can use the quotient rule to find these values.

K-squared must be greater than 0, it turns out that the terms with L in them cancel, and the r-parameter can be taken out of each remaining term. So we are left with a simple expression that can be solved by the quadratic equation:

It so happens that in the case where r = t, the ratio of Adults to Kids will tend toward the Golden Ratio.

Now if that isn’t a little speckled egg, I don’t know what is. It makes me wonder what other fundamental constants are hiding out in sets of ordinary differential equations.

Toodles,
Olivia DeLane, Chicken Philosopher

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Olivia DeLane has a Master’s certificate in normative ethology from Gallus College, a non-accredited online institution. Her writings are for entertainment purposes only and should not be misconstrued as being for any other use. Her newest e-book, ‘Anti-antifragile: Harnessing the Power of Vulnerability’ will be available soon.

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