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Population Fixed Points and Stability.
Let N = p/100, which I define to make it easier to rewrite the new growth rate:
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DP / Dt =
{ (8 - (B+1)N + 8N2 - N3) / (16 + 16N2) }P.
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Down below is a calculator for DP /
Dt (I am continuing to write this as
a discrete rate of change, rather than a continuous (derivative) rate of
change because it makes it easier in the case here with
Dt = one generation). For example,
if you type in P = 275, and B = 18, then you'll get -9.13 for the rate of
change. That means that in the animation on the previous page, if at some
point I have a population of 275 (which I can set by holding the mouse down
on the middle part of the stage), and click the B = 18 button, then click the
new generation button (upper left), I should get a change in population of
about -9.13 (or -9, rounded off) and drop down to about 266. The actual
value you get may vary, but this is the average value.
Now type in P = 66 and B = 16. This yields a population rate of change = -0.07.
This rounds off to 0, which means that the population has a very small probability
of changing from one generation to the next as long as B = 16 and P = 66. So the
pink lover sphere population is very close to fixed at 66 as long as B = 16.
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B in some
sense represents the effectiveness of the predator population, with higher values
of B yielding hungrier, more effective predators, which means smaller growth rates,
or larger death rates (negative growth), for the pink love sphere population. For
example, the values P = 66 when B = 15 is not a fixed point. In that
case the growth rate changes to 1.83, which rounds to 2, so I expect on average that
after one more generation the population will increase to about 68.
Let's go back to P = 66, and B = 16. In particular, consider the following populations,
B-values, and growth rates:
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P = 65
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B = 16
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DP / Dt = 0.16
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P = 67
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B = 16
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DP / Dt = -0.29
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The implication is, if we set B = 16, and decrease the population a little below
the fixed point at P = 66, then the growth rate will be positive and tend to push
the population back up towards 66. Likewise, if we boost the population a little
more than 66, the growther rate will be negative and tend to drop us back to 66.
That means that the fixed point at P = 66 and B = 16 is in some sense STABLE,
meaning that if we change the population a little up or down from P = 66, it will
tend to snap back into place. Ok, now it's up to you ...
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Take a look at P = 200, B = 15. This is another fixed point (growth rate = 0).
Is it stable? Or unstable? How about P = 400, B = 17?
In particular, I'm asking you the reader to devise a definition
of instability generalizing on the notion of stability outlined above.
ONE FINAL REMARK: If you're using Microsoft's Internet Explorer the
"Do The Math" button below may not function, in which case you won't be able
to do any calculations. I'd strongly advise Netscape Navigator 3.0 or better.
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