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Outbreak 4.
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This is a stunning plot of the zero growth rate curve. Don't forget, P = 100N.
In the upper left of the graphic I've indicated a stable fixed point at B = 18,
and P = 53 approximately (N = 0.53). If we decrease B (predator efficacy factor)
we at first get little change in pink love sphere population. I've indicated
another stable fixed point at B = 15 (lazier predators, or fewer of them), and
P = 76 approximately. A relatively small increase in love sphere population. The next point indicated is at about B = 13.8 and P = 121. It's right on the end of the curve where it's turning from stable to unstable. Not surprisingly this point is a mix of stable and unstable. Still, it is a fixed point, and as long as B doesn't change, neither will P - at least theoretically.
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If we even slightly decrease B below this point then we're suddenly in a whole new
area, and area of growth, with D P / D t > 0 all the way up to values of P in the 500's. This is a population outbreak, and it is a consequence of how very nonlinear our expression for D P / D t is (that is, it is not a simple linear function of P (or N), which is of the general form, aP + b). The population is sneaking around the corner through the area indicated by the red arrows, and although the rate of growth is slow at first, it gets much faster in the region between P = 260 and P = 460 before it slows down again and reaches another stable fixed point. I've indicated one of these at B = 13, and P about 583. Note that now everything is changed. Even if we decrease B back to 15, the population will NOT shrink back to 76 (which is where it was the last time we had B = 15), but will slide down this other stable fixed point curve to P = 523 approximatley (indicated by another ball). Eventually we get to the other turning point (B = 17.1, P = 371), and if we go past this into the region indicated by the left pointing green arrows, then the population suddenly and drastically declines, just the opposite of an outbreak. Although our population model is fairly simple, it does illustrate the source of such phenomena as outbreaks and plagues. Nonlinearity is the origin of much that is interesting in Nature, and all that is chaotic, and without the chaotic we would not exist. Without chaos nothing could evolve, and without order nothing could be organized into a complex entity. Complex systems - whether organic or economic - achieve their greatest potential on the edge of order and chaos. We'll begin to explore more chaotic populations on page 10, but first we should go back to that last population animation and play a little more. So head to page 9. |
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