Some Mathematics.

This is the result of one of my trials. And before I get too technical, let me say something about the shape of this curve that may be crucial to our survival.

There is a puzzle story attributed to the French that goes something like this: there was once a pond on which a lily pad grew on day 1. On day 2 there were two lily pads. On day 3 the number doubled again to 4. And so it went on, the number of lily pads doubling every day. Well, on day 29 you stop by and notice that the pond is half full of lily pads, half way to being completely choked with them. How many days must pass after day 29 before the pond is completely choked?

The answer: ONE.

Twenty nine days to get halfway choked, one day to go from halfway to completely choked.

That's because of the shape of the growth curve. As we see in this example, for about 10 generations the population is fairly low, and unalarming. Then for about 5 additional generations it starts to really change, but it's still not too alarming. After that growth becomes explosive, very alarming, and very difficult to handle. Very scary.
Let's get theoretical. Let's say we start with a population of P o. After one generation each member of that population has a 0.5 probability of reproducing, so we expect around Po + 0.5 Po = 1.5 Po = P1 to be the new population. About half of these will reproduce after two generations leading to a new population of 1.5 P1 = 1.52 Po = P2. So after n generations we expect a population given approximately by
Pn = Po 1.5n.
So the population has an exponential dependence on n (which is a discrete time variable (number of generations) taking values 1,2 3,...), but this is theoretical. In reality the vagaries of probability will lead to some variation.

The green curve has the equation

P(t) = Po 1.5t,
where t now represents a continuous time variable and can take any real value, not just positive integers. This will have the same graph every time we run another experiment.

Let's go to a clean page.

Home.
Population Page 1.
Population Page 3.

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