Continuous time exponential growth

Earlier we examined exponential growth in discrete time. Then we examined the concept of continuous flow rates, and informally discussed the concept of modeling a population using continuous variables.

There are two things that can be continuous: the variable (or variables) that represents the state of the system, but also time. In other words, you could have a model where the state of the system is represented by an integer that can only be 0, 1, 2, ... or you could have a model where the state of the system is represented by a continuous quantity. And you could also have a model where we calculate the state variables for specific discrete time points, or we could calculate the state variables for continuous time.

So in applied modeling, you can have

• Discrete time, discrete state
• Discrete time, continuous state
• Continuous time, discrete state
• Continuous time, continuous state

So we're thinking about modeling a population using a continuously varying number, changing smoothly over time. We've seen that we can use continuous flow rates to help understand such processes. What is the flow rate for a simple population process?

Suppose that there are x(t) individuals at a certain time. It is traditional to imagine the tank with a certain amount of water in it, where the water level is x(t). We imagine birth or immigration being mechanically represented by water flowing into the tank, and death or emigration being represented by water flowing out of the tank.

Let's consider a pure-birth process; only birth can happen. Suppose that every one of the x(t) individuals has a chance a dt of giving birth in some tiny interval of time dt. If you make the tiny time interval dt twice as long, you make the probability twice as great. This can't work for large dt, since you would eventually have a probability greater than 1. That's against the law, and we have to restrict ourselves to very small units of time dt. We'll have a lot more to say about this sort of thing later.

So where are we? We say that in some tiny time interval, every individual has a chance a dt of giving birth, and to keep this all simple, let's suppose they give birth to only one new individual. Think of an amoeba dividing. How many new individuals do we expect in the tiny interval of time? We expect x(t)a dt new individuals; that is, the number of individuals times the chance of a birth for each individual.

If we were developing a probabilistic or stochastic model, we would write the probability of there being a certain number at the next time interval. We would treat the number x(t) as a random variable changing over time. But for our purposes, we are going to imagine that x(t) is large enough that we can more or less ignore the randomness in the number of new individuals and simply say that the flow rate is equal to its expected value. In other words, in every little interval of time dt, we have x(t)a dt new individuals.

So the flow rate, the number of individuals per unit time, is x(t)a.

The inflow rate is in fact proportional to the size of the population. It is possible to prove that the actual curve of growth is an exponential under these circumstances. In other words, x(t)=x(0)e^at. The number e is the base of the natural logarithms, and is about 2.718.

To plot an exponential curve, try this in Rweb. Here endtime is the end of the plot, and the variable a we represent by aa. (It is evil to use single letter variable names; try searching for them in a long program using an editor.) You can change these two things and repeat the illustration to get an idea of how this works. Try letting aa be negative; what does this represent? You can also vary the starting value (initial condition).

endtime <- 10 aa <- 0.1 starting.value.of.x <- 1 thetimes <- seq(0,endtime,length=100) amount <- starting.value.of.x * exp(aa*endtime) plot(thetimes,amount,type="l",xlab="t",ylab="x(t)") points(thetimes,amount)

For a somewhat different example of exponential growth, let's think about a nuclear fission bomb. I'm writing this in late 2002, when talk of nuclear terrorism is not being laughed at. But let's leave that aside and think about exponential growth. This is an optional example and you should feel free to skip to the next segment.

How does a fission bomb work? Inside the bomb core, an amount of either plutonium 239 or uranium 235 is assembled together. Whenever the atomic nucleus of one of these isotopes splits in two, several neutrons are released. These can get absorbed by other nuclei, and split them too. They release more neutrons, that split more nuclei, and so on and so on. Every time one of these nuclei splits, energy is released, and the result of this is that you release a tremendous amount of energy in a hurry. You heat up the core to many millions of degrees; you create tremendous pressures, and, well, one of these.

Now if every split nucleus causes more nuclei to split and so on, you can see the sort of runaway growth like we talked about before. You could model this either by discrete time or continuous time. In the book The Effects of Nuclear Weapons by Glasstone and Dolan (put out by the US Government, 1977), we find that the growth rate of the number of neutrons for certain uranium bombs would be about a=10⁸ per second, times the number of neutrons. So we would write N=N₀e^at. Assuming that the chain starts from a single neutron, how long until you get ten kilotons of energy? The book says you get this from 1.45 times 10²⁴ fissions (and thus that many neutrons). How long until you get that many fissions?