Exponential Growth and

Discrete Dynamical Systems

Lesson Objectives:

1. Sequences and Population Models

2. The Exponential Growth Model

3. Definitions: Transition Rule, Exponential growth

Recall from last lesson:

We will be very interested in sequences defined recursively because such sequences are important in modeling the growth of populations over time.

Why???

First, recall why populations are modeled with sequences...

Populations are measured in a discrete way.

- human population, annually
- a population of bacteria, every 10 minutes
- Populations might also be measured in terms of generations!

Hence it is natural to model population growth over time with a sequence:

{P[1], P[2], P[3], … , P[i-1], P[i], …}

where P[i] is the population size during the ith time increment.

To understand why populations are modeled with recursive sequences...

Start with an example. Assume you have two rabbits -- one male, and one female. They reproduce and after one year you find that you now have 8 rabbits.

How many rabbits would you expect to have after two years?

A reasonable answer might be 32, particularly if half of the 8 rabbits are male, the other half female. This is because 8 rabbits can be paired off into four pairs of rabbits. Each of these four pairs can then be expected to reproduce in the same way as the original pair did. (A rabbit is a rabbit is a rabbit...) Since 2 rabbits led to 8 rabbits in the first year, one might expect 4 pairs of rabbits to become 4(8) = 32 rabbits in the third year. Similarly, in the fourth year you might expect to have 4(32) = 128 rabbits. In general, if P[i] is the population of rabbits in year i then P[i+1] = 4P[i] is a reasonable way of predicting the population one year later (i.e., in year i + 1).

In this way, the population of rabbits is represented by the recursive sequence:

P[1] = 2

P[i+1] = 4P[i]

Each iteration is determined by multiplying the previous population level by 4.

More generally, it's easy to see why a recursive sequence is helpful in modeling population growth. Namely, the population level during year i+1 (or generation i+1, etc) depends on the population existing during year i.

The rule describing how the population changes from one year to the next is called the transition rule. Applying the transition rule over and over again, starting with the initial population P[1], we can generate the entire population sequence. Using the terminology from last lesson, we can also say that the population sequence is the orbit of the initial population level under iteration by the transition rule defined by the function f(x). That is, the population sequence can be described by:

P[0],

P[1] = f(P[0]),

P[2] = f(P[1]) = f(f(P[0])),

P[3] = f(P[2]) = f(f(f(P[0]))),

P[4] = f(P[3]) = f(f(f(f(P[0])))),

.

.

.

P[i+1] = f(P[i]) = f(f(f(f(...f(P[0])...)))) (here f is iterated i+1 times)

The important concept to keep in mind is the following:

The same rule (i.e., f(x)) determines the new population level from the old population at each stage. Iterating this rule over and over will produce the entire population sequence.

Example.

A population of snouters starts out (in year 1) at 100, and they double in number every year. How many snouters will there be in 20 years?

ANSWER: The population in year i, P[i], can be modeled using the transition rule f(x) = 2x. That is, P[i] = 2P[i-1]. We get the following population sequence for the snouters:

After 20 years there will be:

Note: we computed P[21] and not P[20] because we were interested in finding the number of snouters in existence after 20 years have past. This means that there will be 20 transitions. Since we started out at i = 1, we must compute the population when i = 21.

The Exponential Growth Model

The previous sequence, P[i+1] = 2P[i], is said to grow exponentially. The reason for this should become clear if you think about the explicit description of the sequence.

More generally, a sequence exhibiting exponential growth will be any sequence of the form P[i+1] = r P[i], where r is some real number.

Definition.

As you will soon see, exponential sequences are very important for the mathematical modeling of biological growth.

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