The Logistic Model of Population Growth is a simple and easy-to-understand mathematical model. Nonetheless, it has some interesting properties that lead directly into the world of Chaos.

The following modules can be explored in any order.

Deriving the Logistic Model
How the logistic model is derived from first principles. No mathematical sophistication needed.

Experimenting with the Logistic Model
Some simple computerized experiments that reveal the nature of the logistic model.

Chaos and Jurassic Park
How the logistic model illustrates the Butterfly Effect.

Further Exploration
Books to read.

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Experiment 1: A model that achieves a stable population.

Experiment 2: A model that oscillates between two population levels

Experiment 3: A model that displays unpredictable behavior

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Scene I: John Hammond's Problem

Scene II: Ian Malcolm's Solution

Populations of fish, insects, people, … usually don't grow in a regular fashion. They grow and decline, grow some more, decline again, and so forth (sort of like the stock market). So they are not modeled well by nice models like the exponential or polynomial models.

One of the chief factors controlling real-world populations is the availability of resources: room to move about in, food, water, enough choices for mating, and so forth. Many of these factors are captured by the Key Idea of the Logistic Model, to be discussed below. But first, we need to introduce some preliminary concepts and terminology.

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consider the population of fish in a pond
if we observe fish population at the beginning of each day
it will appear to grow in discrete jumps
on day 12 there may be 20, on day 13 there may be 15

let P_N represent the population at the beginning of day N
if there are 20 fish in the lake on day 12, we would write:

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Population growth is governed by a growth factor

If, on day 7, there are 50 fish . . .

and the growth factor is 1.5

then on day 8 there will be (50)(1.5) = 75 fish

so we would write

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if the growth factor is denoted by g, then

we have the transition equation

That is, the population tomorrow will be g times what it is today; or, the population on day N+1 will be g times what it is on day N.

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assume we have a pond that can hold at most 100 fish:

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Elbow room is an important concept in the discussion of populations, since the presence or lack of elbow room at a given point in time in an environment has a profound effect on the rate at which the population will grow.

For a pond whose capacity is 100 fish

we define:

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how do you translate "x is directly proportional to y" into math?

let's say that Snickers are 50¢ each

the total amount A that you spend on them is directlyproportional to the number N that you buy -- right? . . .

and the formula is:

If Butterfingers are 75¢, the formula (for them) is

The price of one CB is referred to mathematically as . . .

and may be different for different candy bars

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One would expect that populations will grow slower as they approach maximum capacity, and will grow faster when they are relatively small.  This notion is captured by the Key Idea of the Logistic Model:

For the logistic model, the growth factor will be changing over time

because it depends on (is directly proportional to) elbow room, which changes over time

Mathematically, for growth factor g and elbow room C - P_N, we have

Now, what is R?
It's the constant of proportionality (cost of candy bar?)
different for different kinds of populations:

it's just a number that is determined experimentally by scientists for a given population
and handed to us as a given number
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To summarize what we have so far:

Population tomorrow (day N+1) is expressed as the growth factor times the population today (day N), by way of the transition equation:

Growth factor g is directly proportional to elbow room (key idea)

Substituting, we have the logistic growth model (preliminary version):

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Now to polish up the preliminary version of the logistic growth model,

and put it into its final form by introducing a couple of "mathematical niceties."

Instead of talking about actual population P_N we will, instead, refer to percentage of capacity p_N. Notice that percentage of capacity uses the lower case "p", to distinguish it from actual population "P". Here's the formula (conversion formula) for converting P_N to p_N:

  =  .75     (or 75%)

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The Logistic Growth Model is the following:

p_N+1 = r(1 - p_N)p_N

p_N+1 = population on day N+1 (as a fraction of total capacity)
p_N = population on day N (as a fraction of total capacity)
r = a number depending on the population.

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Example 1:

pond of capacity C = 100 fish
start with 20 fish (20% capacity), so p₁ = .20
let r = 2 (this comes from nowhere).  Find p₁, p₂, p₃, … ,p₇

Solution:

p₁ = .20
p₂ = r(1 - p₁)p₁ = 2(.80)(.20) = .32
p₃ = r(1 - p₂)p₂ = 2(.68)(.32) = .44
p₄ = 2(.56)(.44) = .49
p₅ = 2(.51)(.49) = .50
p₆ = 2(.50)(.50) = .50
p₇ = 2(.50)(.50) = .50

You can see that this population will remain at 50% of capacity after 4 days.

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r = 3.1, p₁ = .2

Solution:

p₁ = .2
p₂ = r(1 - p₁)p₁ = 3.1(.80)(.20) = .50
p₃ = r(1 - p₂)p₂ = 3.1(.50)(.50) = .78
p₄ = 3.1(.22)(.78) = .53
p₅ = 3.1(.47)(.53) = .77
p₆ = 3.1(.23)(.77) = .55
p₇ = 3.1(.45)(.55) = .77

And from this point on, we see that the population will oscillate between 77% of capacity and 55% of capacity forever.  Very strange behavior indeed, but it's what the model predicts!

These calculations are tedious!  Let's get a little help from the computer!  Go to the next section,

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Our starting point is the logistic growth model:

where:

p_N = population on day N (as a fraction of capacity)
p_N+1 = population on day N+1 (as a fraction of capacity)
r = the r-factor (a number characterizing the population; different for different populations)

The numbers the experimenter can "play with" (the parameters of the experiment) are:

• the r-factor r
• p₁, the initial population size (expressed as a fraction of capacity)

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For this experiment, the parameters are: p₁ = .2, r = 2.0.

You will see a display into which these numbers have already been set. "Population 1: start" is where p₁ appears. Click the "Draw" button to see the first ten days of growth develop. Click again to see the next ten days, and so forth. Then you can scroll down for some comments about this experiment.

Click here first.

As we discovered when doing Example 1, this population goes from 20% of capacity to 50% of capacity, and stays there.

You may wish to change the parameters, and see what the model looks like. For example, set p₁ = .8, and then .1, and then .001, to see what happens.

What effect does the starting population value have on its steady-state level?

To return to the experiment, click here.

You can see what the model looks like, at, say, 100 or 1000 days out, by changing the "start day" field, and clicking on "Draw".

To return to the experiment, click here.

You may also experiment with r.  Set it to 2.5, reset the start day to 1, and see what happens. Don't forget to click "Draw".

Is a steady-state population reached?  At what level?

What effect does the value of r have on the steady-state level of population?

To return to the experiment, click here.

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For this experiment, the parameters are: p₁ = .2, r = 3.1.  Click here.

Notice that for this higher value of r, a steady state population is not reached, but rather, it oscillates between two population levels.

Take the model out to 100 days, then 1000 days, to see if this behavior persists.

Click here to go back to the screen.

Try r = 3.5 and some values in between.

What effect do r-values between 3.1 and 3.5 have on population behavior?

Click here to go back to the screen.

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For this experiment, the parameters are: p₁ = .2, r = 3.99. Click here first.

Take the model out to day 100, then day 1000. Does the unpredictable behavior exist? Try it for r = 3.9, 3.8, and others. For what value of r does unpredictable behavior ensue?

Click here to return to the experiment.

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It was a gloriously tropical day on Islar Nublar, but John Hammond, creator of Jurassic Park, was in anything but a sunny mood, and mathematician Ian Malcolm was not helping matters one single bit.

John had been relating to Ian how he had these two ponds, side-by-side, with populations of long-extinct Jurassic piranha.  His scientists had determined that they were genetically equivalent (all clones of a single progenitor), and that conditions in the two ponds were identical, right down to shape, temperature, and amount of water and food.

"I just don't understand it," whined John, "these two ponds started with virtually the same number of piranha, but 15 days later, their population growth patterns are wildly different."

"I'm not surprised," interjected Ian.

"But listen to this: I started one population off at 300 piranha, and the other with 301 --  I meant them to be the same, but later I discovered that I had miscounted.  In any event, on day 14, the pond that started with 301 fish was close to extinction, while the one that started with fewer fish was holding its own with about 300 fish.  I just don't understand it.  If the populations start off similarly-sized, doesn't it just stand to reason that they will more-or-less parallel each other in their population growth and decline patterns?"

Ian answered in an exasperated tone. "No, it absolutely doesn't stand to reason, John."

"Well, you're the mathematician.  Maybe you can explain it to me."

"Unfortunately, John, you are still suffering from the same misconceptions that got Jurassic Park into trouble in the first place. You are still thinking linearly, Newtonically, and deterministically, in spite of your own experience which has proved that nature will, at times, be none of those things."

"Well, I'm not sure I understand what 'Newtonically' means. Could you explain it?"

"Yes. Newton's laws of motion tell us that if we can determine exactly an apple's position, velocity, and acceleration at a given point in time, we can predict where it will be at any point in the future. In fact, even if we don't get the initial conditions quite right, we can still make a 'pretty good' prediction. And the closer we can compute the initial conditions, the closer we will get to the true answer."

John Hammond was approaching his point of frustration. "OK, but how do the laws of population growth differ from the laws of motion? Why can't I predict my piranha population? You're not helping me one whit to understand this!"

"Stay with me. Here's the difference. Some natural behaviors are well-disciplined and regular. That's why scientists can aim a ballistic missile from thousands of miles away and have it land within less than a mile of the intended target. But other natural behaviors are chaotic in nature. They are unpredictable, and the minutest variation in initial conditions can lead to wildly different results. I spoke to you before about the Butterfly Effect: that the flap of a butterfly's wings in New York, or the lack thereof, can make the difference of whether or not there is a tornado in China.  Mathematicians refer to this as extreme sensitivity to initial conditions, and it is a defining feature of chaotic systems, weather being a most notable one."

John was starting to turn red and perspire as a consequence of the mental effort being expended by him. "But are my silly little ponds chaotic?"

"Absolutely. You've been keeping records on the growth of your pond populations. Show me what some of your numbers are, and I will do a little mathematical work-up on them."

So John gave Ian his numbers.  Ian went in to find his lap-top to do a little modeling.

John gathered up a margarita and retired to pool-side.  There he sat, trying to appease the Demons of Mathematics that were assailing him from all quarters.

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A couple of hours later, Ian joined John at the pool, lap-top in tow, and with a gleam in his eye.

"All right, now I have something to show you," he said. "I've applied the logistic model for population growth to this problem. This model needs two parameters in order to do some calculations with it: (1) the starting population, expressed as a fraction of maximum population, and (2) a characteristic number, or r-factor, for the population. The r-factor can be experimentally determined."

"Oooh boy" lamented John, taking another swig of his margarita.

"Your figures show me that the maximum capacity of your ponds is 1000 fish. Therefore, pond 1, at 301 fish, started out at 30.1% of capacity (.301), and pond 2, with 300 fish, at 30% (.300).  I have also figured out that the r-factor for this species of fish in this environment is 4.0.  Now comes the exciting part.  What you will see on my laptop is a graph that projects populations in the two ponds based on the Logistic Model and these parameters."

With that, he showed John the set-up, and then clicked on the "Draw" button. Here is what he saw.

(To see Ian's demo, you need to click here, and then click on the "Draw" button).

"Aha," exclaimed John, "look at that! Chaos indeed! Both graphs are proceeding in parallel fashion. I knew you were …"

"Not so fast!" interrupted Ian. "Take a close look at day 10. See how things are starting to differ? Even after just 10 days?. Let's click again, and see what happens over the next 10 days."

(click here, then click "Draw" twice)

Ian continued. "Look at day 14; just as you observed experimentally: Population 1 close to extinction, and Population 2 at about 30%! Also notice that the behaviors of the two populations are wildly different. Let's see if things settle down at day 100, or day 1000".

(click here, then change start day to 100 and click "Draw", then day 1000 and click)

"I can't believe this," John said. "All because of one bloody fish out of 300. That's an initial condition difference of only 1/300, or … (consulting his calculator) … just .0033!"

"John, there is more to the mathematics of nature than meets the eye."

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You may read more about the logistic model and chaos theory in the following references:

Chaos, Making a New Science, by James Gleick. Viking Penguin Inc. 1987

Exploring Chaos, A Guide to the New Science of Disorder, edited by Nina Hall. W. W. Norton and Co., 1991.

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