Excursions in Modern Mathematics, 7e: 10.6 - 1Copyright © 2010 Pearson Education, Inc.

The Mathematics of Population Growth

Pages 474-491

Mini-Excursion 3

Excursions in Modern Mathematics, 7e: 10.6 - 2Copyright © 2010 Pearson Education, Inc.

• Population growth is a time-dependent process.

• Discrete growth occurs when there are gaps in the growth times.

• Continuous growth occurs without gaps between growth times.

Continuous vs. Discrete Growth

Excursions in Modern Mathematics, 7e: 10.6 - 3Copyright © 2010 Pearson Education, Inc.

• During discrete population growth, there will be sudden changes in the population called a transition.

• A mathematical model of population growth defines the transition rules in the form of an equation.

Discrete Growth

Excursions in Modern Mathematics, 7e: 10.6 - 4Copyright © 2010 Pearson Education, Inc.

• A population sequence is a sequence of numbers that represent the population levels at each transition time. The notation for a population sequence is:

• is called the initial population.

• The transition rule determines the population sequence.

Discrete Growth

,...,, 210 PPP

0P

Excursions in Modern Mathematics, 7e: 10.6 - 5Copyright © 2010 Pearson Education, Inc.

Discrete Growth

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• A time-series graph shows the time of each transition on the horizontal axis and the size of the population on the vertical axis.

Discrete Growth

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Discrete Growth

Excursions in Modern Mathematics, 7e: 10.6 - 8Copyright © 2010 Pearson Education, Inc.

• For a linear growth model, each generation of the population increases (or decreases) by adding (or subtracting) fixed amount called the common difference. For example:

Initial population = 1, common difference = 4

Linear Growth

,...13 ,9 ,5 ,1 3210 PPPP

Excursions in Modern Mathematics, 7e: 10.6 - 9Copyright © 2010 Pearson Education, Inc.

• The time-series graph of a linear growth model makes a straight line.

Linear Growth

Excursions in Modern Mathematics, 7e: 10.6 - 10Copyright © 2010 Pearson Education, Inc.

• If the common difference is denoted d, the linear growth model is

• This is called an arithmetic sequence.

Linear Growth

,... ,..., , , 112010 dPPdPPdPPP NN

Excursions in Modern Mathematics, 7e: 10.6 - 11Copyright © 2010 Pearson Education, Inc.

• We can summarize the linear growth model with the formula:

Linear Growth

NdPPN 0

Excursions in Modern Mathematics, 7e: 10.6 - 12Copyright © 2010 Pearson Education, Inc.

• Problem 4 on page 488

Example

Excursions in Modern Mathematics, 7e: 10.6 - 13Copyright © 2010 Pearson Education, Inc.

• For an exponential growth model, each generation of the population changes by multiplying a fixed amount called the common ratio. This is a geometric sequence. For example:

Initial population = 1, common ratio = 5

Exponential Growth

,...125 ,25 ,5 ,1 3210 PPPP

Excursions in Modern Mathematics, 7e: 10.6 - 14Copyright © 2010 Pearson Education, Inc.

• The time-series graph of an exponential growth model makes an exponential curve.

Exponential Growth

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• If the common ratio is denoted r, the exponential growth model is

• Or:

Exponential Growth

,... ,..., , , 112010 NN rPPrPPrPPP

0PrP NN

Excursions in Modern Mathematics, 7e: 10.6 - 16Copyright © 2010 Pearson Education, Inc.

• A population grows according to an exponential growth model with:

a. Find the common ratio rb. Find c. Give an explicit formula for

(this example is similar to assigned problems 11 and 12)

Example

25 and 20 10 PP

6P

NP

Excursions in Modern Mathematics, 7e: 10.6 - 17Copyright © 2010 Pearson Education, Inc.

• The linear growth model predicts unlimited growth as N gets larger if d is positive.

• The exponential growth model predicts unlimited growth as N gets larger if r is larger than 1.

Logistic Growth

0PrP NN

NdPPN 0

Excursions in Modern Mathematics, 7e: 10.6 - 18Copyright © 2010 Pearson Education, Inc.

• In reality, resources are limited and populations do not grow without limit.

• The logistic growth model predicts resource limited growth as N gets larger.

Logistic Growth

NNN pprp 11

Excursions in Modern Mathematics, 7e: 10.6 - 19Copyright © 2010 Pearson Education, Inc.

• In the logistic growth model represents the fraction of the population out of the total population allowed by the habitat called the carrying capacity of the habitat.

• The value of will always be a number between 0 and 1.

Logistic Growth

Np

Np

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• page 483

Logistic Growth

Excursions in Modern Mathematics, 7e: 10.6 - 21Copyright © 2010 Pearson Education, Inc.

• Solution:

Logistic Growth

Excursions in Modern Mathematics, 7e: 10.6 - 22Copyright © 2010 Pearson Education, Inc.

• Solution:

Logistic Growth

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• Solution:

Logistic Growth

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• page 484

Logistic Growth

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• Solution

Logistic Growth

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• Solution

Logistic Growth

Excursions in Modern Mathematics, 7e: 10.6 - 27Copyright © 2010 Pearson Education, Inc.

• Solution

Logistic Growth

Excursions in Modern Mathematics, 7e: 10.6 - 28Copyright © 2010 Pearson Education, Inc.

• page 484-485

Logistic Growth

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• page 484-485

Logistic Growth

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• page 485

Logistic Growth

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• page 486

Logistic Growth

Excursions in Modern Mathematics, 7e: 10.6 - 32Copyright © 2010 Pearson Education, Inc.

• No pattern (chaos)

Logistic Growth