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**preview**shows half of the first page. to view the full**1 pages of the document.**9.4 Differential Equations

Population Growth Model

Question #2 (Medium): Setting the Logistic Differential Equation

Strategy

The logistic differential equation is:

which shows the rate of population growth over

time incorporating the carrying capacity and growth rate k factor, which can be calculated by

by plugging in the birth rate over death rate as

and initial population of .

Sample Question

Population at a small town by a mountain is about in 2013.

Birth rates range from to per year, and death rates range from to per year.

Assume that the carrying capacity for the town’s population is .

1) Write the logistic differential equation for the given data.

2) Use the logistic model to estimate the town’s population by 2017.

3) How would the future forecast change if the carrying capacity is ?

Solution

1) Given the logistics model of

, where the carrying capacity is

Since

, and when and

denotes population growth rate, birth rates

exceed death rate by on average per year, so

. Thus, the

logistic differential equation modeling the town’s population is:

2)

, where

, then

;

Thus,

. Therefore, the model estimates the population

by 2017 to be .

3) If the carrying capacity is , remains the same because it is not affected by the carrying

capacity. However the factor A would change:

;

; the population estimation for 2017 would change to:

. Therefore the estimate is then

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