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Lecture

9.4 Population Growth Model Question #2 (Medium)


Department
Mathematics
Course Code
MAT136H1
Professor
all

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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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