150100502040603. Suppose that a population develops according to the logisticequation(c) Use the direction field to sketch solutions for initial pop-ulations of 20, 40, 60, 80, 120, and 140. What do thesedP= 0.05P - 0.0005P2dtsolutions have in common? How do they differ? Whichsolutions have inflection points? At what population levelsdo they occur?(d) What are the equilibrium solutions? How are the othersolutions related to these solutions?where t is measured in weeks.(a) What is the carrying capacity? What is the value of k?(b) A direction field for this equation is shown. Whereare the slopes close to 0? Where are they largest?Which solutions are increasing? Which solutions aredecreasing?

Question
Asked Aug 2, 2019
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100
50
20
40
60
3. Suppose that a population develops according to the logistic
equation
(c) Use the direction field to sketch solutions for initial pop-
ulations of 20, 40, 60, 80, 120, and 140. What do these
dP
= 0.05P - 0.0005P2
dt
solutions have in common? How do they differ? Which
solutions have inflection points? At what population levels
do they occur?
(d) What are the equilibrium solutions? How are the other
solutions related to these solutions?
where t is measured in weeks.
(a) What is the carrying capacity? What is the value of k?
(b) A direction field for this equation is shown. Where
are the slopes close to 0? Where are they largest?
Which solutions are increasing? Which solutions are
decreasing?
help_outline

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150 100 50 20 40 60 3. Suppose that a population develops according to the logistic equation (c) Use the direction field to sketch solutions for initial pop- ulations of 20, 40, 60, 80, 120, and 140. What do these dP = 0.05P - 0.0005P2 dt solutions have in common? How do they differ? Which solutions have inflection points? At what population levels do they occur? (d) What are the equilibrium solutions? How are the other solutions related to these solutions? where t is measured in weeks. (a) What is the carrying capacity? What is the value of k? (b) A direction field for this equation is shown. Where are the slopes close to 0? Where are they largest? Which solutions are increasing? Which solutions are decreasing?

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Expert Answer

Step 1

Given that a population develops according to the logistic equation

dP
= 0.05 P
dt
0.0005 P2
where t is measured in weeks.
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Image Transcriptionclose

dP = 0.05 P dt 0.0005 P2 where t is measured in weeks.

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Step 2

We know the Logistic Model equation is

P(1-)
where k is the Malthusian parameter and M is the carrying capacity
(a) By comparing the given differential equation with the equation (1)
dP
kP
-(1)
M
dt
dP
= 0.05 P
dt
0.0005 P2
= 0.05 P(1 0.01P)
P
= 0.05 P 1
100
So, carrying capacity is 100 and the value of k is 0.05
help_outline

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P(1-) where k is the Malthusian parameter and M is the carrying capacity (a) By comparing the given differential equation with the equation (1) dP kP -(1) M dt dP = 0.05 P dt 0.0005 P2 = 0.05 P(1 0.01P) P = 0.05 P 1 100 So, carrying capacity is 100 and the value of k is 0.05

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Step 3

Now,

...
dP
= 0 => 0.05 P ( 1 -
dt
sr(1-100)
P
= 0
=> P = 0 or P
100
11
So, at P 0 and P
100 the slopes are close to 0.
The slopes are
largest at around P=150.
From (0,100) the solutions are increasing and (100,150) the solutions are
decreasing
help_outline

Image Transcriptionclose

dP = 0 => 0.05 P ( 1 - dt sr(1-100) P = 0 => P = 0 or P 100 11 So, at P 0 and P 100 the slopes are close to 0. The slopes are largest at around P=150. From (0,100) the solutions are increasing and (100,150) the solutions are decreasing

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