# SOLUTION: MATH 275 College of San Mateo Week 1 Ordinary Differential Equation HW

I did not middle some figures since I don't own mathlab, so I can't paint the presentation lines.

Logistic Population Models after a while Harvesting
1. Logistic Enlargement after a while Inwavering Harvesting
ππ
π
= ππ (1 β ) β π
ππ‘
π
Represents a logistic design of population enlargement after a while inwavering harvesting at a reprimand a. For a = a1,
what get betide to the fish population for a multitudinous modetrounce stipulations? (Note: This equation is
autonomous, so you can use service of the peculiar techniques that are suited for autonomous
equations.)

2. Logistic Enlargement after a while Occasional Harvesting
ππ
π
= ππ (1 β ) β π(1 + sin ππ‘)
ππ‘
π
Is a non-autonomous equation that considers occasional harvesting. What do the parameters a and b
represent? Let b = 1. If a = a1, what get betide to the fish population for multitudinous modetrounce stipulations?

3. Consider the identical equation as in Part 2 over, but let a=a2. What get betide to the fish
population for multitudinous modetrounce stipulations after a while this appreciate of a?
Use the aftercited choices for the parameters.
Choice

k

N

a1

a2

2

0.50

2

0.21

0.25

8

0.20

5

0.24

0.25

1. Logistic Enlargement after a while Inwavering Harvesting
ππ
π
= ππ (1 β ) β π
ππ‘
π
Represents a logistic design of population enlargement after a while inwavering harvesting at a reprimand a. For a = a1,
what get betide to the fish population for a multitudinous modetrounce stipulations? (Note: This equation is
autonomous, so you can use service of the peculiar techniques that are suited for autonomous
equations.)
ππ
π
= ππ (1 β ) β π
ππ‘
π
π€βπππ:
π = ππππ’πππ‘πππ
π = ππππ€π‘β πππ‘π πππππππππππ‘
π = πππππ¦πππ πππππππ‘π¦ ππ π‘βπ πππ£ππππππππ‘
π = βπππ£ππ π‘πππ πππ‘π
ππ
= πππ‘π ππ πβππππ ππ π‘βπ ππππ’πππ‘πππ π€ππ‘β πππ ππππ‘ π‘π πβππππ ππ π‘πππ
ππ‘
π΅πππ: πππ π‘βππ  πππ’ππ‘πππ, π‘βπ π, π΅ πππ ππ πππ πππππππ‘πππ πππ£ππ ππ π‘βπ π‘ππππ.
The equation is as-well autonomous as the equation is simply hanging on wavering P as waverings k,
N and a=a1 are all inwavering appreciates.
ππ
π
= ππ (1 β ) β π
ππ‘
π
πΉππ πΆβππππ ππ. 2:
ππ
π
= (0.50)π (1 β ) β 0.21
ππ‘
2
πΉππππππ π‘βπ ππππ‘ππππ πππππ‘π :
0 = 0.50π β 0.25π2 β 0.21
[0 = 0.50π β 0.25π2 β 0.21](β100)
0 = 25π2 β 50π + 21
0 = (5π β 3)(5π β 7)
5π β 3 = 0 πππ 5π β 7 = 0
π=

3 7
,
5 5

At p(t)=3/5 and p(t)=7/5, the reprimand of modify in population throughout all modifys of interval, get be
zero.

πππππππππππ πππ ππππππ ππ πππ πππππ:
πππ‘ π‘βπ ππ’πππππ‘ππ πππ’ππ‘πππ πππ‘π ππ‘π  π£πππ‘ππ₯ ππππ π‘π πππππ‘πππ¦ π‘βπ π£πππ‘ππ₯ (π», πΎ)
ππ
= β0.25(π β 1)2 + 0.04
ππ‘
π» = 1 πππ πΎ = 0.04
πβππ  π ππππππππ  π‘βππ‘ ππ‘ π(π‘) = 1, π‘βπ

ππ
ππ  ππ‘ ππ‘π  βππβππ π‘ ππ‘ 0.04
ππ‘

We can then use the equilibria to indicate what would betide to the fish population at multitudinous
modetrounce stipulations (Po).
0 ππ =

3
5

3
5

πβπ πΉππ βππ  π€πππ πππ ππ’π‘.
πβπ πΉππ β ππππ’πππ‘πππ π€πππ ππππππ ππππ π‘πππ‘.

3
7
πβπ πΉππ β ππππ’πππ‘πππ π€πππ πππππππ π.
5
5
7
ππ =
πβπ πΉππ β ππππ’πππ‘πππ π€πππ ππππππ ππππ π‘πππ‘.
5
7
ππ >
πβπ πΉππ β ππππ’πππ‘πππ π€πππ ππππππ ππππ π‘πππ‘.
5
πΉππ πΆβππππ ππ. 8:
ππ
π
= (0.20)π (1 β ) β 0.24
ππ‘
5
πΉππππππ π‘βπ ππππ‘ππππ πππππ‘π :
0 = 0.20π β 0.04π2 β 0.24
[0 = 0.20π β 0.04π2 β 0.24](β100)
0 = 4π2 β 20π + 24
0 = (2π β 6)(2π β 4)
2π β 6 = 0 πππ 2π β 4 = 0
π = 3 ,2
At p(t)=3...