# 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
𝑑𝑝
𝑝
= 𝑘𝑝 (1 − ) − 𝑎
𝑑𝑡
𝑁
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
𝑑𝑝
𝑝
= 𝑘𝑝 (1 − ) − 𝑎(1 + sin 𝑏𝑡)
𝑑𝑡
𝑁
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
𝑑𝑝
𝑝
= 𝑘𝑝 (1 − ) − 𝑎
𝑑𝑡
𝑁
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.)
𝑑𝑝
𝑝
= 𝑘𝑝 (1 − ) − 𝑎
𝑑𝑡
𝑁
𝑤ℎ𝑒𝑟𝑒:
𝑝 = 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
𝑘 = 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡
𝑁 = 𝑐𝑎𝑟𝑟𝑦𝑖𝑛𝑔 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑛𝑣𝑖𝑟𝑜𝑛𝑚𝑒𝑛𝑡
𝑎 = ℎ𝑎𝑟𝑣𝑒𝑠𝑡𝑖𝑛𝑔 𝑟𝑎𝑡𝑒
𝑑𝑝
= 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑡𝑖𝑚𝑒
𝑑𝑡
𝑵𝒐𝒕𝒆: 𝑓𝑜𝑟 𝑡ℎ𝑖𝑠 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛, 𝑡ℎ𝑒 𝒌, 𝑵 𝑎𝑛𝑑 𝒂𝟏 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟𝑠𝑔𝑖𝑣𝑒𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒.
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.
𝑑𝑝
𝑝
= 𝑘𝑝 (1 − ) − 𝑎
𝑑𝑡
𝑁
𝐹𝑜𝑟 𝐶ℎ𝑜𝑖𝑐𝑒 𝑛𝑜. 2:
𝑑𝑝
𝑝
= (0.50)𝑝 (1 − ) − 0.21
𝑑𝑡
2
𝐹𝑖𝑛𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑜𝑖𝑛𝑡𝑠:
0 = 0.50𝑝 − 0.25𝑝2 − 0.21
[0 = 0.50𝑝 − 0.25𝑝2 − 0.21](−100)
0 = 25𝑝2 − 50𝑝 + 21
0 = (5𝑝 − 3)(5𝑝 − 7)
5𝑝 − 3 = 0 𝑎𝑛𝑑 5𝑝 − 7 = 0
𝑝=

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.

𝒅𝒆𝒕𝒆𝒓𝒎𝒊𝒏𝒊𝒏𝒈 𝒕𝒉𝒆 𝒗𝒆𝒓𝒕𝒆𝒙 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒖𝒓𝒗𝒆:
𝑆𝑒𝑡 𝑡ℎ𝑒 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑖𝑐 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑖𝑛𝑡𝑜 𝑖𝑡𝑠 𝑣𝑒𝑟𝑡𝑒𝑥 𝑓𝑜𝑟𝑚 𝑡𝑜 𝑖𝑑𝑒𝑛𝑡𝑖𝑓𝑦 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑒𝑥 (𝐻, 𝐾)
𝑑𝑝
= −0.25(𝑝 − 1)2 + 0.04
𝑑𝑡
𝐻 = 1 𝑎𝑛𝑑 𝐾 = 0.04
𝑇ℎ𝑖𝑠 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑒𝑠 𝑡ℎ𝑎𝑡 𝑎𝑡 𝑝(𝑡) = 1, 𝑡ℎ𝑒

𝑑𝑝
𝑖𝑠 𝑎𝑡 𝑖𝑡𝑠 ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑎𝑡 0.04
𝑑𝑡

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

3
5

3
5

𝑇ℎ𝑒 𝐹𝑖𝑠ℎ𝑒𝑠 𝑤𝑖𝑙𝑙 𝑑𝑖𝑒 𝑜𝑢𝑡.
𝑇ℎ𝑒 𝐹𝑖𝑠ℎ 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑙𝑙 𝑟𝑒𝑚𝑎𝑖𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.

3
7
𝑇ℎ𝑒 𝐹𝑖𝑠ℎ 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑙𝑙 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒.
5
5
7
𝑃𝑜 =
𝑇ℎ𝑒 𝐹𝑖𝑠ℎ 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑙𝑙 𝑟𝑒𝑚𝑎𝑖𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
5
7
𝑃𝑜 >
𝑇ℎ𝑒 𝐹𝑖𝑠ℎ 𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑤𝑖𝑙𝑙 𝑟𝑒𝑚𝑎𝑖𝑛 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡.
5
𝐹𝑜𝑟 𝐶ℎ𝑜𝑖𝑐𝑒 𝑛𝑜. 8:
𝑑𝑝
𝑝
= (0.20)𝑝 (1 − ) − 0.24
𝑑𝑡
5
𝐹𝑖𝑛𝑑𝑖𝑛𝑔 𝑡ℎ𝑒 𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑝𝑜𝑖𝑛𝑡𝑠:
0 = 0.20𝑝 − 0.04𝑝2 − 0.24
[0 = 0.20𝑝 − 0.04𝑝2 − 0.24](−100)
0 = 4𝑝2 − 20𝑝 + 24
0 = (2𝑝 − 6)(2𝑝 − 4)
2𝑝 − 6 = 0 𝑎𝑛𝑑 2𝑝 − 4 = 0
𝑝 = 3 ,2
At p(t)=3...