Pop'n Model equation

**fifthrapiers** · October 13th 2007, 12:22 PM

Given that the population of a certain fish species increases according to the model below:

$\frac{dP}{dt}=0.3\left(1-\frac{P}{200}\right)\left(\frac{P}{50}-1\right)P$

1.) Determine the values for P where the pop'n is at equilibrium.

2.) Use Euler's Method to approximate the time it would take for an initial pop'n of 58,000 to grow to 175,000.

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So, for #1 we just set dP/dt = 0, and solve for P

$0= 0.3\left(1-\frac{P}{200}\right)\left(\frac{P}{50}-1\right)P$

P = 0 or P = 50 or P = 200.

Not sure for #2!

**CaptainBlack** · October 13th 2007, 01:36 PM

Originally Posted by fifthrapiers

Given that the population of a certain fish species increases according to the model below:

$\frac{dP}{dt}=0.3\left(1-\frac{P}{200}\right)\left(\frac{P}{50}-1\right)P$

1.) Determine the values for P where the pop'n is at equilibrium.

2.) Use Euler's Method to approximate the time it would take for an initial pop'n of 58,000 to grow to 175,000.

------

So, for #1 we just set dP/dt = 0, and solve for P

$0= 0.3\left(1-\frac{P}{200}\right)\left(\frac{P}{50}-1\right)P$

P = 0 or P = 50 or P = 200.

Not sure for #2!

You start with $P(0)=58000$ , then use the stepping formula:

$<br /> P(t+\delta t) = P(t)+\delta t P'(t)=P(t)+\delta t \left[ 0.3\left(1-\frac{P(t)}{200}\right)\left(\frac{P(t)}{50}-1\right)P(t) \right]$

to step forward untill $P\approx175000$ , best done in a spreadsheet.

RonL

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