Two ODE problems not sure about

[clynne21]

Homework Statement

consider a lake that is stocked with walleye pike and that the pike population is governed by P'=.1P(1-P/10) where time is measured in days and P is thousands of fish. Suppose that fishing is started in this lake and that 100 fish are removed daily. modify the logistic model to account for the fishing

Homework Equations

P'=.1P(1-P/10)

The Attempt at a Solution

I am thinking it's just P'=.1P(1-P/10)-.1 but that seems too easy LOL. any thoughts?

Homework Statement

Suppose a population is growing according to the logistic eqn

dP/dt=rP(1-P/K)

Prove that the rate at which the population is increasing is at its greatest when the population is at one-half of it's carrying capacity. Hint: Consider the second derivative of P

Homework Equations

dP/dt=rP(1-P/K)

The Attempt at a Solution

Absolutely no idea where to start with this one :-(

 

[vela]

Homework Statement

consider a lake that is stocked with walleye pike and that the pike population is governed by P'=.1P(1-P/10) where time is measured in days and P is thousands of fish. Suppose that fishing is started in this lake and that 100 fish are removed daily. modify the logistic model to account for the fishing

Homework Equations

P'=.1P(1-P/10)

The Attempt at a Solution

I am thinking it's just P'=.1P(1-P/10)-.1 but that seems too easy LOL. any thoughts?

Looks good to me.

Homework Statement

Suppose a population is growing according to the logistic eqn

dP/dt=rP(1-P/K)

Prove that the rate at which the population is increasing is at its greatest when the population is at one-half of it's carrying capacity. Hint: Consider the second derivative of P

Homework Equations

dP/dt=rP(1-P/K)

The Attempt at a Solution

Absolutely no idea where to start with this one :-(

the rate at which the population is increasing = dP/dt
is at its greatest = is maximized

You should recall from calculus that when a function attains a local maximum, its derivative is equal to 0. In this problem, the function is dP/dt, and its derivative is therefore d²P/dt². You want to set that equal to 0 and solve for P.

Next, you want to find the carrying capacity of the system in terms of r and K. Do you know how to find this?

Finally, you just need to show that your first answer is twice your second answer.

 

[clynne21]

Looks good to me.

the rate at which the population is increasing = dP/dt
is at its greatest = is maximized

You should recall from calculus that when a function attains a local maximum, its derivative is equal to 0. In this problem, the function is dP/dt, and its derivative is therefore d²P/dt². You want to set that equal to 0 and solve for P.

Next, you want to find the carrying capacity of the system in terms of r and K. Do you know how to find this?

Finally, you just need to show that your first answer is twice your second answer.

Got through the first question fine and did take the second derivative of the equation, but once I do that all variables are codependent on each other so setting it equal to zero makes everything zero.

d²P/dt²= -2Pr/K

so I'm kind of at a loss of how to make that a maximum. I must be missing something. Thank you for the help! I do know how to find the carrying capacity.

 

[vela]

You calculated the second derivative incorrectly. You have to use the product rule, or you can just multiply dP/dt out first:

$$\frac{dP}{dt} = rP - \frac{r}{K} P^2$$

and then differentiate each term separately. Don't forget you're differentiating with respect to t, not P.

 

[clynne21]

and then differentiate each term separately. Don't forget you're differentiating with respect to t, not P.

That's where I messed up! Was very tired last night LOL. Thanks for straightening me out!