Solve Problem 5: Investigate Harvesting Model (a=5, b=1, h=25/4)

In summary, In problem 5, investigate the harvesting model qualitatively and analytically. Determine whether the population becomes extinct in finite time. If so, find that time.
  • #1
Synergyx 26
4
0
Here is the question:

Investigate the harvesting model in problem 5 both qualitatively and analytically in the case a=5, b=1, and h=25/4. Determine whether the population becomes extinct in finite time. If so, find that time.

The information from problem 5 is:

dP/dt= P(a-bP)-h , P(0)=P0


Attempt:
I subbed in the values for a,b, and h into the problem. Used variable separable and integrated both sides. My solution to the integration came out to be:

-2/(2P-5) = -t + C

I have absolutely no clue where to go from there. My roommate who took the class last semester couldn't figure out what to do either. Any help would be much appreciated.
 
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  • #2
As a next step, I would:
  • Solve for P in terms of t
  • Take the derivitive of P, and verify that it is a solution to the differential equation
 
  • #3
by solving for P, I get: P= 1/t + 5/2. With what respect am I taking the derivative because if you do P you will get 1. Otherwise dP/dt= -1/t2. I'm still stuck either way because our professor doesn't explain topics very well. He just writes things out and doesn't actually say what he did so following it isn't easy.
 
Last edited:
  • #4
Synergyx 26 said:
by solving for P, I get: P= 1/t + 5/2.
Hmm, what happened to the C you had before? That should be in this equation somewhere. (Getting P(0)=P0=∞ at t=0 is a sign that something is wrong.)
With what respect am I taking the derivative because if you do P you will get 1. Otherwise dP/dt= -1/t2.
The differential equation you had was "dP/dt=...". So you need dP/dt in order to verify it.
After you account for the C term from before, compare dP/dt with the RHS of the differential equation you had.
I'm still stuck either way because our professor doesn't explain topics very well. He just writes things out and doesn't actually say what he did so following it isn't easy.
 
  • #5
I guess I forgot to write the C in. but I've come up with something else but I don't know if it's correct.

Taking P= 5/2 + 1/(C+t) I used P(0)=P0 = 1/C +5/2. then solved for C and got.
C=1/(P0-(5/2))
Then: used P(t1)=0 and plugged in my C value into the original P= 5/2 + 1/(C+t) getting:
5/2 +1/(1/P0-(5/2))+t1)=0. I then solved for t getting
t1= (2/5)(-5/(2P0+5)-2/5.

My next thought I had was to do: lim (t->infinity)P(t)= 5/2 + 1/(1/P0-(5/2))+t1). When I take the limit my answer for P(t)= 5/2. I am not sure what that answer actually gives me.
 
  • #6
Guys, I'm confused:

[tex]\frac{dP}{dt} = aP - bP^2 - h[/tex]

To me, this equation looks non-linear and non-separable. How did you solve it by straight integration?
 
  • #7
Synergyx 26 said:
I guess I forgot to write the C in. but I've come up with something else but I don't know if it's correct.

Taking P= 5/2 + 1/(C+t) I used P(0)=P0 = 1/C +5/2. then solved for C and got.
C=1/(P0-(5/2))
Then: used P(t1)=0 and plugged in my C value into the original P= 5/2 + 1/(C+t) getting:
5/2 +1/(1/P0-(5/2))+t1)=0. I then solved for t getting
t1= (2/5)(-5/(2P0+5)-2/5.
Looks good. For a question like this, we're concerned only concerned with positive time ... i.e. does P ever become zero, after starting out at some initial value P0?

My next thought I had was to do: lim (t->infinity)P(t)= 5/2 + 1/(1/P0-(5/2))+t1). When I take the limit my answer for P(t)= 5/2. I am not sure what that answer actually gives me.
It tells you the solution to the equation at infinite times, but does not tell you what you need: does the solution ever become zero at any time?

cepheid said:
Guys, I'm confused:

[tex]\frac{dP}{dt} = aP - bP^2 - h[/tex]

To me, this equation looks non-linear and non-separable. How did you solve it by straight integration?
But it is separable. Time appears only in the dt term, so it's trivial to separate.
 

1. How do I determine the optimal value for a in the harvesting model?

The optimal value for a in the harvesting model can be determined by finding the value that maximizes the sustainable yield of the resource. This can be done using mathematical equations and techniques such as optimization algorithms.

2. What is the significance of b in the harvesting model?

The parameter b in the harvesting model represents the natural growth rate of the resource. It is used to calculate the growth of the resource in the model and affects the sustainability of the harvest.

3. How does changing the value of h impact the harvesting model?

The parameter h in the harvesting model represents the cost of harvesting the resource. Changing the value of h will affect the sustainability of the resource and may impact the optimal value for a and the maximum sustainable yield.

4. Is the harvesting model a accurate representation of real-world harvesting?

The harvesting model is a simplified mathematical model and may not fully capture all the complexities of real-world harvesting. However, it can provide valuable insights and predictions about the sustainability of a resource.

5. Can the harvesting model be applied to different types of resources?

Yes, the harvesting model can be applied to a variety of natural resources such as fish, timber, and wildlife. However, the specific parameters and equations used may vary depending on the characteristics of the resource being studied.

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