Logistic Model with Harvesting Questions

bluejay87

New member
Joined
Sep 13, 2009
Messages
8
Hey Guys,

I'm doing a question related to logistic model with harvesting.

It goes as follows:

Fishery A uses the following model for the absolute growth rate
dP/dt= r P(1-P/K)-h
where h is the constant absolute harvest rate.

(a)If r = 1.04 and K = 100 calculate the harvest rate h which gives
an equilibrium at Peq = 55.

(b)Using the above values of r, K and h show that the model can be
rewritten in the factorised form
dP/dt=r/K(P ? a)(b ? P).
and find the values of a and b.

By my calculations h = 25.74

But I'm having trouble working out how they got to the factorised form. I haven't done maths in ages so I'm a bit scratchy on factorization, which may be where my problem lies. I specifically don't understand where the h dissapeared to. What also confuses me is I dont understand how you can introduce a and b- which I thought is what you do when using the partial fraction technique and seperation of variable technique- when this problem doesnt seem to be a fraction or, in their factorised form, have the variables separated.

Sorry if that confusing, but I guess it's a reflection of my general attitude toward this question!

Would be really appreciative if anyone could point me in the right direction!

Cheers,
 
bluejay87 said:
Hey Guys,

I'm doing a question related to logistic model with harvesting.

It goes as follows:

Fishery A uses the following model for the absolute growth rate
dP/dt= r P(1-P/K)-h

dP/dt = r/K * (-P[sup:2jvilgrj]2[/sup:2jvilgrj] + KP - h\r)

Now either use "completing the square" or "quadratic formula" to factorize the expression above

for a quick refresher go to:

http://www.purplemath.com/modules/solvquad3.htm


where h is the constant absolute harvest rate.

(a)If r = 1.04 and K = 100 calculate the harvest rate h which gives
an equilibrium at Peq = 55.

(b)Using the above values of r, K and h show that the model can be
rewritten in the factorised form
dP/dt=r/K(P ? a)(b ? P).

and find the values of a and b.

By my calculations h = 25.74

But I'm having trouble working out how they got to the factorised form. I haven't done maths in ages so I'm a bit scratchy on factorization, which may be where my problem lies. I specifically don't understand where the h dissapeared to. What also confuses me is I dont understand how you can introduce a and b- which I thought is what you do when using the partial fraction technique and seperation of variable technique- when this problem doesnt seem to be a fraction or, in their factorised form, have the variables separated.

Sorry if that confusing, but I guess it's a reflection of my general attitude toward this question!

Would be really appreciative if anyone could point me in the right direction!

Cheers,
 
I'm sorry, I'm not exactly how sure how you rearranged the equation like that. Is that just by expanding out the brackets?

Also I dont see how either technique will lead you to the form they give you, with the a and the b.

Sorry I'm a bit slow with the math, but thanks for your effort!
 
bluejay87 said:
I'm sorry, I'm not exactly how sure how you rearranged the equation like that. Is that just by expanding out the brackets? Yes

Also I dont see how either technique will lead you to the form they give you, with the a and the b. - These techniques are taught in high-school algebra - you just need to brush-up. Did you visit the referenced web-site?

Sorry I'm a bit slow with the math, but thanks for your effort!
 
Hey,

I get dP/dt = r/K * (-P^2 + KP - hk\r)- which makes the k and the r cancel out to get - h as in the original form.

After this though, I dont get how to get it down to the a and b- I understand completing the square and the quadratic formula, but I dont see how it applies in this case. a and b was I thought involved with he technique of partial fractions. In this case though, I can't seem to get the equation in a form which partial fractions technique would be suitable- the complicated form of the fraction is on the top, and if I do the inverse to get it on bottom it stuff up the derivative sign. Plus i've never seen the a and b introduced in quadratic form.

Hope that makes sense.

Thanks for the help.
 
bluejay87 said:
Hey,

I get dP/dt = r/K * (-P^2 + KP - hK\r) <<< Correct

- which makes the K and the r cancel out to get - h as in the original form.

After this though, I dont get how to get it down to the a and b- I understand completing the square and the quadratic formula, Then do it ...and factorize...

Ax[sup:xjfrfeoj]2[/sup:xjfrfeoj] + Bx + C = A * (x - x[sub:xjfrfeoj]1[/sub:xjfrfeoj])(x - x[sub:xjfrfeoj]2[/sub:xjfrfeoj])

where,

x[sub:xjfrfeoj]1,2[/sub:xjfrfeoj] = [-B ± ?(B[sup:xjfrfeoj]2[/sup:xjfrfeoj] - 4AC)]/(2A)




but I dont see how it applies in this case. a and b was I thought involved with he technique of partial fractions. In this case though, I can't seem to get the equation in a form which partial fractions technique would be suitable- the complicated form of the fraction is on the top, and if I do the inverse to get it on bottom it stuff up the derivative sign. Plus i've never seen the a and b introduced in quadratic form.

Hope that makes sense.

Thanks for the help.
 
I just dont get how you can factorize that. Where does the partial fractions come in??
 
bluejay87 said:
I just dont get how you can factorize that. Where does the partial fractions come in??

Right here...

Ax[sup:2niy4yzu]2[/sup:2niy4yzu] + Bx + C = A *
(x - x[sub:2niy4yzu]1[/sub:2niy4yzu])(x - x[sub:2niy4yzu]2[/sub:2niy4yzu])

You need to get:

dP/dt = r/K *
(P ? a) * (b ? P).

here

P = x

a = x[sub:2niy4yzu]1[/sub:2niy4yzu]

b = x[sub:2niy4yzu]2[/sub:2niy4yzu]

What I don't get it is - why are you using the term "partial fractions"?
 
Well partial fractions is the the new technique we have learned in lectures. It's where you split up a complex fraction into 2 smaller components so as to allow integration. This technique is the only technique we have used which introduces a and b, so it makes sense that this question has something to do partial fractions given the a and b come up.

I just dont understand how it's possible to do it the way you are saying. I'm sorry, i'm just a bit sketchy on this stuff.
 
bluejay87 said:
Well partial fractions is the the new technique we have learned in lectures. It's where you split up a complex fraction into 2 smaller components so as to allow integration. This technique is the only technique we have used which introduces a and b, so it makes sense that this question has something to do partial fractions given the a and b come up.

I just dont understand how it's possible to do it the way you are saying. I'm sorry, i'm just a bit sketchy on this stuff.

Did you find the expressions for x[sub:135z60du]1[/sub:135z60du] and x[sub:135z60du]2[/sub:135z60du] - from the quadratic function you had derived?

If you have - tell me what are those?

If not do it....

You'll use partial fraction after this step - when you will re-write your ODE as follows:

\(\displaystyle \frac{dP}{dt} \, = \, \frac{r}{K} * (P-a) * (b-P)\)

\(\displaystyle \frac{dP}{(P-a) \cdot (b-P)} \, = \, \frac{r}{K} dt\)

Here you'll use partial fractions to break up the left-hand-side
 
Hey I am doing the same problem and I got the answers 55 and 45 for the quadratic

Now here is where I got confused
normally I would sub that into (X - a)(X - b)
But in this situation it wants it in the form (X - a)(b - X)
Is this a problem or is there something very simple I am over looking.

Thanks.

Oh never mind I think I just worked it out when I was subbing back I forgot to take into account the -1 from the A which will swap the signs around for one of the things in the brackets. Sheesh right after I posted, could have had this brain wave half an hour ago XD
 
ahh I figured out what I was doing wrong- I was trying to get the quadratic expression of the algebraic symbols- turned out alot easier to just sub in the values, get that quadratic expression and then turn the equate a and b.

now, however, I'm stuck on part c)

You are meant to to show how dP/dt= r/K(45-P)(P-55) goes to, through seperation of variables and partial fractions, ln[P-45]-ln[P-55]=kAt + C (with kA meant to symbolise K with a subscript A)

I understand how the RHS integrates to ln[P-45]-ln[P-55], but I don't get how you integrate dt*r/K to kA + C
And why does K even turn into kA after integration?

If anyone could help clear this up, would be much appreciated!

Cheers
 
bluejay87 said:
ahh I figured out what I was doing wrong- I was trying to get the quadratic expression of the algebraic symbols- turned out alot easier to just sub in the values, get that quadratic expression and then turn the equate a and b.

now, however, I'm stuck on part c)

You are meant to to show how dP/dt= r/K(45-P)(P-55) goes to, through seperation of variables and partial fractions, ln[P-45]-ln[P-55]=kAt + C (with kA meant to symbolise K with a subscript A)

I understand how the RHS integrates to ln[P-45]-ln[P-55], but I don't get how you integrate dt*r/K to kA + C <<< Are you sure ?? not kt + C
And why does K even turn into kA after integration? <<< assume r/K = k
If anyone could help clear this up, would be much appreciated!

Cheers
 
yeah sorry, it's kAt+ C

hmm yeah I kinda assumed it might be like that, but shouldnt integrating change it so that r/K is different to kAt? So what's the rule for integrating when u have two variables for r/K? Can't seem to find the rule for that
 
The intergral becomes
(r/K)t + c

you had to write iti n the form
kat + c
ka being one thing and not two things multiplied toghether
ka = r/k
 
Thanks. So the integral of r/K is just r/K, but in this example we just rename that as kA?
If so, that makes the next question ('what is the value of kA?') alot easier.

I actually more confused than I thought I was about using partial fractions and integrating the other side :

I get 1/((P-45)(55-P))*dP= a/(P-45)+ b/(55-P)

which, when put over a common denominator, amounts to 1= a(55-P)+b(P-45)

So, therefore using P = 55, 45 I get a=b=0.1

so wouldn't the integral then be 10ln[p-45]-10ln[P-55] rather than ln[P-45]-ln[P-55] (which is what it's meant to be)??

Can you please show me where I've gone wrong??

Oh actually there's a hint on the sheet which is relevant

It says the follow result may be useful

b-a/(P-a)(b-P)= 1/(P-a)-1/(P-b)

Using that the integrals work out fine. But I've got no idea where that result came from! Could you please show me how that result came about??

Many Thanks!
 
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