# Population Dynamics

• Jul 26th 2009, 07:56 PM
penney21
Population Dynamics
I'm having some trouble with a Differential Equations Population problem.

Here is the background for the question..

The DE governing a fish pop. P(t) with harvesting proportional to the population is given by:
P'(t)=(b-kP)P-hP
where b>0 is birthrate, kP is deathrate, where k>0, and h is the harvesting rate. Model assumes that the death rate per individual is proportional to the pop. size. An equilibrium point for the DE is a value of P so that P'(t)=0.

I worked out the integral to be the following:

1 / [(b-h)-kP]P dp = dt

equals:
(lnP - ln(b-h-kP)) / (b-h) + C

However, I'm having trouble answering this part of the question..
Determine h so that Y is maximized, and find this Y. This is the maximum sustainable yield.

How would I go about solving that part? Any help would be greatly appreciated.
• Jul 27th 2009, 10:27 AM
Subhotosh Khan
Quote:

Originally Posted by penney21
I'm having some trouble with a Differential Equations Population problem.

Here is the background for the question..

The DE governing a fish pop. P(t) with harvesting proportional to the population is given by:
P'(t)=(b-kP)P-hP
where b>0 is birthrate, kP is deathrate, where k>0, and h is the harvesting rate. Model assumes that the death rate per individual is proportional to the pop. size. An equilibrium point for the DE is a value of P so that P'(t)=0.

I worked out the integral to be the following:

1 / [(b-h)-kP]P dp = dt

equals:
(lnP - ln(b-h-kP)) / (b-h) + C

However, I'm having trouble answering this part of the question..
Determine h so that Y is maximized, and find this Y. This is the maximum sustainable yield.

How would I go about solving that part? Any help would be greatly appreciated.

Moderator edit: This question has been posted and answered elsewhere. Thread closed.