General solution(s) to Logistic model with harvesting?

[clarkie_49]

Homework Statement

Hi guys! I am trying to show the limiting population for this model:

Homework Equations

P'(t) = P(S - P) + H

The harvesting, is actually immigration, so its a positive unknown. I have shown this before (without harvesting) by solving for P(t) using separation of variables. Then taking the limit t -> infinity. However i can't work this out, without an actual value for H. Problem is there isn't one!

The Attempt at a Solution

So far i attempted to change the formula to use separation and method of partial fractions:

P'(t) = [ (S + sqrt{S^2 + 4H} - 2P)/2 ] [ (2P- S - sqrt{S^2 + 4H} )/2 ]

then (2AP - AS - Asqrt{S^2 + 4H} + BS + Bsqrt{S^2 + 4H} - 2BP)/2

is the numerator in my attempt at a partial fraction. Here's about as far is i get. Assuming S^2 + 4H is positive i can't seem to group to find a partial fraction. All i can obviously see is P(A - B) = 0

Any help would be greatly appreciated.

 

[clarkie_49]

I managed to solve it, after a few more hours :)

I just subbed in variables for the equilibria, rather than leave them in square root form. Taking the limit of t -> infinity showed population approaching the equilibrium value which made

P''(equilibrium_value) < 0