Hi, can anyone help me solve a differential equation for the logistic growth model?
M'(t) = M(S-M) + I, where M(t) represents the growth of a biomass. "I" represents immigration (in a coral reef) and there is no breeding.
The Attempt at a Solution
I've treated it as a separable differential equation, but get the term
(integral of) 1/(M(S-M)+I) dM
A solution is possible using wolfram mathematica, but it doesn't isolate M.
I'd solve the equation using partial fractions, but the "I" term seems to make it impossible. Am I right in saying that?
Also, going back a step, can anyone explain how to use partial fractions to derive the general logistic equation. Wikipedia has a section on it: http://en.wikipedia.org/wiki/Partial_fraction
but there's a step I don't understand, namely: A=B, A=1/M, B=1/M. On what basis was this assumed?
In trying to solve it myself, I let A=0 and got B= 1/P, then let B=0 and got A=1/(M-P).
When these figures for A and B are substituted back into the original equation, I get 2/(P(M-P)) .
Can anyone explain what I'm doing wrong?