# Integrating for logistic growth model

## Homework Statement

Hi, can anyone help me solve a differential equation for the logistic growth model?

## Homework Equations

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?

Thanks,

Darkmisc

## The Attempt at a Solution

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LCKurtz
Homework Helper
Gold Member
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?
Say you are trying to set up this partial fraction expansion and figure out A and B:

$$\frac 1 {m(s-m)} = \frac A m+ \frac B {s-m}$$

Add the two fractions on the right to get:

$$\frac 1 {m(s-m)} = \frac {A(s-m)+Bm}{(m)(s-m)}$$

For these to be equal, the numerators must be equal:

$$1 = A(s-m)+Bm$$

If you put m = 0 you get A = 1/s and if you put m =s you get B = 1/s. So your partial fraction expansion becomes:

$$\frac 1 {m(s-m)} = \frac {\frac 1 s} m+ \frac {\frac 1 s} {s-m}$$