Integrating for logistic growth model

In summary, the conversation is about solving a differential equation for the logistic growth model, where M(t) represents the growth of a biomass and "I" represents immigration. The attempt at a solution involves using partial fractions, but there is confusion about how to derive the general logistic equation and the values of A and B. Further clarification is needed.
  • #1
Darkmisc
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Homework Statement



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

Homework Equations





It reads:

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


 
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  • #2
Darkmisc said:
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:

[tex]\frac 1 {m(s-m)} = \frac A m+ \frac B {s-m}[/tex]

Add the two fractions on the right to get:

[tex]\frac 1 {m(s-m)} = \frac {A(s-m)+Bm}{(m)(s-m)}[/tex]

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

[tex] 1 = A(s-m)+Bm[/tex]

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:

[tex]\frac 1 {m(s-m)} = \frac {\frac 1 s} m+ \frac {\frac 1 s} {s-m}[/tex]
 

1. What is the purpose of integrating for logistic growth model?

The purpose of integrating for logistic growth model is to accurately predict the growth of a population or system over time, taking into account factors such as initial population size, carrying capacity, and growth rate. It allows for a more realistic and gradual growth curve, as opposed to a linear or exponential one.

2. How is the logistic growth model different from other growth models?

The logistic growth model differs from other growth models in that it takes into account the concept of carrying capacity. This means that as the population approaches its maximum capacity, the growth rate slows down and eventually reaches a stable equilibrium. This is in contrast to exponential growth models, where the growth rate remains constant.

3. What is the mathematical equation for the logistic growth model?

The mathematical equation for the logistic growth model is: dN/dt = rN(1 - N/K), where dN/dt represents the rate of change in population size over time, r is the intrinsic growth rate, N is the current population size, and K is the carrying capacity.

4. How is the logistic growth model used in real-world applications?

The logistic growth model has many real-world applications, including predicting the growth of animal populations, forecasting the spread of diseases, and analyzing the growth of human populations. It can also be used in economics to model the growth of industries or markets.

5. What are the limitations of the logistic growth model?

While the logistic growth model is a useful tool for predicting growth in many scenarios, it does have its limitations. For example, it assumes a constant carrying capacity, which may not always be the case in real-world situations. It also does not take into account external factors such as environmental changes or competition for resources, which can affect population growth. Additionally, the model may not accurately represent sudden or unexpected changes in population size.

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