can you show how these equations of logistic curve method are derived?

Question

In population forecasting, logistic curve method is used in long term estimates. Can you show me how the formula for saturated population, the constants a and b are derived?

Answer 1

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Answer 2

Alex,

 

Since you have not specified what a and b represent, I will have to make a few assumptions. First, I do not know if you know about differential equations yet, but I will assume you have been exposed to them (you will be eventually, anyway).

 

The logistic differential equation has the following properties:

 

1) There is a population size limit, called the carrying capacity, which I will denote by M.

2) When the time variable, t, is small, the population behaves like exponential growth; that is, its rate of change is proportional to its size when t is small.

3) As t → ∞, the derivative approaches zero.

 

The simplest differential equation under these assumptions is:

 

dP/dt = kP(1 - P/M), where k and M are constants.

 

Solving this is quite involved, but the solution is:

 

P(t) = M/(1 + Ae^−kt), where A results from the integration constant.

 

Now, I ask again, what are a and b? When you submit a response, I will post another reply.