# Help with logistic-like function

1. Nov 26, 2012

### agentsanta

Hey, I am doing a project on population growth modelling, and I have hit a brick wall in trying to derive a function to represent an aspect of my population

I understand that the derivative of a logistic function is
P'(t)=P(t)(1-P(t))

and from there one can obtain the function P(t)= 1/(1+e^(-t))

However, it seems that the logistic function is too simple for my purposes and I need a function with the following derivative

P'(t)=P(t)(1-P(t)) - P(t-a)(1-P(t-a)) where a is a constant

My problem is to find the function with the above derivative

Anyone have ideas?
I've literally sat down and stared at this thing for the last week and got nothing.

FYI I'm at a senior grade high school level

2. Nov 26, 2012

### Dick

You've got a separable differential equation. dP/dt=P(1-P). You find a general solution by writing dP/(P(1-P))=dt and integrating both sides. Do the left side by parts. You have had calculus, right? How much do you know? If not enough, keep asking questions.

3. Nov 26, 2012

### agentsanta

Hey
I'll try to digest what you said
I've only done enough calculus to barely understand what's going on
I haven't done integration by parts but I use wolframalpha for that XD

Thanks

4. Nov 26, 2012

### agentsanta

Alright I tried what you suggested
...I have no idea what that is x.X

I know how to do integration and stuff but I've never integrated dt or dP before

Any tips?
Or am I biting off more than I can chew and should simplify my model?

5. Nov 27, 2012

### Staff: Mentor

x -> dx
t -> dt
P -> dP

You can use for your variable any symbol you want. There are no separate rules for dx, dt, dP - it is exactly the same integration.