Help with logistic-like function

[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

 

[Dick]

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

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.

 

[agentsanta]

Hey
I'll try to digest what you said
Thanks for the reply though
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

 

[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?

 

[Borek]

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.