Differential Equation of a Population Model

Oct 2012
12
0
Cambridge Ontario
"An example of a differential equation that serves as a slightly more realistic model
of population growth than the exponential growth model is the Lotka-Volterra DE,
dy/dt= ky − ay^2, (1)
Once again, y(t) denotes the population of a species at time t. The term −ay^2, which decreases the rate of growth, represents
competition between population members for resources. In the special case a = 0, there is no competition, resulting in exponential growth.

(a) Show that the constant population y(t) = k/a is a solution to Eq. (1).
(b) Show that the following function, y(t) = y0/[(1 − by0)e^(−kt) + by0], t >= 0, (2)
where b = a/k, is a solution to DE in (1). Also show that it satisfies the
(positive) initial condition y(0) = y0 > 0"

I've been working on this question for what seems like an eternity and I just can't fully understand how to prove. I need a hint, or something, please someone give me a push in the right direction.
 
Last edited:
Jan 2008
484
109
UK
To show that a function satisfies a differential equation you may simply substitute in the function and any necessary derivatives.
 
Oct 2012
12
0
Cambridge Ontario
Yes, but it has never specified whether a is a constant or not - so i don't know what the derivative of y(t)=k/a is!
 
Jan 2008
484
109
UK
a is a constant.

If it doesn't say so in your notes or book then check another source.

By the way, writing e-kt instead of \(\displaystyle e^{-kt}\) is not a good idea. You could at least write e^(-kt) but better still, use LaTeX. LaTeX Help
 
Oct 2012
12
0
Cambridge Ontario
i didnt even notice that, thank u - but at this point im really looking for steps. Ive exhausted my knowledge and my text book and sources online I am in need of a walkthorugh of this question. Would be greatly appreciated
 
Oct 2012
12
0
Cambridge Ontario
i believe i got a) down, but b) is troublemaker.

i found dy/dt of y(t) to be ((e^(-kt))(1-by0))/[((e^(-kt))-by0(e^(-kt))+(by0))]^2

and honestly everything I've tried to work it out has been a failure and it just doesnt work out.

The closest method i have had was setting the derivative i found above equal to the derivative given, with the y values substituted with y(t)... which makes it massive and bulky. But, essentially, with a bit of algebra, ive found ((e^(-kt))(1-by0)) = ((e^(-kt))(ky0-a(y0^2))) but i feel like that is wrong
 
Jan 2008
484
109
UK
Your dy/dt is wrong. It's also almost unreadable. Use the quotient rule.