This is the last part of a long problem I am working on. We have come to this system of differential equations:
$\frac{dX}{dt}=2X-2XY$ and $\frac{dY}{dt}=Y-XY$,
where we are modeling the populations of two species ($X$ and $Y$) interacting in the same habitat.
The fixed points clearly are $(X,Y)=(0,0)$ and $(X,Y)=(1,1)$.
From here, we are asked to use an initial condition that is close to each fixed point, and integrate the equations to see whether the solution converges toward or diverges away from the fixed point over time.
We are specifically asked to find the stabilities of the points in the above way and to not do a linearized analysis of the equations, but I have never found stabilities through integration.
How can I proceed?
A simple Ljapunov function $x^2/2+y^2$ will prove instability for $(0,0)$.
As $(1,1)$ is hyperbolic, no ljapunov function can be found. However because of this almost all solutions will diverge to either one of the axis.
Now on solving: You could easily try numeric solving, which can be done here http://www.falstad.com/vector/ or elsewhere.
I havent figured out how to solve it analyticly yet, but it seams extraordinarily hard for a exersice if it is one.