I know how to solve nullclines, the following link is very helpful
http://mcb.berkeley.edu/courses/mcb137/exercises/Nullclines.pdf
However I don't understand how to solve equations that have only variables assuming from the Lotka–Volterra equations.
Take as example the following equations
dx/dt=ax+axy
dy/dt=ax^2+ay
how would I find the nullclines, and the fixed points?
You can still proceed in the usual way, but your results will contain unspecified constants $ \ a \ , \ b \ , \ c \ , \ e \ , \ \text{and} \ f \ $ (only $ \ x \ $ and $ \ y \ $ are the variables):
$$ \frac{dx}{dt} \ = \ 0 \ = \ x \ (a - by - ex) \ \ , \ \ \frac{dy}{dt} \ = \ 0 \ = \ y \ (cx - f) \ . $$
So the nullclines are the $ \ x-$ and $ \ y-$ axes (the lines $ \ y = 0 \ $ and $ \ x = 0 \ $ ), the vertical line $ \ x = \frac{f}{c} \ , $ and the line $ \ ex + by = a \ . $ The fixed or equilibrium points are the intersection points of these various lines, where both derivatives are zero. EDIT: There are three points for which that is the case (correcting an oversight I made -- thank you, Amzoti, for alerting me to this gaffe). So some of the coordinates of the points are going to be expressions involving those unspecified constants, rather than particular numbers.
EDIT (3/27) -- Here are a couple of possible arrangements of the nullclines and fixed points (it is assumed here that the quantities represented require that $ \ x \ge 0 \ $ and $ \ y \ge 0 \ $ .
For $ \ a, b, c, e, f \ > \ 0 \ , $ with $ \ \frac{a}{e} > \frac{f}{c} > 1 \ , $ we have (You are not yet asked to work out the "flow" in each region, but that will come up soon...)
By contrast, another possibility is to have $ \ a, c, e, f \ > \ 0 \ \ \text{and} \ \ b < 0 \ $ with $ \ \frac{f}{c} > \frac{a}{e} > 1 \ ; $ here, we have four fixed points, since the sloping nullcline does not intersect $ \ x = 0 \ $ for $ \ y \ge 0 \ . $