What are the equilibrium point of this coupled ODE?

Question

Consider $$\dot x = x(a - bx - cy)$$ $$\dot y = y(-d + ex - fy)$$

$$a,b,c,d,e >0, f \geq 0$$

Find all the equilibrium points in the set $\mathbb{R}^2_{\geq 0}$

I can find by inspection the equilibrium points

$A = (0,0)$

and

$B = (\dfrac{a}{b}, 0)$

Are there any more that can be found?

Answer 1

To solve $0 = x(a - bx - cy)$ and $0 = y(-d + ex - fy)$ for the non-zero solutions $x \ne 0, y \ne 0$ you have to solve $(a - bx - cy) = 0$ and $(-d + ex - fy)$. This is two equations in two unknowns so you can solve for $x, y$ (in fact they are the equations of two lines and you are looking for their intersection). But yes, you have to go through all the solutions systematically.