Examine the nonlinear model:
$$\triangle x_t = rx_t(1-\frac{x_t}{K})-sx_ty_t$$ $$\triangle y_t = -dy_t+\epsilon x_ty_t$$
Find the equilibrium and their stability. Here all the parameters are positive.
The first step is to find the stability, and then set up a jacobian matrix to identify if these are stable or unstable.
Step one: find the equilibrium: so we do this by setting both equations equal to zero.
$$0 = rx(1-\frac{x}{k})-sxy$$ $$0 =x(r(1-\frac{x}{k})-sy)$$ This is the first equation setting equal to zero The next equation setting equal to zero is $$0=y(-d+\epsilon x)$$ By inspection, one equilibrium is at $x_t=y_t=0$
Now there are two other equilibrium. from $0=y(-d+ \epsilon x)$ we get two options to plug into the first equation. We can either plug in zero for $y$ or we can plug in $-d+ \epsilon x$ Doing this we get $$x(r(1-\frac{x}{k})-s(0) \rightarrow x(r-\frac{x}{k})=0$$ The solution we get is $$x=k$$ and this is our second equilibrium.
The third equilibrium we get is when we have $y=-d+ \epsilon x$ plugging this back into the $x$ equation, we get $$x(r(1-\frac{x}{k})-S(-d + \epsilon x)$$
Now we have to solve for when this equation equals zero. This is where I am stuck, I know it is just algebra.
