I have a question with regards to finding the critical points and making sense of it in the context the prey-predator model. So I have a system given by the following:
$$x' = 10(1-\frac{x}{k})x-\frac{xy}{1+x}$$
$$y' = -y+\frac{10}{4}\frac{xy}{1+x}$$
Linearizing and setting the LHS equal to 0 results in:
$x = 0, y = 0, x = \frac{2}{3}, y = (1-\frac{x}{k})(1+x)$
Further solving, one eventually gets $ x = k $.
So the equilibrium points are (0,0), ($\frac{2}{3}$,0), (k,0) where k is some constant. In the model, it follows that $x(t)$ represents the prey population while $y(t)$ represents the predator population.
So I am asked for the equilibrium points, which I am assuming are the critical points obtained through linearizing. However, does it make sense for all of the points to be at $y = 0$? I am having a difficult time interpreting the result of a critical value and wanted to seek guidance to see if what I am doing is correct.
Any advice is appreciated.
Thank you
The fixed points for this particular model are as follows,
$(0,0),~(\frac{2}{3},\frac{150k - 100}{9k}),~(k,0)$
From this, you can easily interpret that when there are no predator or prey, the fixed point is $(0,0)$ (may be because of extinction). In addition, it can also be seen that there is a point, where the evolution of predator-prey gets constant and stays in an invariant set and attains the equilibrium (this corresponds to the $2^{nd}$ fixed point). Moreover, $3^{rd}$ fixed point depicts the unbounded growth of the prey (because of the absence of predator), which is again upper bounded by the constant $k$.