
PredatorPrey Model
From the following equations, set $\displaystyle H1$ at a fixed positive value. Show that as $\displaystyle H2 > c^{}$, the equilibrium point inside the population quadrant approaches the point $\displaystyle (0, \frac{a+H1}{b})$ on the yaxis and that if
H2 = c, all points on the yaxis are equilibrium points of system (9).
$\displaystyle x' = (a  H1 + by)x$
$\displaystyle y' = (c  H2  kx)y$
I decided to let H1 = 1.
It would seem that as $\displaystyle H2 > c^{}$ , the equation would reduce to $\displaystyle kxy$.
I don't know how to solve for the equilibrium point, however.
The only mention in my book I find is the following excerpt:
"If the harvest coefficients in the above 2 equations are too large, the internal equilibrium point $\displaystyle \frac{c  H2}{k}, \frac{a+H1}{b}$ crosses the positive yaxis, and one (or both) species becomes extinct, as we see in the next example."
Where should I start in proving the equilibrium point?
Thanks!

The equilibrium points occur when $\displaystyle \displaystyle \frac{dy}{dt}=0, \text{ and } \frac{dx}{dt}=0$
If you set
$\displaystyle \frac{dx}{dt}=(aH_1+by)x=0$ and
$\displaystyle \frac{dy}{dt}=kyx=0$
Since the population of both is positive we reject $\displaystyle (x,y)=(0,0)$
And solve
$\displaystyle aH_1+by=0$ and
$\displaystyle kyx=0$
This gives the desired conclusion.