# Ratio dependent predator prey model

The system is

where a, K, c, m, f, d are positive constants that stand for prey intrinsic growth rate, carrying capacity, capturing rate, half capturing saturation constant, conversion rate, predator death rate, respectively. $x$ are prey and $y$ are predator population.
This system has there steady states $(0,0), (K,0)$ and a positive steady state $E^*=(x^*,y^*)$ there are few things that I don't understand in this article.

1. In the discussion section it is mentioned that "Even when there is no positive steady state, both prey and predator can become extinct. Such extinction occurs in two cases. In one case, both species become extinct regardless of the initial data. In the other case, both species will die out only if initial prey/predator ratio is too low."
I want to know if the case which says

"both species become extinct regardless of the initial data" happen because of (0,0) is a globally asymptotically stable?

Also, how is it concluded that "In the other case, both species will die out only if initial prey/predator ratio is too low". Is it through the theorem in the article which says "if $f$ is closer to $d$ then $E^*$ is locally asymptomatically stable."
I thought of this because f represents the conversion rate (rate in which predators are born) and d is the death of predators. However, $E^*$ is positive steady state. So it can't be that the populations go extinct here. And I don't understand how it relates to 'initial data'. I don't understand based on what result they have come up with this conclusion.

1. Although in this article it talks about solutions, I tried solving this in Mathematica. But it doesn't find a solution even when I let it run for a long time. Isn't it possible to find solutions to this system? is it because I have to specify $F(0,0)=0$ and $G(0,0)=0$.
This is the Mathematica code.

DSolve[{x'[t] == ax[t](1 - (x[t]/k)) - c*x[t]y[t]/(my[t] + x[t]), y'[t] == -dy[t] + fx[t]y[t]/(my[t] + x[t])}, {x[t], y[t]}, t]