I am reading a paper and I am stuck understanding some argument. I have the following system of equations (different that the paper one). all parameters are assumed to be positive.
$$x' = x(1-x) - \frac{cxy}{A+x},$$ $$y' = By(1-\frac{\delta y}{ax+b}).$$
I want to prove that $\lim_{t\rightarrow \infty} \sup x(t) \leq 1$. I have that $$x' \leq x(1-x), $$ but i don't know how using this fact, we obtain $\lim_{t\rightarrow \infty} \sup x(t) \leq 1$. Moreover, they show that there exists $T>0$ such that $x(t) < M,$ for $t > T,$ where $M>1$. Now the equation that the paper has is the following: $$y' = \delta y(\beta -\frac{y}{x}),$$ Again they claim that for $t>T$, we have: $$y' \leq \delta y(\beta -\frac{y}{M}),$$ so that $\lim_{t\rightarrow \infty} \sup y(t) \leq M\beta.$ It is valid to say in my case that for $t>T,$ we have: $$y' \leq By(1-\frac{\delta y}{aM+b}),$$ and conclude that $\lim_{t\rightarrow \infty} \sup y(t) \leq \frac{aM+b}{\delta}$. I am not understanding at all how they obtain the $\lim_{t\rightarrow \infty} \sup y(t)$ and $\lim_{t\rightarrow \infty} \sup x(t).$