Lotka-Volterra First Integral and Fixed Point I have the following problem that I am dealing with, quite a long time, I must say. Let us assume that we have a predator-prey, Lotka-Volterra system given to us by:
\begin{align} 
& \frac{dx}{dt}={\alpha}x-{\beta}xy \\
& \frac{dy}{dt}=-{\delta}y+{\gamma}xy
\end{align}
with all the parameters ${\alpha}, {\beta}, {\gamma}, {\delta}$ to be positive integers.
A First Integral of this particular system would be given by:
\begin{equation}
F(x,y)={\gamma}x-{\delta}\ln(x)+{\beta}y-{\alpha}\ln(y)
\end{equation}
Next we consider the sets defined as:
\begin{equation}
{\Sigma}_c=\{(x,y)\in \mathbb{R}^2/ x>0, y>0, F(x,y)\leq c\}, \quad c\in \mathbb{R}^{+}
\end{equation}
Edit: Reversed the Inequality
and we would like to prove that those ${\Sigma}_c$ sets are closed, bounded and convex, in order to use the Kakutani-Markov Fixed Point Theorem (not really sure if that is the theorem I must use, since the professor is not giving that away) and prove that, at the stable equilibrium of this particular Lotka Volterra the function $F(x,y)$ acquires its maximum value, that is $K_0=\max {F(x,y)}=F(\bar{x},\bar{y})$, where $(\bar{x},\bar{y})$ denotes the stable equilibrium. 
$$$$Any assistance would be much appreciated! Thank you all for your time!

I don't see why you need to use a fixed point theorem.  The function $F$ is strictly convex.  At the equilibrium point it has a minimum, not a maximum.
