\documentclass[tikz]{standalone}

\usetikzlibrary{decorations.pathreplacing,decorations.markings}
\usetikzlibrary{patterns}

\begin{document}

\begin{tikzpicture}

%diagonals + diagonal labels
\draw[gray,thick,dashed] (0,1) to node[midway,right,yshift=2pt,xshift=-2pt]{ $1$} (1,0);
\draw[gray,thick,dashed] (1,1) to node[midway,right,yshift=2pt,xshift=-2pt]{ $1$} (2,0);
\draw[gray,thick,dashed] (1,2) to node[midway,right,yshift=2pt,xshift=-2pt]{ $2$} (2,1);
\draw[gray,thick,dashed] (2,2) to node[midway,right,yshift=2pt,xshift=-2pt]{ $2$} (1,3);



%snake graph outline
\draw[thick, green] (0,0) to (1,0) ;
\node [below,scale=1,yshift=-2pt] at (0.5,0) {$x_2$};
\draw[thick] (1,0)to node[midway,right ,scale=1,yshift=-7pt, xshift = -2pt]{$\mu x_1$} (1,1);
\node [above,scale=1,yshift=-2pt] at (0.5,1) {$x_1$};
\draw[thick, green] (1,1) to  (0,1) ;
\draw[thick] (0,1) to node[midway,left,xshift=2pt]{$x_a$}(0,0);
\draw[thick] (1,0) to node[midway,below,yshift=2pt]{$x_1$} (2,0) ;
\draw[thick, green] (2,0) to (2,1);
\node [right ,scale=1,yshift=-2pt] at (2,0.5) {$x_2$};
\draw[thick] (1,2) to node[midway,above,yshift=-3pt]{$\lambda x_2$} (2,2) to node[midway,right,xshift=-1pt]{$x_2$} (2,1) to node[midway,above,yshift=-2pt]{$x_a$} (1,1) to node[midway,left,xshift=3pt]{$x_1$} (1,2);
\draw[thick, green] (1,2) to (1,3); \node [left,scale=1,yshift=-2pt] at (1,2.5) {$x_2$};
\draw[thick] (1,3) to node[midway,above,xshift=-1pt]{$x_1$} (2,3) to node[midway,right,yshift=-2pt]{$x_a$} (2,2);

\draw[fill=black] (0,0) circle [radius=2pt];
\draw[fill=black] (0,1) circle [radius=2pt];
\draw[fill=black] (2,2) circle [radius=2pt];
\draw[fill=black] (2,3) circle [radius=2pt];

\end{tikzpicture}



\end{document}