I attempted all the parts.
For part (a), $E_1 = P + R + S$, so $ \dot E_1 = \dot P + \dot R + \dot S$. Plugging these values and rearranging the terms would get the desired expression. From any 3D plotter, we can see that $P+R+S=1$ would give us a plane.
For part (b), when $P, R, S <1$ and $P+R+S=1$, then this is also a plane but this time it is closed in the positive octant. So, the shape is a triangular shape closed plane. If we draw the phase plane, it's quite obvious that it is invariant. But I don't see how to prove that mathematically?
For part (c), (e), (f) and (g), from the following linked picture, the desired result can be obtained. The link: https://i.stack.imgur.com/L8fAJ.jpg.
For part (d), the proof is similar to part(a). We need to plug in and rearrange and assume that $P+R+S=(P+R+S)^2=1$.
My question is whether there is a detailed method to prove the same result without the help of that picture. Can't we have a purely mathematical treatment of it?