[For part I see the preceding review, Zbl 0809.92017).] The stability of a positive equilibrium \(U\) \((U = (Q_ 0, P_ 0)\), \(Q_ 0 > 0\), \(P_ 0 > 0)\) of a one-dimensional reaction-diffusion system with zero-flux boundary conditions is studied under natural constraints. The diffusion coefficients \(q\) (prey diffusion rate) and \(p\) (predator diffusion rate) are nonnegative. Positive constants \(q_ 0\) and \(q_ 1\) are given so that \(U\) is asymptotically stable for \(q \geq q_ 1\), \(p>0\), but for \(q_ 1 > q \geq q_ 0\) the equilibrium \(U\) undergoes a Turing bifurcation at \(p = p_ 0 (q) \geq q\). In the latter case, a nonconstant stationary solution also exists for \(p\) near to \(p_ 0(q)\) and its stability is studied.