Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1;=;d_2,;{alpha}_{12};=;{alpha}_{21};=;0$, and (ii) $0;<;{alpha}_{21};<;8_{alpha}_{11},;0;<;{alpha}_{12};<;8_{alpha}_{22}$. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when $t;{ o};{infty}$ via suitable Liapunov functionals.x????