The time-global unique solvability on the reaction--diffusion equations
for preypredator models and dormancy on predators is established. The
crucial step is to construct time-local nonnegative classical solutions
by using a new approximation associated with time-evolution operators.
Although the system does not equip usual comparison principles, a priori
bounds are derived, so solutions are extended time-globally. Via
observations to the corresponding ordinary differential equations,
invariant regions and asymptotic behaviors of solutions are also
investigated.