We mainly discuss the limit set and periodic solutions of the n-variablc Volterra prey-predator systems where e

i

and p

ij

are real numbers. Under the assumption that P=(P

ij

) is stably admissible

we reduce the problem of the limit set and periodic solutions of （·） to that of the corresponding equations with lower dimensions. We conclude that for certain kinds of systems

if the positive critical point is not globally asymptotically stable

then （*） has nonconstant periodic solution.