  We consider a Rosenzweig-MacArthur predator-prey system which incorporates
logistic growth of the prey in the absence of predators and a Holling type II
functional response for interaction between predators and preys. We assume that
parameters take values in a range which guarantees that all solutions tend to a
unique limit cycle and prove estimates for the maximal and minimal predator and
prey population densities of this cycle. Our estimates are simple functions of
the model parameters and hold for cases when the cycle exhibits small predator
and prey abundances and large amplitudes. The proof consists of constructions
of several Lyapunov-type functions and derivation of a large number of
non-trivial estimates which are also of independent interest.
