Estimates of size of cycle in a predator-prey system

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