A Periodic Lotka-volterra System

In this paper periodic time-dependent Lotka-Volterra systems are considered. It is shown that such a system has positive periodic solutions. It is done without constructive conditions over the period and the parameters. 1. The Periodic Lotka-Volterra System. Consider the Predator-Prey model (see Volterra [1]) N ′ 1 = (ε1 − γ1N2)N1 N ′ 2 = (−ε2 + γ2N1)N2. (1) The functions N1 and N2 measure the sizes of the Prey and Predator populations respectively. The coefficients ε1, ε2, γ1, γ2 are assumed as nonnegative ω-periodic functions of time t. The period ω > 0 is arbitrary chosen and fixed. This periodicity assumption is natural; one may see for instance the work of J. Cushing [2] in which is given a satisfactory justification on it. We still recount (due to [2]) some periodic factors like seasonal effects of weather, food supply, mating habits, hunting or harvesting seasons, etc. Here one may add any unidirectional ω-periodic influence of another predator over the prey. We will look for ω-periodic positive solutions for the conservative system (1) that corresponds to the nature of N1 and N2. 1991 Mathematics Subject Classification: 34A25, 92B20