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Individual-based simulations of a simple prey-predator system of Lotka-Volterra type were carried out on a tessellation of identical squares with discrete time steps. The particles representing individuals moved freely along (roughly) straight lines with constant (on the average) velocity, and changed their movement during a collision with another particle. ; Individuals were of two types: preys (with free exponential population growth) and predators (with exponential population decrease in the absence of a prey, they attack with probability one and are characterized by zero handling and gestation times). Therefore the system might be also interpreted as a chemical reaction in a gas. For this simple system, a spontaneous generation of complex spatio-temporal pattern was observed with wavy spatial patterns and tendency for preys to form clusters surrounded by predators if the population density was high. ; The oscillations of the system were investigated at different spatial scales, and the phase lag between the oscillations in different local observation windows was demonstrated. The parameters of the classical Lotka-Volterra equations were estimated and the impact of the migration and the oscillation phase drift on the parameter values was discussed.