Undamped oscillations in prey–predator models on a finite size lattice
Sustained oscillation is frequently observed in population dynamics of biospecies. The oscillation comes not only from deterministic but also from stochastic characteristics. In the present article, we deal with a finite size lattice which contains prey and predator. The interaction between a pair of lattice points is carried out by two different methods; local and global interactions. In the former, interaction occurs between adjacent sites, while in the latter interaction takes place between any pair of lattice sites. It is found that both systems exhibit undamped oscillations. The amplitude of oscillation decreases with the increase of the total lattice sites. In the case of global interaction, we can present a stochastic differential equation which is composed of two factors, i.e., the Lotka–Volterra equation with density dependence and noise term. The quantitative agreement between theory and simulation results of global interaction is almost perfect. The stochastic theory qualitatively expresses characteristics of sustainable oscillation for local interaction.