Numerical Study on a Coupled-Logistic Map as a Simple Model for a Predator-Prey System

Dynamics of an iterated map of an interval, x n +1 = a x n (1- x n )+ c x n y n , y n +1 = b y n (1- y n )- d y n x n is studied numerically. Such a coupled-logistic map serves as a simple model for a predator-prey system in ecology. A generated sequence { X n } (where X n =( x n , y n )) is attracted to a periodic, or a torus, or a chaotic attractor depending on parameters a , b , c and d . A phase diagram which classifies the attractors in the a - b plane is investigated varying the coupling strength c (where c = d ). It is found that the global structure of the phase diagram is rather simple, whereas the detailed structure of the phase diagram exhibits remarkable self-similarity. Properties of the attractors are empirically studied on the Fourier and fractal-dimension analysis. Initial value dependence of the attractors is also discussed.