ITERATION THEORY (ECIT ‘04) W. F¨org-Rob, L. Gardini, D. Gronau, L. Reich, J. Sm´ ıtal (Eds.) Grazer Math. Ber., ISSN 1016–7692 Bericht Nr. 350 (2006), 82 – 95 Route to chaos in three-dimensional maps of logistic type Dani` ele Fournier-Prunaret, Ricardo Lopez-Ruiz and AbdelKaddous Taha Abstract A route to chaos is studied in 3-dimensional maps of logistic type. Mech- anisms of doubling for invariant closed curves (ICC) are found for specific 3-dimensional maps. These bifurcations cannot be observed for ICC in the 2-dimensional case. When the parameter of the system is modified, local- ized oscillations occur on the ICC that give rise to weakly chaotic rings, then to chaotic attractors, which finally disappear by contact bifurcations. These maps can be considered as models for the symbiotic interaction of three species. 1 Introduction Many papers have been devoted to the study of two-dimensional coupled logistic maps ([5][6][7][2][3]). Some of them can be considered as biological models, corre- sponding to interactions between species ([5][6][7][2]). In this paper, we consider two models of the same kind in the 3-dimensional case. The first one, given by the map T 1 , corresponds to a symbiotic interaction between correlative pairs of species:    x k+1 = λ(3y k + 1)x k (1 - x k ) y k+1 = λ(3z k + 1)y k (1 - y k ) z k+1 = λ(3x k + 1)z k (1 - z k ) (1) where λ is a real positive parameter, (x, y, z ) represent the species populations. Mathematics Subject Classification 2000: Primary 39B12, Secondary 26A18, 39B52. Keywords and phrases: bifurcation, basin, logistic map, three-dimensional map, noninvertible, chaos.