Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species - Ricardo López-Ruiz; Danièle Fournier-Prunaret (Mathematical Biosciences and Engineering)
A symmetrical cubic discrete coupled logistic equation is proposed to modelthe symbiotic interaction of two isolated species. The coupling depends on the population sizeof both species and on a positive constant $λ$, called the mutual benefit.Different dynamical regimes are obtained when the mutual benefit is modified.For small $λ$, the species become extinct. For increasing $λ$,the system stabilizes in a synchronized state or oscillates in a two-periodic orbit.For the greatest permitted values of $λ$, the dynamics evolves into a quasiperiodic,into a chaotic scenario, or into extinction. The basins for these regimes are visualizedas colored figures on the plane. These patterns suffer different changes as consequenceof basins' bifurcations. The use of the critical curves allows us to determine the influenceof the zones with different numbers of first-rank preimages in those bifurcation mechanisms.