Complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species.

A symmetrical cubic discrete coupled logistic equation is proposed to model the symbiotic interaction of two isolated species. The coupling depends on the population size of both species and on a positive constant lambda, called the mutual benefit. Different dynamical regimes are obtained when the mutual benefit is modified. For small lambda, the species become extinct. For increasing lambda, the system stabilizes in a synchronized state or oscillates in a two-periodic orbit. For the greatest permitted values of lambda, the dynamics evolves into a quasiperiodic, into a chaotic scenario, or into extinction. The basins for these regimes are visualized as colored figures on the plane. These patterns suffer different changes as consequence of basins' bifurcations. The use of the critical curves allows us to determine the influence of the zones with different numbers of first-rank preimages in those bifurcation mechanisms.

[1]  Y. Pomeau,et al.  Intermittent transition to turbulence in dissipative dynamical systems , 1980 .

[2]  FRACTAL AGGREGATION OF BASIN ISLANDS IN TWO-DIMENSIONAL QUADRATIC NONINVERTIBLE MAPS , 1995 .

[3]  Van Buskirk R,et al.  Observation of chaotic dynamics of coupled nonlinear oscillators. , 1985, Physical review. A, General physics.

[4]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[5]  Carroll,et al.  Desynchronization by periodic orbits. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  Laura Gardini,et al.  A DOUBLE LOGISTIC MAP , 1994 .

[7]  J. Eckmann.  Roads to turbulence in dissipative dynamical systems , 1981 .

[8]  Christian Mira,et al.  Basin bifurcations of two-dimensional noninvertible maps : Fractalization of basins , 1994 .

[9]  C. Mira,et al.  Chaotic Dynamics: From the One-Dimensional Endomorphism to the Two-Dimensional Diffeomorphism , 1987 .

[10]  Christian Mira,et al.  On Some Properties of Invariant Sets of Two-Dimensional Noninvertible Maps , 1997 .

[11]  M. Feigenbaum.  Quantitative universality for a class of nonlinear transformations , 1978 .

[12]  Christian Mira,et al.  Chaotic Dynamics in Two-Dimensional Noninvertible Maps , 1996 .

[13]  B. Kendall,et al.  Spatial structure, environmental heterogeneity, and population dynamics: analysis of the coupled logistic map. , 1998, Theoretical population biology.

[14]  金子 邦彦.  Collapse of tori and genesis of chaos in nonlinear nonequilibrium systems , 1984 .

[15]  J. Eckmann,et al.  Iterated maps on the interval as dynamical systems , 1980 .

[16]  Ricardo López-Ruiz,et al.  Dynamics of maps with a global multiplicative coupling , 1991 .

[17]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[18]  P. Verhulst.  Recherches mathématiques sur la loi d’accroissement de la population , 1845, Nouveaux mémoires de l'Académie royale des sciences et belles-lettres de Bruxelles.

[19]  B. A. Huberman,et al.  Generic behavior of coupled oscillators , 1984 .

[20]  Wright,et al.  Symmetric and nonsymmetric coupled logistic maps. , 1987, Physical review. A, General physics.

[21]  Kapral.  Pattern formation in two-dimensional arrays of coupled, discrete-time oscillators. , 1985, Physical review. A, General physics.

[22]  A. Lichtenberg,et al.  NONLINEAR DYNAMICS OF SELF-SYNCHRONIZING SYSTEMS , 1991 .

[23]  Spiegel,et al.  On-off intermittency: A mechanism for bursting. , 1993, Physical review letters.

[24]  Ricardo López-Ruiz,et al.  Complex Patterns on the Plane: Different Types of Basin Fractalization in a Two-Dimensional Mapping , 2003, Int. J. Bifurc. Chaos.

[25]  Ricardo López-Ruiz,et al.  Basin Bifurcations in a Two-Dimensional Logistic Map , 2003, nlin/0304059.

[26]  A. Hübler,et al.  Adaptation to the edge of chaos in the self-adjusting logistic map. , 2000, Physical review letters.