Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species

  • Received: 01 February 2004 Accepted: 29 June 2018 Published: 01 July 2004
  • MSC : 92D25, 70K50, 37M20.

  • 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.

    Citation: Ricardo López-Ruiz, Danièle Fournier-Prunaret. Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species[J]. Mathematical Biosciences and Engineering, 2004, 1(2): 307-324. doi: 10.3934/mbe.2004.1.307

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  • 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.
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    © 2004 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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