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 modelthe symbiotic interaction of two isolated species. The coupling depends on the population sizeof both species and on a positive constant $\lambda$, called the mutual benefit.Different dynamical regimes are obtained when the...
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AIMS Press
2004-06-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2004.1.307 |
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| author | Ricardo López-Ruiz Danièle Fournier-Prunaret |
| author_facet | Ricardo López-Ruiz Danièle Fournier-Prunaret |
| author_sort | Ricardo López-Ruiz |
| collection | DOAJ |
| description | 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 $\lambda$, 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 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. |
| format | Article |
| id | doaj-art-4997c27b6592495dbc7fca2bef43a1bf |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2004-06-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-4997c27b6592495dbc7fca2bef43a1bf2025-01-24T01:46:53ZengAIMS PressMathematical Biosciences and Engineering1551-00182004-06-011230732410.3934/mbe.2004.1.307Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two SpeciesRicardo López-Ruiz0Danièle Fournier-Prunaret1Department of Computer Science and BIFI, Facultad de Ciencias-Edificio B, Universidad de Zaragoza - 50009 ZaragozaSystèmes Dynamiques (SYD), L.E.S.I.A., Institut National de Sciences Appliquées - 31077 Toulouse CedexA 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 $\lambda$, 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 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.https://www.aimspress.com/article/doi/10.3934/mbe.2004.1.307population dynamicscomplex patternssynchronizationbasins.coupled logistic mapssymbiotic speciescritical curvesinvariant sets |
| spellingShingle | Ricardo López-Ruiz Danièle Fournier-Prunaret Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species Mathematical Biosciences and Engineering population dynamics complex patterns synchronization basins. coupled logistic maps symbiotic species critical curves invariant sets |
| title | Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species |
| title_full | Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species |
| title_fullStr | Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species |
| title_full_unstemmed | Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species |
| title_short | Complex Behavior in a Discrete Coupled Logistic Model for the Symbiotic Interaction of Two Species |
| title_sort | complex behavior in a discrete coupled logistic model for the symbiotic interaction of two species |
| topic | population dynamics complex patterns synchronization basins. coupled logistic maps symbiotic species critical curves invariant sets |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2004.1.307 |
| work_keys_str_mv | AT ricardolopezruiz complexbehaviorinadiscretecoupledlogisticmodelforthesymbioticinteractionoftwospecies AT danielefournierprunaret complexbehaviorinadiscretecoupledlogisticmodelforthesymbioticinteractionoftwospecies |