Darwinian dynamics of a juvenile-adult model
The bifurcation that occurs from the extinction equilibrium in a basicdiscrete time, nonlinear juvenile-adult model for semelparous populations,as the inherent net reproductive number $R_{0}$ increases through $1$,exhibits a dynamic dichotomy with two alternatives: an equilibrium withoverlapping gen...
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AIMS Press
2013-05-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.1017 |
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| author | J. M. Cushing Simon Maccracken Stump |
| author_facet | J. M. Cushing Simon Maccracken Stump |
| author_sort | J. M. Cushing |
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| description | The bifurcation that occurs from the extinction equilibrium in a basicdiscrete time, nonlinear juvenile-adult model for semelparous populations,as the inherent net reproductive number $R_{0}$ increases through $1$,exhibits a dynamic dichotomy with two alternatives: an equilibrium withoverlapping generations and a synchronous 2-cycle with non-overlappinggenerations. Which of the two alternatives is stable depends on theintensity of competition between juveniles and adults and on the directionof bifurcation. We study this dynamic dichotomy in an evolutionary settingby assuming adult fertility and juvenile survival are functions of aphenotypic trait $u$ subject to Darwinian evolution. Extinction equilibriafor the Darwinian model exist only at traits $u^{\ast }$ that are criticalpoints of $R_{0}\left( u\right) $. We establish the simultaneous bifurcationof positive equilibria and synchronous 2-cycles as the value of $R_{0}\left(u^{\ast }\right) $ increases through $1$ and describe how the stability ofthese dynamics depend on the direction of bifurcation, the intensity ofbetween-class competition, and the extremal properties of $R_{0}\left(u\right) $ at $u^{\ast }$. These results can be equivalently stated in termsof the inherent population growth rate $r\left( u\right) $. |
| format | Article |
| id | doaj-art-85e5ed94781643daada61d69a86d2eaf |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2013-05-01 |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-85e5ed94781643daada61d69a86d2eaf2025-01-24T02:26:20ZengAIMS PressMathematical Biosciences and Engineering1551-00182013-05-011041017104410.3934/mbe.2013.10.1017Darwinian dynamics of a juvenile-adult modelJ. M. Cushing0Simon Maccracken Stump1Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721The bifurcation that occurs from the extinction equilibrium in a basicdiscrete time, nonlinear juvenile-adult model for semelparous populations,as the inherent net reproductive number $R_{0}$ increases through $1$,exhibits a dynamic dichotomy with two alternatives: an equilibrium withoverlapping generations and a synchronous 2-cycle with non-overlappinggenerations. Which of the two alternatives is stable depends on theintensity of competition between juveniles and adults and on the directionof bifurcation. We study this dynamic dichotomy in an evolutionary settingby assuming adult fertility and juvenile survival are functions of aphenotypic trait $u$ subject to Darwinian evolution. Extinction equilibriafor the Darwinian model exist only at traits $u^{\ast }$ that are criticalpoints of $R_{0}\left( u\right) $. We establish the simultaneous bifurcationof positive equilibria and synchronous 2-cycles as the value of $R_{0}\left(u^{\ast }\right) $ increases through $1$ and describe how the stability ofthese dynamics depend on the direction of bifurcation, the intensity ofbetween-class competition, and the extremal properties of $R_{0}\left(u\right) $ at $u^{\ast }$. These results can be equivalently stated in termsof the inherent population growth rate $r\left( u\right) $.https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.1017equilibriumstructured population dynamicsevolutionary game theory.bifurcationdynamicdichotomysemelparitydarwinian dynamicssynchronous cyclesjuvenile-adult populationmodel |
| spellingShingle | J. M. Cushing Simon Maccracken Stump Darwinian dynamics of a juvenile-adult model Mathematical Biosciences and Engineering equilibrium structured population dynamics evolutionary game theory. bifurcation dynamicdichotomy semelparity darwinian dynamics synchronous cycles juvenile-adult populationmodel |
| title | Darwinian dynamics of a juvenile-adult model |
| title_full | Darwinian dynamics of a juvenile-adult model |
| title_fullStr | Darwinian dynamics of a juvenile-adult model |
| title_full_unstemmed | Darwinian dynamics of a juvenile-adult model |
| title_short | Darwinian dynamics of a juvenile-adult model |
| title_sort | darwinian dynamics of a juvenile adult model |
| topic | equilibrium structured population dynamics evolutionary game theory. bifurcation dynamicdichotomy semelparity darwinian dynamics synchronous cycles juvenile-adult populationmodel |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2013.10.1017 |
| work_keys_str_mv | AT jmcushing darwiniandynamicsofajuvenileadultmodel AT simonmaccrackenstump darwiniandynamicsofajuvenileadultmodel |