Mathematical Biosciences and Engineering, 2013, 10(4): 1017-1044. doi: 10.3934/mbe.2013.10.1017

Primary: 92D25, 92D15; Secondary: 37N25.

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Darwinian dynamics of a juvenile-adult model

1. Department of Mathematics, Interdisciplinary Program in Applied Mathematics, 617 N Santa Rita, University of Arizona, Tucson AZ 85721
2. Department of Ecology and Evolutionary Biology, 1041 E. Lowell St, University of Arizona, Tucson, AZ 85721

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) $.
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