Research article
Special Issues
Discrete time darwinian dynamics and semelparity versus iteroparity

Department of Mathematics, Interdisciplinary Program in Applied Mathematics, University of Arizona, 617 N Santa Rita Av, Tucson, AZ 85721

Received:
16 November 2018
Accepted:
07 February 2019
Published:
06 March 2019




We derive and analyze a Darwinian dynamic model based on a general di erence equation population model under the assumption of a tradeo between fertility and survival. Both inherent and density dependent terms are functions of a phenotypic trait (subject to Darwinian evolution) and its population mean. We prove general theorems about the existence and stability of extinction equilibria and the bifurcation of positive equilibria when extinction equilibria destabilize. We apply these results, together with the Evolutionarily Stable Strategy (ESS) Maximum Principle, to the model when both semelparous and iteroparous traits are available to individuals in the population. We find that if the density terms in the population model are trait independent, then only semelparous equilibria are ESS. When density terms do depend on the trait, then in a neighborhood of a bifurcation point it is again the case that only semelparous equilibria are ESS. However, we also show by simulations that ESS iteroparous (and also nonESS semelparous) equilibria can arise outside a neighborhood of bifurcation points when density e ects depend in a hierarchical manner on the trait.
Citation: J. M. Cushing. Discrete time darwinian dynamics and semelparity versus iteroparity[J]. Mathematical Biosciences and Engineering, 2019, 16(4): 18151835. doi: 10.3934/mbe.2019088

Abstract
We derive and analyze a Darwinian dynamic model based on a general di erence equation population model under the assumption of a tradeo between fertility and survival. Both inherent and density dependent terms are functions of a phenotypic trait (subject to Darwinian evolution) and its population mean. We prove general theorems about the existence and stability of extinction equilibria and the bifurcation of positive equilibria when extinction equilibria destabilize. We apply these results, together with the Evolutionarily Stable Strategy (ESS) Maximum Principle, to the model when both semelparous and iteroparous traits are available to individuals in the population. We find that if the density terms in the population model are trait independent, then only semelparous equilibria are ESS. When density terms do depend on the trait, then in a neighborhood of a bifurcation point it is again the case that only semelparous equilibria are ESS. However, we also show by simulations that ESS iteroparous (and also nonESS semelparous) equilibria can arise outside a neighborhood of bifurcation points when density e ects depend in a hierarchical manner on the trait.
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