Discrete time darwinian dynamics and semelparity versus iteroparity

We derive and analyze a Darwinian dynamic model based on a general difference equation population model under the assumption of a trade-off 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 non-ESS semelparous) equilibria can arise outside a neighborhood of bifurcation points when density effects depend in a hierarchical manner on the trait.