Abstract
We derive and analyze a Darwinian dynamic model based on a general di erence equation population model under the assumption of a trade-o 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 e ects depend in a hierarchical manner on the trait.
Highlights
Life history strategies that individuals adopt play a central and important role in the dynamics of biological populations [1, 2]
The Darwinian equations (2.3) constitute a dynamic evolutionary model based on a scalar difference equation for the dynamics of a population in which a phenotypic trait, and the model coefficients, change over time due to Darwinian evolutionary principles
The key assumption with regard to the dependence of the inherent fertility and survival rates on the phenotypic trait is that they suffer a trade off, i.e. a change in an individual’s trait that results in an increase in fertility is accompanied by a decrease in survival
Summary
Life history strategies that individuals adopt play a central and important role in the dynamics of biological populations [1, 2]. In his review of the large literature on this topic, Hughes states “that mathematical models purporting to explain the general evolution of semelparous life histories from iteroparous ones (or vice versa) should not assume that organisms can only display either an annual-semelparous life history or a perennial-iteroparous one” and argues in favor of a “continuum of modes of parity” [5] In keeping with this point of view, our goal here is derive and analyze an evolutionary game theoretic model in which fertility and survival depend on a continuous trait that is subject to the Darwinian principles of evolution and to use the model to investigate the evolutionary stability or instability of semelparity and iteroparity.
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