A bifurcation theorem for Darwinian matrix models and an application to the evolution of reproductive life-history strategies

Abstract

We prove bifurcation theorems for evolutionary game theoretic (Darwinian dynamic) versions of nonlinear matrix equations for structured population dynamics. These theorems generalize existing theorems concerning the bifurcation and stability of equilibrium solutions when an extinction equilibrium destabilizes by allowing for the general appearance of the bifurcation parameter. We apply the theorems to a Darwinian model designed to investigate the evolutionary selection of reproductive strategies that involve either low or high post-reproductive survival (semelparity or iteroparity). The model incorporates the phenotypic trait dependence of two features: population density effects on fertility and a trade-off between inherent fertility and post-reproductive survival. Our analysis of the model determines conditions under which evolution selects for low or for high reproductive survival. In some cases (notably an Allee component effect on newborn survival), the model predicts multiple attractor scenarios in which low or high reproductive survival is initial condition dependent.

Keywords:

• Population dynamics
• evolutionary dynamics
• bifurcation
• equilibrium
• stability
• semelparity
• evolutionary stable strategies

AMS Subject Classifications:

• 92D15
• 39A11
• 37G35

Acknowledgments

The author thanks an anonymous reviewer for a very careful reading of the paper and for numerous suggestions that clarified several issues.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

Author supported by National Science Foundation grant DMS-1407564.