Two-Sex Models: Chaos, Extinction, and Other Dynamic Consequences of Sex

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Abstract
Most demographic models consider only one sex, usually the female. The widespread occurrence of sexual dimorphism in life history traits and the occurrence of skewed and fluctuating sex ratios suggest that one-sex models or those dominated by one sex may often be less appropriate than two-sex models. Reproduction in two-sex models is a frequency-dependent nonlinear function (the birth or marriage function) of the relative abundance of males and females. In this paper, we examine the population dynamics resulting from three different two-sex, discrete-time, population-projection models. For a large class of birth functions, models without inter-stage mate competition are shown to converge to a locally stable adult sex ratio. Formulas for the stable population structure, stable sex ratio, and reproductive value at equilibrium are derived. When individuals of different stages compete for mates, the equilibrium population structure may become unstable. A sequence of bifurcations then occurs, leading to periodic oscillations, quasi-periodic fluctuations, and chaos as the intensity of competition increases. Finally, when per capita fecundity is a sigmoid function of the relative abundance of the other sex, perturbations of the sex ratio may lead to extinction.