Abstract
This paper studies the properties of the sex ratio in two-period models of threshold (e.g., polygenic or temperature-dependent) sex determination under heavy-tailedness in the framework of possibly skewed stable distributions and their convolutions. We show that if the initial distribution of the sex determining trait in such settings is moderately heavy-tailed and has a finite first moment, then an excess of males (females) in the first period leads to the same pattern in the second period. Thus, the excess of one sex over the other one accumulates over two generations and the sex ratio in the total alive population in the second period cannot stabilize at the balanced sex ratio value of 1/2. These properties are reversed for extremely heavy-tailed initial distributions of sex determining traits with infinite first moments. In such settings, the sex ratio of the offspring oscillates around the balanced sex ratio value and an excess of males (females) in the first period leads to an excess of females (males) in the second period. In addition, the sex ratio in the total living population in the second period can stabilize at 1/2 for some extremely heavy-tailed initial distributions of the sex determining trait. The results in the paper are shown to also hold for bounded sex determining phenotypes.
Highlights
Introduction and discussion1.1 Objectives and key resultsA number of modern species exhibit, to a larger or smaller extent, threshold systems of sex determination
This paper studies the properties of the sex ratio rt in two-period models (1), (2) under heavy-tailedness in the framework of stable distributions and their convolutions
We show that if the initial distribution of the sex-determining trait is moderately heavy-tailed and has a finite first moment, the behavior of the sex ratio rt in two-period models (1), (2) is the same as in the case of log-concave densities analyzed by Karlin (1984, 1992)
Summary
A number of modern species exhibit, to a larger or smaller extent, threshold systems of sex determination. We show that if the initial distribution of the sex-determining trait is moderately heavy-tailed and has a finite first moment, the behavior of the sex ratio rt in two-period models (1), (2) is the same as in the case of (extremely light-tailed) log-concave densities analyzed by Karlin (1984, 1992) Under such assumptions, an excess of males (females) in the initial period leads to the same pattern in the second period (Theorem 1). We further demonstrate that the above properties are reversed in two-period models (1), (2) for extremely heavy-tailed distributions of sex-determining traits with infinite first moments In such settings, the sex ratio of the offspring oscillates around the balanced sex ratio value and an excess of males (females) in the initial period leads to an excess of females (males) in the second period (Theorem 2).
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