# Modelling the evolution of traits in a two-sex population, with an application to grandmothering

## TL;DR

This paper introduces a mathematical model for the evolution of heritable traits affecting life-history timing in two-sex populations, demonstrating how grandmothering influences sex ratios and longevity through fitness landscape analysis.

## Contribution

It provides a novel formula for the fitness landscape incorporating two sexes, age structure, and multiple traits, extending the Price Equation framework.

## Key findings

- Grandmothering can increase sex ratios due to longer female post-fertile lifespan.
- Longer lifespans lead to higher fitness for both sexes.
- Grandmothering alone can drive evolution towards increased longevity.

## Abstract

We present a mathematical simplification for the evolutionary dynamics of a heritable trait within a two-sex population. This trait is assumed to control the timing of sex-specific life-history events, such as the age of sexual maturity and end of female fertility, and each sex has a distinct fitness tradeoff associated with the trait. We provide a formula for the fitness landscape of the population and show a natural extension of the result to an arbitrary number of heritable traits. Our method can be viewed as a dynamical systems generalisation of the Price Equation to include two sexes{, age structure} and multiple traits. We use this formula to examine the effect of grandmothering, whereby post-fertile females subsidize their daughter's fertility by provisioning grandchildren. Grandmothering can drive a shift towards higher sex ratios due to lengthening female post-fertile longevity, leading to changes in fitness for both sexes. For males, increased longevity is accompanied by a substantially longer fertile lifespan resulting in higher sex ratios in the fertile ages. Our fitness landscapes show a net increase in fitness for both males and females at longer lifespans, and as a result, we find that grandmothering alone provides an evolutionary trajectory to higher longevities.