  We use a two-sex partial differential equation (PDE) model based on the
Grandmother Hypothesis. We build on an earlier model by Kim et al. [27] by
allowing for evolution in both longevity and age at last birth, and also
assuming that post-fertile females support only their daughters' fertility.
Similarly to Kim et al. [27], we find that only two locally stable equilibria
exist: one corresponding to great ape-like longevities and the other
corresponding to hunter-gatherer longevities. Our results show that
grandmothering enables the transition between these two equilibria, without
extending the end of fertility. Moreover, sensitivity analyses of the model
show that male competition, arising from a skew in the mating sex ratio towards
males, plays a significant role in determining whether the transition from
great ape-like longevities to higher longevities is possible and the
equilibrium value of the average adult lifespan. Whereas grandmothering effects
have a significant impact on the equilibrium value of the average age at last
birth and enable the transition to higher longevities, they have an
insignificant impact on the equilibrium value of the average adult lifespan.
