The demographic models I reviewed in previous chapters are all one-sex models, in which the sex referred to is usually the female. This setting can be justified if we assume either that the life-cycle vital rates (as functions of state variables) for both sexes are the same or that the population dynamics are determined by one sex alone, independent of the possibly relative abundance of the other sex. However, at least for human population, neither assumption is valid. The ratio of newborn girls and newborn boys is close to one, but is less than one for almost all countries in the world. The age-specific mortality rates of women are also lower than those of men worldwide. This is called sexual dimorphism in the demography literature. Such a dimorphism makes the study of two-sex models indispensable. If we look at the male and female vital rates, we find that the differences are small. Despite this small difference, population dynamics derived solely from male vital rates and those derived solely from female vital rates will show ever-increasing differences with the passage of time. Furthermore, because the intrinsic growth rates derived from male and female lines, respectively, are distinct, we cannot avoid the undesirable conclusion that, if we do not incorporate males and females in a unified model, eventually the sex ratio will become either zero or infinity, which is never the case in reality. This is the inconsistency we have to overcome while dealing with population models with two sexes. Another technical difficulty with two-sex modeling has to do with the irreducibility of the state-transition matrix. I mentioned in chapters 2 and 3 that in an age-specific one-sex model, because people older than a particular age, say β, are not fertile anymore, the age group older than β is an absorbing set; hence, our focus of population dynamics can be restricted to the age set [0, β]. This is why we can transform the n × n Leslie matrix to a Lolka renewal equation. In a two-sex model, however, there does not exist a common upper bound for the reproduction of both sexes, for a male older than β can marry a female younger than β and become fertile again.
New Islands of Tractability of Cost-Optimal Planning