Reproductive value, the stable stage distribution, and the sensitivity of the population growth rate to changes in vital rates

The population growth rate, or intrinsic rate of increase, measures the potential rate of growth of a population with specified and fixed vital rates.The sensitivity of population growth rate to changes in the vital rates can be written in terms of the stable stage or age distribution and the reproductive value distribution. If the vital rate measures the rate of production of one type of individual by another, then the sensitivity of growth rate is proportional to the reproductive value of the destination type and the representation in the stable stage distribution of the source type. This formal relationship exists in three forms: one limited to age-classified populations, a second that applies to stage- or age-classified populations, and a third that uses matrix calculus. Each uses a different set of formal demographic techniques; together they provide a relationship that beautifully cuts across different types of demographic models.