Beyond carrying capacity: Space-limited density dependence in a resource-based human population model

To understand the interaction between human ecology and life history evolution requires models that incorporate age structure, behavior and environmental forces. One such effort is the food ratio model developed by Lee and Tuljapurkar, which makes vital rates functions of food availability. Food abundance is in turn a function of age structure, behavior and environmental factors. Here we explore food-dependent demographic dynamics in a regime where forage or agricultural area limits population growth, which provides an explicit mechanism for density dependence as it is likely to have affected preindustrial populations. The area utilized for food procurement or production becomes a fraction of the maximum area and an asymptotic function of population size and structure. We show that this system has at most one non-trivial equilibrium in a constant environment, and we provide a means of approximating it. This equilibrium is found to be locally stable if the responses of fertility and mortality to caloric restriction are not extreme. We examine the effect of stochastic yield and find that this variability is magnified in the food ratio, a measure of caloric restriction, due to direct and indirect causes. This work provides insight into what the life of the first humans and of early agriculturalists might have been like. It also allows us to examine the effect of climate on population stability in the face of both famine and temporary abundance.