Chapter 8: Extending Analytical Solutions to Age–Mass Models of a Population
There has been a long history of population models accounting for variations in physiological characteristics of individuals in the mathematical literature. Classically represented as the McKendrick partial differential equation (PDE) in time and age, there are multiple extensions that include, for example, mass and size. These extended equations rapidly become intractable analytically, without further simplifying assumptions, due to interconnections between the characteristics and fecundity. Here, we consider population dynamics using a three-dimensional PDE incorporating time, age, and an additional characteristic, which we consider to be mass, but without renewal. Such a scenario could represent pathogen development in a host prior to transmission. In advancement of previous work, our growth function remains a function of all three variables. Under conditions of separability, we obtain analytical results for the population dynamics. We confirm and extend these results with numerical simulations of our three-dimensional PDE. Our results can be used to understand age and characteristic, e.g., mass or size-dependent growth of populations.