Now the development of each individual from birth till death is often described by an ode where the change in time of the state of the individual is its growth rate. Complemented with birth and death models, the state of a population is not described by a single number but by some distribution over the individual state, such as age or size. When the number of individuals is large, this distribution is taken continuous instead of discrete. When such a distribution is smooth the resulting population model, derived straightforwardly from the individual model using balance laws, is a partial differential equation (pde).
Initially age was used to describe the state of the individuals. However, for many organisms not age, or not age alone, characterizes its state but also size or energy reserves are important. In addition, the life cycle of the individuals consists of a number of life-stages, such as egg, juvenile and adult or the distribution of the number of individuals with respect to the state variable is irregular. This lead to physiologically structured population models psp. The introduction of these two extra components, namely several life-stages and non-smooth distributions, gave mathematical difficulties which were difficult to overcome with the pde formulation. Therefore, recently a cumulative formulation was proposed where the governing formulation is in terms of renewal integral equations.