5 - Renewal Equation in the McKendrick–von Foerster Model

Summary

If equipped with an additional operation of multiplication of vectors by other vectors (to give yet other vectors), an operation that is well-intertwined with the existing linear and topological structures, a Banach space becomes a Banach algebra. Such an additional operation is naturally defined in the space of continuous functions by pointwise multiplication. In the space of integrable functions on the positive half-axis, however, the role of multiplication is most naturally played by convolution. We use this additional algebraic structure in the discussed space to study the McKendrick–von Foerster model of population dynamics. The existence and uniqueness of the renewal equation that is a key to the model turns out – surprise, surprise! – to be the result of the completeness of the underlying space.