Abstract
Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller’s classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.
Highlights
Renewal equations are a class of integral equations which, in their simplest form, look as b(t) = b(t − a)k(a)da for t > 0
The interpretation of these inequalities is that the contribution to the population birth rate of the individuals born before time 0 tends to zero exponentially as time tends to infinity
We describe the asymptotic behaviour of the population birth rate b under the following additional assumption
Summary
Renewal equations are a class of integral equations which, in their simplest form, look as. We focus on situations in which individuals give birth at a certain rate, i.e., with a certain probability per unit of time In such situations it is natural to work with rates at the population level as well, as in (1.1). Models of physiologically structured populations are often formulated in terms of PDEs describing i-state development and survival together with a description of the reproduction process [41, 42]. There is a rich body of literature on Volterra integral equations in Banach spaces, see for instance the book [45] by Prüß and the review article [14] by Corduneanu and the references therein Most of these works focus on situations in which the kernel is an unbounded operator and, as a consequence, already proving existence and uniqueness of solutions can be a formidable task. The techniques presented in these works are not suitable for renewal equations with measurevalued solutions
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