Numerical computation of periodic solutions of renewal equations from population dynamics

We describe a piecewise collocation method for computing periodic solutions of renewal equations, obtained as an extension of the corresponding method in [K. Engelborghs et al., SIAM J. Sci. Comput., 22 (2001), pp. 1593--1609] for retarded functional differential equations. Then, we rigorously prove its convergence under the abstract framework proposed in [S. Maset, Numer. Math., 133 (2016), pp. 525--555], as previously done in [A.A. and D.B., SIAM J. Numer. Anal., 58 (2020), pp. 3010--3039] for general retarded functional differential equations. Finally, we show some numerical experiments on models from populations dynamics which confirm the order of convergence obtained theoretically, as well as a few applications in view of bifurcation analysis.