In this paper we explore the eco-evolutionary dynamics of a predator-prey model, wherethe prey population is structured according to a certain life history trait. The traitdistribution within the prey population is the result of interplay between geneticinheritance and mutation, as well as selectivity in the consumption of prey by thepredator. The evolutionary processes are considered to take place on the same time scaleas ecological dynamics, i.e. we consider the evolution to be rapid. Previously publishedresults show that population structuring and rapid evolution in such predator-prey systemcan stabilize an otherwise globally unstable dynamics even with an unlimited carryingcapacity of prey. However, those findings were only based on direct numerical simulationof equations and obtained for particular parameterizations of model functions, whichobviously calls into question the correctness and generality of the previous results. Themain objective of the current study is to treat the model analytically and considervarious parameterizations of predator selectivity and inheritance kernel. We investigatethe existence of a coexistence stationary state in the model and carry out stabilityanalysis of this state. We derive expressions for the Hopf bifurcation curve which can beused for constructing bifurcation diagrams in the parameter space without the need for adirect numerical simulation of the underlying integro-differential equations. Weanalytically show the possibility of stabilization of a globally unstable predator-preysystem with prey structuring. We prove that the coexistence stationary state is stablewhen the saturation in the predation term is low. Finally, for a class of kernelsdescribing genetic inheritance and mutation we show that stability of the predator-preyinteraction will require a selectivity of predation according to the life trait.

We show how results by Diekmann et al. (2007) on the qualitative behaviour of solutionsof delay equations apply directly to a resource-consumer model with age-structured consumerpopulation.

We study the mathematical properties of a general model of cell division structuredwith several internal variables. We begin with a simpler and specific model with two variables, wesolve the eigenvalue problem with strong or weak assumptions, and deduce from it the long-timeconvergence. The main difficulty comes from natural degeneracy of birth terms that we overcomewith a regularization technique. We then extend the results to the case with several parameters andrecall the link between this simplified model and the one presented in [6]; an application to thenon-linear problem is also given, leading to robust subpolynomial growth of the total population.