In this paper we explore the eco-evolutionary dynamics of a predator-prey model, where
the prey population is structured according to a certain life history trait. The trait
distribution within the prey population is the result of interplay between genetic
inheritance and mutation, as well as selectivity in the consumption of prey by the
predator. The evolutionary processes are considered to take place on the same time scale
as ecological dynamics, i.e. we consider the evolution to be rapid. Previously published
results show that population structuring and rapid evolution in such predator-prey system
can stabilize an otherwise globally unstable dynamics even with an unlimited carrying
capacity of prey. However, those findings were only based on direct numerical simulation
of equations and obtained for particular parameterizations of model functions, which
obviously calls into question the correctness and generality of the previous results. The
main objective of the current study is to treat the model analytically and consider
various parameterizations of predator selectivity and inheritance kernel. We investigate
the existence of a coexistence stationary state in the model and carry out stability
analysis of this state. We derive expressions for the Hopf bifurcation curve which can be
used for constructing bifurcation diagrams in the parameter space without the need for a
direct numerical simulation of the underlying integro-differential equations. We
analytically show the possibility of stabilization of a globally unstable predator-prey
system with prey structuring. We prove that the coexistence stationary state is stable
when the saturation in the predation term is low. Finally, for a class of kernels
describing genetic inheritance and mutation we show that stability of the predator-prey
interaction will require a selectivity of predation according to the life trait.