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.

Results of a 3-y study on the populations of three evergreen woody species (Byrsonima crassifolia, Palicourea rigida and Bowdichia virgilioides), from a regularly burnt savanna, in Venezuela, are reported. Tree density, size structure, fruit-seed production per tree of different size classes and soil seed bank were estimated. Viable seed production was one order of magnitude higher in B. crassifolia, which was also the only species with a permanent soil seed bank. Seeds of B. virgilioides germinated readily after dispersal, whilst seeds of P. rigida showed an innate dormancy synchronously released after a year. Both of the latter species seemed to maximize the first growth season by germinating at the beginning of the favourable season. In B. crassifolia, germination proceeds with an early small burst of germination, possibly followed by a second peak at the beginning of the next rainy season and little or no germination thereafter. The results showed that despite the high fire frequency, the studied species produce a substantial number of viable propagules, and have the capacity to germinate in field conditions (readiness depending on the species). Sexual reproduction is expected, therefore, to play an important role, as suggested by their reproductive investment, in long-term population maintenance.