The perpetuation of three-trophic level ecosystems where the three species exhibit
unpredictable time-varyingsurvival strategies is described by a specific set, the viability
kernel, gathering all states from which there exists at least one trajectory safeguarding
each species over a given density threshold. The strategies permitting this property are
delineated and called viable strategies. All solutions starting outside the viability kernel
lead to too low densities or extinction. The viability approach highlights the timing of
strategy changes necessary for a system to perpetuate itself or alternatively to lead
one species to extinction.
The study of the dependence of the viability kernel on the admissible sets of strategies
reveals the minimal flexibilities necessary for the existence of the system. The shape of
the viability kernel determines whether the exogenous addition or substraction of prey
or predator will endanger the system or not, thus gathering different experiments with
opposite results. The comparison of the coexistence kernel with viability kernels for
one, two or three species points out the importance of repeated strategies, not necessarily
in a periodic manner, thus emphasizingthe concept of repetitions in ecosystems
instead of cycles as a key feature of coexistence.