Population viability in three trophic-level food chains

The perpetuation of three-trophic level ecosystems where the three species exhibit unpredictable time-varying survival 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 emphasizing the concept of repetitions in ecosystems instead of cycles as a key feature of coexistence.