Characterizing the Phase Transitions between Stable Equilibrium and Periodic Oscillations in Predator-prey Population Dynamics: A Theoretical Appraisal from an Extended Nicholson & Bailey Model

Multi-phase patterns with more or less sharp phase transitions, first highlighted in thermodynamics, have progressively revealed having wider relevance, being encountered in various other contexts, for example fluid mechanics, and can even occur in the interactive dynamics in biological populations involving two or more species that share opposite interests, such as predator-prey or parasite-host pairs of species. In the latter, the pattern of abundances of both interacting species usually reaches an equilibrium level which can be either stable or cyclic (with large periodic oscillations in the latter case). Both alternative modes are separated by well-define boundaries and, accordingly, can relevantly be described in terms of phases and phase transitions. While this has recently been approached from very general perspectives, a more focused analysis is still lacking, regarding the nature of the phase transitions between stable and oscillatory equilibria and – still more importantly – how the nature of these phase transitions may possibly depend (or not) on the biological and contextual factors driving the parasite-host interactive dynamics. These issues are addressed hereafter, on a theoretical basis, yet intimately related to the real field context, by taking advantage of a newly derived extension of the classical Nicholson & Bailey model of parasite-host interactions. Highlighted in particular are the possibility of either first-order, second-order or continuous phase-transitions, depending on (i) the respective own dynamics of both host and parasite, (ii) the density of feeding resource for the host, (iii) the level of migration exchange in a meta-population context.