Dynamics of a Stoichiometric discrete producer-grazer model

In the last decade, several theoretical models based on stoichiometric principles as well as field and laboratory experiments have shown that nutritional quality of the prey can have dramatic and counterintuitive impact. For example, the predator can become extinct while having plentiful prey in a completely deterministic system. The explanation lies in the bad nutritional quality of the prey that precludes the predator from efficiently converting the consumed food into its own biomass. Another effect is the halt of oscillations that are ubiquitous to predator-prey systems, which happens when bad prey quality drives the system through a saddle-node bifurcation. We note that all the existing models exhibiting these effects are continuous in time. However, in experiments, data are collected on discrete time intervals and many producers in nature have non-overlapping generations. Such scenarios call for discrete equation models. Hence we ask: (1) to what degree stoichiometric effects are just artifacts of continuous time models? (2) Can novel stoichiometric effects arise in discrete systems? Here, by comparing a continuous stoichiometric model to its discrete analog, we show that stoichiometric impacts of prey quality persist in discrete system. Moreover, not only bad prey quality can pull the system out of oscillations but also it can halt chaotic dynamics that surfaces in the discrete system. Stoichiometric mechanisms become increasingly important in our understanding of food web dynamics and our results suggest the robustness of these mechanisms to the discretization of time.