Stability and Bifurcation in a Stoichiometric Producer-Grazer Model with Knife Edge

All organisms are composed of multiple chemical elements such as nitrogen (N), phosphorus (P), and carbon (C). P is essential to build nucleic acids (DNA and RNA) and N is needed for protein production. To keep track of the mismatch between the P requirement in the consumer (grazer) and the P content in the provider (producer), stoichiometric models have been constructed to explicitly incorporate food quality and quantity. In addition to their fundamental applications in ecology and biology, stoichiometric models are especially suitable for medical applications where stoichiometrically distinct pathogens or cancer cells are competing with normal cells and suffer a higher death rate due to excessive chemotherapy agent or radiation uptake. Most stoichiometric models have suggested that the consumer dynamics heavily depends on the P content in the provider when the provider has low nutrient content (low P: C ratio). Motivated by recent lab experiments, researchers explored the effect of excess producer nutrient content (extremely high P: C ratio) on the consumer dynamics. This phenomenon is called the stoichiometric knife edge and its rich dynamics is yet to be appreciated due to the fact that a global analysis of a knife-edge model is challenging. The main challenge stems from the phase plane fragmentation and parameter space partitioning in order to carry out a detailed and complete case by case analysis of the model dynamics. The aim of this paper is to present a sample of a complete mathematical analysis of the dynamics of this model and to perform a bifurcation analysis for the model with Holling type-II functional response.