Stoichiometric knife-edge model on discrete time scale

Ecological stoichiometry is the study of the balance of multiple elements in ecological interactions and processes (Sterner and Elser in Ecological Stoichiometry: The Biology of Elements from Molecules to the Biosphere, 2002). Modeling under this framework enables us to investigate the effect nutrient content on organisms whether the imbalance involves insufficient or excess nutrient content. This phenomenon is called the “stoichiometric knife-edge”. In this paper, a discrete-time predator–prey model that captures this phenomenon is established and qualitatively analyzed. We systematically expound the similarities and differences between our discrete model and the corresponding continuous analog. Theoretical and numerical analyses show that while the discrete and continuous models share many properties, differences also exist. Under certain parameter sets, the models exhibit qualitatively different dynamics. While the continuous model shows limit cycle, Hopf bifurcation, and saddle-node bifurcation, the discrete-time model exhibits richer dynamical behaviors, such as chaos. By comparing the dynamics of the continuous and discrete model, we can conclude that stoichiometric effects of low food quality on predators are robust to the discretization of time. This study can possibly serve as an example for pointing to the importance of time scale in ecological modeling.