Mechanistically derived Toxicant-mediated predator-prey model under Stoichiometric constraints.

Studies in ecological stoichiometry highlight that grazer dynamics are affected by insufficient food nutrient content (low phosphorus (P)/carbon (C) ratio) as well as excess food nutrient content (high P:C). Contaminant stressors affect all levels of the biological hierarchy, from cells to organs to organisms to populations to entire ecosystems. Eco-toxicological modeling under the framework of ecological stoichiometry predicts the risk of bio-accumulation of a toxicant under stoichiometric constraints. In this paper, we developed and analyzed a Lotka-Volterra type predator- prey model which explicitly tracks the environmental toxicant as well as the toxicant in the populations under stoichiometric constraints. Analytic, numerical, slow-fast steady state and bifurcation theory are employed to predict the risk of toxicant bio-accumulation under varying food conditions. In some cases, our model predicts different population dynamics, including wide amplitude limit cycles where producer densities exhibit very low values and may be in danger of stochastic extinction.

Seasonal Variation of Nutrient Loading in a Stoichiometric Producer-Consumer System.

Recent discoveries in ecological stoichiometry have indicated that food quality in terms of the phosphorus/carbon (P/C) ratio affects consumers whether the imbalance involves insufficient or excess nutrients. This phenomenon is called the "stoichiometric P/C knife-edge." In this study, we develop and analyze a producer-consumer model which captures this phenomenon. It assesses the effects of (external) nutrient (P) loading on consumer dynamics in an aquatic environment by mechanistically deriving and accounting for seasonal variation in nutrient loading. In the absence of seasonal effects, previous models suggest that the dynamics are Hopf bifurcation, saddle-node bifurcations, and limit cycles. However, seasonal effects can have major implications on the predicted solutions and enrich population dynamics. Bifurcation analyses demonstrate that seasonal forcing can cause both periodic and quasi-periodic solutions.

Dynamics of a Producer-Grazer Model Incorporating the Effects of Phosphorus Loading on Grazer's Growth.

Phosphorus is an essential element for all life forms, and it is also a limiting nutrient in many aquatic ecosystems. To keep track of the mismatch between the grazer's phosphorus requirement and producer phosphorus content, stoichiometric models have been developed to explicitly incorporate food quality and food quantity. Most stoichiometric models have suggested that the grazer dynamics heavily depends on the producer phosphorus content when the producer has insufficient nutrient content [low phosphorus (P):carbon (C) ratio]. However, recent laboratory experiments have shown that the grazer dynamics are also affected by excess producer nutrient content (extremely high P:C ratio). This phenomenon is known as the "stoichiometric knife edge." While the Peace et al. (Bull Math Biol 76(9):2175-2197, 2014) model has captured this phenomenon, it does not explicitly track P loading of the aquatic environment. Here, we extend the Peace et al. (2014) model by mechanistically deriving and tracking P loading in order to investigate the growth response of the grazer to the producer of varying P:C ratios. We analyze the dynamics of the system such as boundedness and positivity of the solutions, existence and stability conditions of boundary equilibria. Bifurcation diagram and simulations show that our model behaves qualitatively similar to the Peace et al. (2014) model. The model shows that the fate of the grazer population can be very sensitive to P loading. Furthermore, the structure of our model can easily be extended to incorporate seasonal P loading.