Enrichment in a general class of stoichiometric producer–consumer population growth models

A mathematical model between keystone species: Bears, salmon and vegetation

We study an ecosystem of three keystone species: salmon, bears and vegetation. Bears consume salmon and vegetation for energy and nutrient intake, but the food quality differs significantly due to the nutritional level difference between salmon and vegetation. We propose a stoichiometric predator-prey model that not only tracks the energy flow from one trophic level to another but also nutrient recycling in the system. Analytical results show that bears may coexist with salmon and vegetation at a steady state, but the abundance of salmon may differ under different regimes. Numerical simulations reveal that a smaller vegetation growth rate may drive the vegetation population to extinction, whereas a large vegetation growth rate may drive the salmon population to extinction. Moreover, a large vegetation growth rate may stabilize the system where the bear, salmon and vegetation populations oscillate periodically.

Noise-Induced Transitions in a Nonsmooth Producer-Grazer Model with Stoichiometric Constraints

Stoichiometric producer-grazer models are nonsmooth due to the Liebig's Law of Minimum and can generate new dynamics such as bistability for producer-grazer interactions. Environmental noises can be extremely important and change dynamical behaviors of a stoichiometric producer-grazer model. In this paper, we consider a stochastically forced producer-grazer model and study the phenomena of noise-induced state switching between two stochastic attractors in the bistable zone. Namely, there is a frequent random hopping of phase trajectories between attracting basins of the attractors. In addition, by applying the stochastic sensitivity function technique, we construct the confidence ellipse and confidence band to find the configurational arrangement of equilibria and a limit cycle, respectively.

Quasi-equilibrium reduction in a general class of stoichiometric producer-consumer models

This article compares a general closed nutrient, stoichiometric producer-consumer model to a two-dimensional 'quasi-equilibrium' approximation. We demonstrate that the quasi-equilibrium system can be rigorously analysed, resulting in nullcline-based criteria for the local stability of system equilibria and for the non-existence of periodic orbits. These results are applied to a study of the dependence of the reduced system on nutrient and energy enrichment. When energy and nutrient enrichment are considered together, the associated bifurcation structures of the two models are seen to share the same essential qualitative characteristics. However, numerical simulations of the three-dimensional parent model show highly complex domains of the persistence and extinction that by Poincare-Bendixson theory are not possible for the two-dimensional reduction. This complexity demonstrates a major difference between the two models, and suggests potential challenges in the use of either model for predicting the long-term behaviour of real-world systems at specific nutrient and energy levels.