Generalized Structural Kinetic Modeling: A Survey and Guide

Many current challenges involve understanding the complex dynamical interplay between the constituents of systems. Typically, the number of such constituents is high, but only limited data sources on them are available. Conventional dynamical models of complex systems are rarely mathematically tractable and their numerical exploration suffers both from computational and data limitations. Here we review generalized modeling, an alternative approach for formulating dynamical models to gain insights into dynamics and bifurcations of uncertain systems. We argue that this approach deals elegantly with the uncertainties that exist in real world data and enables analytical insight or highly efficient numerical investigation. We provide a survey of recent successes of generalized modeling and a guide to the application of this modeling approach in future studies such as complex integrative ecological models.

The impact of additional food on plankton dynamics in the absence and presence of toxicity

This article represents a three-dimensional dynamical system which studies the impact of additional food on the plankton system in the absence and presence of toxicity. It is determined that available additional food and toxin produced by phytoplankton play a crucial role in the termination of harmful algal blooms. The significance of the parameter, additional food, lies in the fact that its limited availability can terminate planktonic blooms when the concentration of toxicity is high. The positivity, boundedness, and permanence of the system are proved to make the model biologically valid. The existence of different biological feasible equilibria and their local stability criteria are determined analytically under certain conditions. The Hopf-bifurcation analysis is carried out by considering additional food as a bifurcation parameter. All the theoretical results are verified numerically. Our results indicate that the prey and predator population are never extinct and survive at a stable level when the additional food is available for the predator population.

Ontogenetic asymmetry modulates population biomass production and response to harvest

Patterns in biomass production are determined by resource input (productivity) and trophic transfer efficiency. At fixed resource input, variation in consumer biomass production has been related to food quality, metabolic type and diversity among species. In contrast, intraspecific variation in individual body size because of ontogenetic development, which characterizes the overwhelming majority of taxa, has been largely neglected. Here we show experimentally in a long-term multigenerational study that reallocating constant resource input in a two-stage consumer system from an equal resource delivery to juveniles and adults to an adult-biased resource delivery is sufficient to cause more than a doubling of total consumer biomass. We discuss how such changes in consumer stage-specific resource allocation affect the likelihood for alternative stable states in harvested populations as a consequence of stage-specific overcompensation in consumer biomass and thereby the risk of catastrophic collapses in exploited populations.

The effect of intraspecific body size variation on the efficiency with which energy is transferred between trophic levels is not well understood. Here, Reichstein et al. show that biasing resource delivery toward less efficient consumer life stages can lead to a doubling of consumer biomass.

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.

Enrichment in a stoichiometric model of two producers and one consumer

We consider a stoichiometric population model of two producers and one consumer. Stoichiometry can be thought of as the tracking of food quality in addition to food quantity. Our model assumes a reduced rate of conversion of biomass from producer to consumer when food quality is low. The model is open for carbon but closed for nutrient. The introduction of the second producer, which competes with the first, leads to new equilibria, new limit cycles, and new bifurcations. The focus of this paper is on the bifurcations which are the result of enrichment. The primary parameters we vary are the growth rates of both producers. Secondary variable parameters are the total nutrients in the system, and the producer nutrient uptake rates. The possible equilibria are: no-life, one-producer, coexistence of both producers, the consumer coexisting with either producer, and the consumer coexisting with both producers. We observe limit cycles in the latter three coexistence combinations. Bifurcation diagrams along with corresponding representative time series summarize the behaviours observed for this model.