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.     

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

This paper presents the derivation and partial analysis of a general producer-consumer model. The model is stoichiometric in that it includes the growth constraints imposed by species-specific biomass carbon to nutrient ratios. The model unifies the approaches of other studies in recent years, and is calibrated from an extensive review of the algae-Daphnia literature. Numerical simulations and bifurcation analysis are used to examine the impact of energy enrichment under nutrient and stoichiometric constraints. Our results suggest that the variety of system responses previously cited for related models can be attributed to the size of the total system nutrient pool, which is here assumed fixed. New, more complicated bifurcation sequences, such as multiple homoclinic bifurcations, are demonstrated as well. The mechanistic basis of the model permits us to show the robustness of the system’s dynamics subject to alternate approaches to modeling producer and consumer biomass production.     

Autocatalysis in a stoichiometric model of energy metabolism]

A simple kinetic model of the energy metabolism with autocatalytic stoichiometric structure is analysed. The model includes the participation of ATP (a product of energy metabolism) in the activation of oxidation substrate. It is shown that the energy metabolism displays multiple steady states and autooscillations in the absense of all kinds of non-stoichiometric (allosteric, isosteric and cooperative) interactions. The results of the analysis are represented in the form of a parametric portrait. The domains of the parameter values of the oxidation substrate source are separated, which correspond to qualitatively different patterns of the model dynamic behavior. Under specified conditions the model is reduced to various versions of the Lotka model.     

Dynamics of a mechanistically derived stoichiometric producer-grazer model

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol. 62 (2000) pp. 1137-1162.] (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE models become the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers.     

Dynamics of a mechanistically derived stoichiometric producer-grazer model

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol. 62 (2000) pp. 1137–1162.] (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE models become the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers.     

A stoichiometric organic matter decomposition model in a chemostat culture

Biodegradation, the disintegration of organic matter by microorganism, is essential for the cycling of environmental organic matter. Understanding and predicting the dynamics of this biodegradation have increasingly gained attention from the industries and government regulators. Since changes in environmental organic matter are strenuous to measure, mathematical models are essential in understanding and predicting the dynamics of organic matters. Empirical evidence suggests that grazers’ preying activity on microorganism helps to facilitate biodegradation. In this paper, we formulate and investigate a stoichiometry-based organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. We determine the criteria for the uniform persistence and extinction of the species and chemicals. Our results show that (1) if at the unique internal steady state, the per capita growth rate of bacteria is greater than the sum of the bacteria’s death and dilution rates, then the bacteria will persist uniformly; (2) if in addition to this, (a) the grazers’ per capita growth rate is greater than the sum of the dilution rate and grazers’ death rate, and (b) the death rate of bacteria is less than some threshold, then the grazers will persist uniformly. These conditions can be achieved simultaneously if there are sufficient resources in the feed bottle. As opposed to the microcosm decomposition models’ results, in a chemostat culture, chemicals always persist. Besides the transcritical bifurcation observed in microcosm models, our chemostat model exhibits Hopf bifurcation and Rosenzweig’s paradox of enrichment phenomenon. Our sensitivity analysis suggests that the most effective way to facilitate degradation is to decrease the dilution rate.

     

Stability of cluster solutions in a cooperative consumer chain model

We study a cooperative consumer chain model which consists of one producer and two consumers. It is an extension of the Schnakenberg model suggested in Gierer and Meinhardt [Kybernetik (Berlin), 12:30–39, 1972] and Schnakenberg (J Theor Biol, 81:389–400, 1979) for which there is only one producer and one consumer. In this consumer chain model there is a middle component which plays a hybrid role: it acts both as consumer and as producer. It is assumed that the producer diffuses much faster than the first consumer and the first consumer much faster than the second consumer. The system also serves as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir. In the small diffusion limit we construct cluster solutions in an interval which have the following properties: The spatial profile of the third component is a spike. The profile for the middle component is that of two partial spikes connected by a thin transition layer. The first component in leading order is given by a Green’s function. In this profile multiple scales are involved: The spikes for the middle component are on the small scale, the spike for the third on the very small scale, the width of the transition layer for the middle component is between the small and the very small scale. The first component acts on the large scale. To the best of our knowledge, this type of spiky pattern has never before been studied rigorously. It is shown that, if the feedrates are small enough, there exist two such patterns which differ by their amplitudes.We also study the stability properties of these cluster solutions. We use a rigorous analysis to investigate the linearized operator around cluster solutions which is based on nonlocal eigenvalue problems and rigorous asymptotic analysis. The following result is established: If the time-relaxation constants are small enough, one cluster solution is stable and the other one is unstable. The instability arises through large eigenvalues of order $O(1)$ . Further, there are small eigenvalues of order $o(1)$ which do not cause any instabilities. Our approach requires some new ideas: (i) The analysis of the large eigenvalues of order $O(1)$ leads to a novel system of nonlocal eigenvalue problems with inhomogeneous Robin boundary conditions whose stability properties have been investigated rigorously. (ii) The analysis of the small eigenvalues of order $o(1)$ needs a careful study of the interaction of two small length scales and is based on a suitable inner/outer expansion with rigorous error analysis. It is found that the order of these small eigenvalues is given by the smallest diffusion constant ${\epsilon }_2^2$ .     

Enrichment technique for the selection of catabolite repression-resistant mutants of Trichoderma as producers of cellulase

Abstract The enrichment technique for the preparation of catabolite repression-resistant producers of cellulase from Trichoderma reesei is based on the submerged cultivation of mutagenized conidia on 2% (w/v) cellobiose or carboxymethyl-cellulose and in the presence of 0.5% (w/v) 2-deoxyglucose as the catabolite repressor. Conidia that are resistant towards the catabolite repressor can produce enzymes necessary for hydrolysis of used substrates and grow under the given conditions. They can be separated from the ungerminated conidia by filtration and used for the production of new conidia which are already enriched with catabolite repression-resistant mutants.     

A stoichiometric model of early plant primary succession

The relative importance of plant facilitation and competition during primary succession depends on the development of ecosystem nutrient pools, yet the interaction of these processes remains poorly understood. To explore how these mechanisms interact to drive successional dynamics, we devised a stoichiometric ecosystem-level model that considers the role of nitrogen and phosphorus limitation in plant primary succession. We applied this model to the primary plant community on Mount St. Helens, Washington State, to check the validity of the proposed mechanisms. Our results show that the plant community is colimited by nitrogen and phosphorus, and they confirm previous suggestions that the presence of a nitrogen-fixing legume, Lupinus lepidus, can enhance community biomass. In addition, the observed nutrient supply rates may promote alternative successional trajectories that depend on the initial plant abundances, which may explain the observed heterogeneity in community development. The model further indicates the importance of mineralization rates and other ecosystem parameters to successional rates. We conclude that a model framework based on ecological stoichiometry allows integration of key biotic processes that interact nonlinearly with biogeochemical aspects of succession. Extension of this approach will improve the understanding of the process of primary succession and its application to ecosystem rehabilitation.     

Thermodynamic and kinetic studies of a stoichiometric model of energetic metabolism under starvation conditions

Abstract A stoichiometric model of anaerobic glycolysis is presented and the influence on its dynamics by the ATP-consuming membrane transport processes and substrate input rate are studied. The model is represented by a system of four ODE (ordinary differential equations), mass conservation equations and functions of state variables, such as thermodynamic efficiency. A low substrate input rate provokes damped oscillations while a high enrgy load determines sustained oscillations in all the metabolites and in thermodynamic efficiency. Due to the lack of linearity between fluxes and forces in the oscillatory region it may be stated that oscillations appear when the system is kinetically controlled.     

Coexistence and community structure in a consumer resource model with implicit stoichiometry

We combine stoichiometry theory and optimal foraging theory into the MacArthur consumer-resource model. This generates predictions for diet choice, coexistence, and community structure of heterotroph communities. Tradeoffs in consumer resource-garnering traits influence community outcomes. With scarce resources, consumers forage opportunistically for complementary resources and may coexist via tradeoffs in resource encounter rates. In contrast to single currency models, stoichiometry permits multiple equilibria. These alternative stable states occur when tradeoffs in resource encounter rates are stronger than tradeoffs in elemental conversion efficiencies. With abundant resources consumers exhibit partially selective diets for essential resources and may coexist via tradeoffs in elemental conversion efficiencies. These results differ from single currency models, where adaptive diet selection is either opportunistic or selective. Interestingly, communities composed of efficient consumers share many of the same properties as communities based on substitutable resources. However, communities composed of relatively inefficient consumers behave similarly to plant communities as characterized by Tilman’s consumer resource theory. The results of our model indicate that the effects of stoichiometry theory on community ecology are dependent upon both consumer foraging behavior and the nature of resource garnering tradeoffs.     

A model for the stoichiometric regulation of blood coagulation

We have developed a model of the extrinsic blood coagulation system that includes the stoichiometric anticoagulants. The model accounts for the formation, expression, and propagation of the vitamin K-dependent procoagulant complexes and extends our previous model by including: (a) the tissue factor pathway inhibitor (TFPI)-mediated inactivation of tissue factor (TF).VIIa and its product complexes; (b) the antithrombin-III (AT-III)-mediated inactivation of IIa, mIIa, factor VIIa, factor IXa, and factor Xa; (c) the initial activation of factor V and factor VIII by thrombin generated by factor Xa-membrane; (d) factor VIIIa dissociation/activity loss; (e) the binding competition and kinetic activation steps that exist between TF and factors VII and VIIa; and (f) the activation of factor VII by IIa, factor Xa, and factor IXa. These additions to our earlier model generate a model consisting of 34 differential equations with 42 rate constants that together describe the 27 independent equilibrium expressions, which describe the fates of 34 species. Simulations are initiated by "exposing" picomolar concentrations of TF to an electronic milieu consisting of factors II, IX, X, VII, VIIa, V, and VIIII, and the anticoagulants TFPI and AT-III at concentrations found in normal plasma or associated with coagulation pathology. The reaction followed in terms of thrombin generation, proceeds through phases that can be operationally defined as initiation, propagation, and termination. The generation of thrombin displays a nonlinear dependence upon TF, AT-III, and TFPI and the combination of these latter inhibitors displays kinetic thresholds. At subthreshold TF, thrombin production/expression is suppressed by the combination of TFPI and AT-III; for concentrations above the TF threshold, the bolus of thrombin produced is quantitatively equivalent. A comparison of the model with empirical laboratory data illustrates that most experimentally observable parameters are captured, and the pathology that results in enhanced or deficient thrombin generation is accurately described.     

A unified stoichiometric model for oxidative and oxidoreductive growth of yeasts

Yeasts degrade glucose through different metabolic pathways, where the choice of the pathway is dependent on the nature of the limitation in the various substrates. When oxygen is limiting in addition to glucose, yeasts often grow according to a mixture of oxidative and reductive metabolism. Oxygen may be limiting either by supply or by inherent biological restrictions such as the respiratory bottleneck in Saccharomyces cerevisiae or by both. A unified model incorporating both supply and biological limitations is proposed for the quantitative prediction of growth rates, consumption and production rates, as well as key metabolite concentrations during mixed oxidoreductive metabolism occuring as a result of such oxygen limitations. This simple unstructured model can be applied to different yeast strains while at the same time requiring a minimum number of measured parameters. "Estimators" are utilized in order to predict the presence of supply-side or biological limitations. The values of these estimators also characterize the relative importance of oxidative to total metabolism. Results from the aerobic and oxygen-limited chemostat cultures were used to corroborate the model predictions. During these experiments, the heat released by the yeast cultures was also monitored on-line. The model correctly predicted the overall stoichiometry, steady-state concentrations, and rates including heat dissipation rates measured in the various situations of oxygen limitations. Direct continuous measurements such as heat can be used in conjunction with the unified model for on-line proces control. (c) 1992 John Wiley & Sons, Inc.     

Optimal feeding under stoichiometric constraints: a model of compensatory feeding with functional response

When the nutrient content of food is limited, herbivores often increase their feeding rates. Such an increase in the feeding rate is called ‘compensatory feeding’. Although it has a number of implications for herbivore population and plant–forager dynamics, the compensatory feeding is not yet functionally formulated especially in relation with ecological stoichiometry. Therefore, we constructed a simple mathematical model by incorporating the optimal feeding rate into the type II functional response to maximize a forager's growth rate under constraints of carbon or nutritionally important element like phosphorus (P). We used the planktonic herbivore Daphnia as a model herbivore. The model revealed that the optimal feeding rate increased by using excess carbon when relative P content of food was less than a certain level, which is known as the threshold elemental ratio. This level changed with the change of food abundance. It also showed that whether or not foragers should exhibit compensatory feeding depends on their stoichiometric characteristics and digestive traits, and also on the assimilability of a given food. These findings are helpful to test the feeding conditions under which compensatory feeding is advantageous for a given animal. Our model can be easily incorporated into forager population dynamics and prey‐consumer interaction models because the optimal feeding rate can be analytically given.