The impact of variable stoichiometry on predator-prey interactions: a multinutrient approach

A model for prey and predators is formulated in which three essential nutrients can limit growth of both populations. Prey take up dissolved nutrients, while predators ingest prey, assimilate a fraction of ingested nutrients that depends on their current nutrient status, and recycle the balance. Although individuals are modeled as identical within populations, amounts of nutrients within individuals vary over time in both populations, with reproductive rates increasing with these amounts. Equilibria and their stability depend on nutrient supply conditions. When nutrient supply increases, unusual results can occur, such as a decrease of prey density. This phenomenon occurs if, with increasing nutrient, predators sequester rather than recycle nutrients. Furthermore, despite use of a linear functional response for predators, high nutrient supply can destabilize equilibria. Responses to nutrient supply depend on the balance between assimilation and recycling of nutrients by predators, which differs depending on the identity of the limiting nutrient. Applied to microbial ecosystems, the model predicts that the efficiency of organic carbon mineralization is reduced when supply of mineral nutrients is low and when equilibria are unstable. The extent to which predators recycle or sequester limiting nutrients for their prey is of critical importance for the stability of predator-prey systems and their response to enrichment.     

Nutrient flows between ecosystems can destabilize simple food chains

Dispersal of organisms has large effects on the dynamics and stability of populations and communities. However, current metacommunity theory largely ignores how the flows of limiting nutrients across ecosystems can influence communities. We studied a meta-ecosystem model where two autotroph-consumer communities are spatially coupled through the diffusion of the limiting nutrient. We analyzed regional and local stability, as well as spatial and temporal synchrony to elucidate the impacts of nutrient recycling and diffusion on trophic dynamics. We show that nutrient diffusion is capable of inducing asynchronous local destabilization of biotic compartments through a diffusion-induced spatiotemporal bifurcation. Nutrient recycling interacts with nutrient diffusion and influences the susceptibility of the meta-ecosystem to diffusion-induced instabilities. This interaction between nutrient recycling and transport is further shown to depend on ecosystem enrichment. It more generally emphasizes the importance of meta-ecosystem theory for predicting species persistence and distribution in managed ecosystems.     

Simulations in a Plankton Model

A technique for analyzing the stability of a zooplankton, phytoplankton, nutrient interaction model is described. This is an extension of a two-dimensional predator-prey model, incorporating a nutrient food source. In this study difference equations, rather than differential equations, are used to simulate the system since the systems are more easily studied in this formulation. Using only a few reasonable constraints on the system, a remarkable stability is obtained.     

How far details are important in ecosystem modelling: the case of multi-limiting nutrients in phytoplankton-zooplankton interactions

We try to answer the question of to what extent details in nutrient uptake and phytoplankton physiology matter for population and community dynamics. To this end, we study how two nutrients interact in limiting phytoplankton growth. A popular formulation uses a product-rule for nutrient uptake, which we compare with that on the basis of synthesizing units. We first fit different nutrient uptake models to a dataset and conclude that the quantitative differences between the models are small. Then we study the sensitivity of phytoplankton growth and zooplankton-phytoplankton interactions (ZPi) models to uptake formulations. Two population models are compared; they are based on different assumptions on the relation between nutrient uptake and phytoplankton growth. We find that the population and community models are sensitive to uptake formulations. According to the uptake formulation used in the ZPi models, qualitative differences can be observed. Indeed, although two models based on functions with similar shapes have close equilibria, these can differ in stability properties. Since stability involves the derivatives of formulas, even if two formulas provide close values, large numerical differences in the stability criterion may occur after derivation. We conclude that mechanistic details can be of importance for community modelling.     

具有抑制剂的恒化器竞争模型的分析（英文）

考虑了具有抑制剂的两种群竞争一种微生物的恒化器模型,其中吸收函数和功能反应函数都是营养的一般单调递增函数,并且一种微生物能够分泌一种对另一种微生物起致命影响的抑制剂.利用常微分方程定性理论,首先得到了平衡点的存在条件和局部渐进稳定性;然后讨论了全局渐进稳定性,以及极限环和Hopf分支的存在性.     

Enhanced Arginine Methylation of Programmed Cell Death 4 Protein during Nutrient Deprivation Promotes Tumor Cell Viability

The role of programmed cell death 4 (PDCD4) in tumor biology is context-dependent. PDCD4 is described as a tumor suppressor, but its coexpression with protein arginine methyltransferase 5 (PRMT5) promotes accelerated tumor growth. Here, we report that PDCD4 is methylated during nutrient deprivation. Methylation occurs because of increased stability of PDCD4 protein as well as increased activity of PRMT5 toward PDCD4. During nutrient deprivation, levels of methylated PDCD4 promote cell viability, which is dependent on an enhanced interaction with eIF4A. Upon recovery from nutrient deprivation, levels of methylated PDCD4 are regulated by phosphorylation, which controls both the localization and stability of methylated PDCD4. This study reveals that, in response to particular environmental cues, the role of PDCD4 is up-regulated and is advantageous for cell viability. These findings suggest that the methylated form of PDCD4 promotes tumor viability during nutrient deprivation, ultimately allowing the tumor to grow more aggressively.     

Plankton-toxin interaction with a variable input nutrient

A simple model of phytoplankton–zooplankton interaction with a periodic input nutrient is presented. The model is then used to study a nutrient–plankton interaction with a toxic substance that inhibits the growth rate of plankton populations. The effects of the toxin upon the existence, magnitude, and stability of the periodic solutions are discussed. Numerical simulations are also provided to illustrate analytical results and to compare more complicated dynamical behaviour.     

受干扰的生态系统Ⅰ.柞蚕林的养分循环

本文通过田间实验与测试揭示了柞蚕林养分循环状态和干扰因素对养分平衡的影响。结果表明,地下部储存的养分占总积累的比重较大。同时叶部养分向树体转移能力较强。柞蚕林二级生产力是以收获蚕茧为主要形式,其养分输出水平对系统的养分平衡产生的影响主要表现对树体养分内循环方面。林木养分储量的下降是造成生产力衰退的重要原因。对养分循环格局的认识有助于对该系统干扰强度进行调控,以增强柞蚕林生态系统的稳定性。     

Turing-Hopf instability in biochemical reaction networks arising from pairs of subnetworks

A three dimensional nutrient-plant-herbivore model was proposed and conditions for boundedness, positive invariance, existence and stability of different equilibrium points, Hopf-bifurcation and global stability were obtained. We performed numerical simulations to observe the simultaneous effect of the top-down and the bottom-up mechanism on the system. It was found that nutrient enrichment destroyed the coexistence steady state of the system. This nutrient enrichment could be due to high nutrient input rate or high nutrient recycling rate. In both cases the system showed instability. Moreover, these results were independent of the grazing pressure and the predation functional form.     

Persistence and coexistence in zooplankton-phytoplankton-nutrient models with instantaneous nutrient recycling

We consider plankton-nutrient interaction models consisting of phytoplankton, herbivorous zooplankton and dissolved limiting nutrient with general nutrient uptake functions and instantaneous nutrient recycling. For the model with constant nutrient input and different constant washout rates, conditions for boundedness of the solutions, existence and stability of non-negative equilibria, as well as persistence are given. We also consider the zooplankton-phytoplankton-nutrient interaction models with a fluctuating nutrient input and with a periodic washout rate, respectively. It is shown that coexistence of the zooplankton and phytoplankton may arise due to positive bifurcating periodic solutions.Research has been supported in part by a University of Alberta Ph.D. Scholarship and is in part based on the author's Ph.D. thesis under the supervision of Professor H. 1. Freedman, to whom the author owes a debt of appreciation and gratitude for his kind advice, helpful comments and continuous encouragement     

Paradoxes of enrichment: effects of increased light versus nutrient supply on pelagic producer-grazer systems

Energy-based plant-herbivore models produce the "paradox of enrichment," a destabilizing influence of enrichment on population dynamics. Because many plants change their carbon : nutrient stoichiometry in response to the light : nutrient supply ratio, enrichment with light can cause a mismatch between the elemental compositions of plants and their herbivores. Herbivore growth rates may then decrease with increased light supply, which is termed the "paradox of energy enrichment." I present a stoichiometric phytoplankton-grazer model that accounts for the dynamical vertical light gradient and explore how algal and grazer densities, mineral nutrient concentration, algal nutrient stoichiometry, and system stability respond to enrichment with light (through changes in irradiance, background turbidity, and water column depth) versus enrichment with nutrients. Parameterized for Daphnia, the model produces several "unusual" phenomena: multiple equilibria (with grazers extinct in spite of high algal biomass at one equilibrium), inconsistent light enrichment effects on stability (light enrichment first destabilizes and then stabilizes), and the paradox of energy enrichment. These phenomena are restricted to the low end of realistic nutrient supplies except in very shallow systems, where high sedimentation rates effectively deplete the water column of nutrients. At higher nutrient supplies, light enrichment produces the classical paradox of enrichment, leading first to an increase in grazers at a stable equilibrium and then to algae-grazer oscillations.     

How enrichment,ecosystem size,and their effects on species richness co‐determine the stability of microcosm communities

Nutrient enrichment, ecosystem size, and richness each may directly affect the stability of both populations and communities. Alternatively, nutrient enrichment and ecosystem size each may directly affect richness, which in turn may affect stability. No previous studies, however, have tested empirically how these three factors interact and co‐determine stability. We manipulated nutrient input and ecosystem size in replicate microcosms containing a diverse bacterial flora, and a range of green algae and heterotrophic protozoa, and used these manipulations and the resulting variation in species richness to measure their combined effects on temporal stability of both populations and communities. Results showed that nutrient enrichment and ecosystem size controlled protist richness, and their effects on stability could be mediated by richness. In addition, both community‐level and population‐level stability increased with protist richness. Furthermore, mean species evenness and mean species richness was negatively related. Effects of statistical averaging, overyielding, and component population stability were identified as possible mechanisms involved explaini ng the stabilizing effects of richness on community stability. Their relative strength in influencing stability, however, is likely to change as mean evenness decreased with increasing richness. This decrease in evenness would tend to weaken the strength of the statistic averaging effect, but increase the strength of the other two mechanisms due to relatively lower population variability (component population stability) and higher mean biovolumes of dominant protists (overyielding).     

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.     

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.     

Stability in chemostat equations with delayed nutrient recycling

The growth of a species feeding on a limiting nutrient supplied at a constant rate is modelled by chemostat-type equations with a general nutrient uptake function and delayed nutrient recycling. Conditions for boundedness of the solutions and the existence of non-negative equilibria are given for the integrodifferential equations with distributed time lags. When the time lags are neglected conditions for the global stability of the positive equilibrium and for the extinction of the species are provided. The positive equilibrium continues to be locally stable when the time lag in recycling is considered and this is proved for a wide class of memory functions. Computer simulations suggest that even in this case the region of stability is very large, but the solutions tend to the equilibrium through oscillations.Work performed within the activity of the research group Equazioni di evoluzione e applicazioni Fisico-Matematiche, MPI (Italy), and under the auspices of GNFM, CNR (Italy)     

Effects of time lags on transient characteristics of a nutrient cycling model.

A simple ecosystem with limiting nutrient cycling is modeled by chemostat equations with an integral term describing the continuous time lag involved in the process of nutrient regeneration from organic sediments. The same model has already been proposed in a previous paper, where conditions for boundedness of the solutions and stability of the equilibria were given. This paper is concerned with the relationships between resilience, that is, the speed with which the system returns to a stable equilibrium following a perturbation, and the time lag in the nutrient recycling process. Simple algorithms are given for the numerical calculation of the characteristic return time toward the stable equilibrium following a small perturbation. These methods also allow us to distinguish the case of monotone convergence from that of oscillatory convergence toward equilibrium. The numerical results obtained show that the presence of the time lag causes both qualitative and quantitative modifications in the dependence of equilibrium resilience on some relevant ecological parameters, such as the input nutrient concentration and the recycling extent. Analytical results for "quasi-closed" ecosystems are given that show that such stable systems are characterized by a very low resilience.     

Interspecific competition in natural plant communities: mechanisms, trade-offs and plant-soil feedbacks

Interspecific competition in natural plant communities is highly dependent on nutrient availability. At high levels of nutrient availability, competition is mainly for light. As light is a unidirectional resource, high nutrient habitats are dominated by fast-growing perennials with a tall stature and a rather uniform vertical distribution of leaf area. Moreover, these species have high turnover rates of leaves and roots and a high morphological plasticity during the differentiation of leaves. There is less consensus, however, about the importance and intensity of interspecific competition in nutrient-poor environments. It is argued that selection in nutrient-poor habitats is not necessarily on a high competitive ability for nutrients and a high growth rate, but rather on traits which reduce nutrient losses (low tissue nutrient concentrations, slow tissue turnover rates, high nutrient resorption efficiency). Due to evolutionary trade-offs plants can not maximize both growth rate and nutrient retention. Thus, the low growth rate of species from nutrient-poor habitats should be considered as the consequence of nutrient retention rather than as a feature on which direct selection takes place. The contrasting traits of species from nutrient-poor and nutrient-rich habitats mutually exclude them from each others' habitats. Moreover, these traits have severe consequences for litter decomposability and thereby also for nutrient cycling. This leads both in nutrient-poor and nutrient-rich habitats to a positive feedback between plant species dominance and nutrient availability, thereby promoting ecosystem stability.     

Phenotypic diversity and stability of ecosystem processes.

The resistance of an ecosystem to perturbations and the speed at which it recovers after the perturbations, which is called resilience, are two important components of ecosystem stability. It has been suggested that biodiversity increases the resilience and resistance of aggregated ecosystem processes. We test this hypothesis using a theoretical model of a nutrient-limited ecosystem in a heterogeneous environment. We investigate the stability properties of the model for its simplest possible configuration, i.e. , a system consisting of two plant species and their associated detritus and local resource depletion zones. Phenotypic diversity within the plant community is described by differences in the nutrient uptake and mortality rates of the two species. The usual measure of resilience characterizes the system as a whole and thus also applies to aggregated ecosystem processes. As a rule this decreases with increased diversity, though under certain conditions it is maximum for an intermediate value of diversity. Resistance is a property that characterizes each system component and process separately. The resistance of the inorganic nutrient pools, hence of nutrient retention in the ecosystem, decreases with increased diversity. The resistance of both total plant biomass and productivity either monotonically decreases or increases over part of the parameter range with increased diversity. Furthermore, it is very sensitive to parameter values. These results support the view that there is no simple relationship between diversity and stability in equilibrium deterministic systems, whether at the level of populations or aggregated ecosystem processes. We discuss these results in relation to recent experiments.     

The foams of fermentation broths. I. Some parameters of the foaming of fermentation media

The stability of foam formed during fermentation is decisively affect ed by the nature of the nutrient media used. In froth-flotation models, (a) the foam formation time, characteristic of the tendency to foam, and (b) foam subsistence time, characteristic of the stability of foams formed, have been studied. With the utilization of these two parameters, the stability of foam from aqueous solution of several surface active components of nutrient media has been noted as a function of concentrations. Further, but, without attempting completeness, the viscosity enhancing effect of carbohydrate components, and the effect of the subsistence time of their foam, upon the stability of foam have been studied together with the correlation between “standing” time after sterilization and tendency to foam. Taking soy-bean meal as a model, the stability of foam films in function of pi I, at constant concentration, has been studied. It seems that though a proper control of the factors mentioned, nutrient media with a low tendency of foaming can be formulated.