Dynamics of a Producer–Grazer Model Incorporating the Effects of Excess Food Nutrient Content on Grazer’s Growth

Theory of Stoichiometric Intraguild Predation: Algae, Ciliate, and Daphnia

Consumers respond differently to external nutrient changes than producers, resulting in a mismatch in elemental composition between them and potentially having a significant impact on their interactions. To explore the responses of herbivores and omnivores to changes in elemental composition in producers, we develop a novel stoichiometric model with an intraguild predation structure. The model is validated using experimental data, and the results show that our model can well capture the growth dynamics of these three species. Theoretical and numerical analyses reveal that the model exhibits complex dynamics, including chaotic-like oscillations and multiple types of bifurcations, and undergoes long transients and regime shifts. Under moderate light intensity and phosphate concentration, these three species can coexist. However, when the light intensity is high or the phosphate concentration is low, the energy enrichment paradox occurs, leading to the extinction of ciliate and Daphnia. Furthermore, if phosphate is sufficient, the competitive effect of ciliate and Daphnia on algae will be dominant, leading to competitive exclusion. Notably, when the phosphorus-to-carbon ratio of ciliate is in a suitable range, the energy enrichment paradox can be avoided, thus promoting the coexistence of species. These findings contribute to a deeper understanding of species coexistence and biodiversity.

Stoichiometric microplastics models in natural and laboratory environments

Microplastics pose a severe threat to marine ecosystems; however, relevant mathematical modeling and analysis are lacking. This paper formulates two stoichiometric producer-grazer models to investigate the interactive effects of microplastics, nutrients, and light on population dynamics under different settings. One model incorporates optimal microplastic uptake and foraging behavior based on nutrient availability for natural settings, while the other model does not include foraging in laboratory settings. We establish the well-posedness of the models and examine their long-term behaviors. Our results reveal that in natural environments, producers and grazers exhibit higher sensitivity to microplastics, and the system may demonstrate bistability or tristability. Moreover, the influences of microplastics, nutrients, and light intensity are highly intertwined. The presence of microplastics amplifies the constraints on grazer growth related to food quality and quantity imposed by extreme light intensities, while elevated phosphorus input enhances the system's resistance to intense light conditions. Furthermore, higher environmental microplastic levels do not always imply elevated microplastic body burdens in organisms, as organisms are also influenced by nutrients and light. We also find that grazers are more vulnerable to microplastics, compared to producers. If producers can utilize microplastics for growth, the system displays significantly greater resilience to microplastics.

Dynamics of competition model between two plants based on stoichiometry

The dynamics of two-plant competitive models have been widely studied, while the effect of chemical heterogeneity on competitive plants is rarely explored. In this study, a model that explicitly incorporates light and total phosphorus in the system is formulated to characterize the impacts of limited carbon and phosphorus on the dynamics of the two-plant competition system. The dissipativity, existence and stability of boundary equilibria and coexistence equilibrium are proved, when the two plants compete for light equally. Our simulations indicate that, with equal competition for light ($ b_{12} = b_{21} $) and a fixed total phosphorus in the system ($ T $), plants can coexist with moderate light intensity ($ K $). A higher $ K $ tends to favor the plant with a lower phosphorus loss rate ($ d_1 $ vs $ d_2 $). When $ K $ is held constant, a moderate level of $ T $ leads to the dominance of the plant with a lower phosphorus loss rate ($ d_1 $ vs $ d_2 $). At high $ T $ levels, both plants can coexist. Moreover, our numerical analysis also shows that, when the competition for light is not equal, the low level of total phosphorus in the system may lead the model to be unstable and have more types of bistability compared with the two-dimensional Lotka-Volterra competition model.

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.

Stoichiometry and environmental change drive dynamical complexity and unpredictable switches in an intraguild predation model

We incorporate stoichiometry (the balance of key elements) into an intraguild predation (IGP) model. Theoretical and numerical results show that our system exhibits complex dynamics, including chaos and multiple types of both bifurcations and bistability. Types of bifurcation present include saddle-node, Hopf, and transcritical bifurcations, and types of bistability present include node-node, node-cycle, and cycle-cycle bistability; cycle-cycle bistability has never been observed in IGP ordinary differential equation models. Stoichiometry can stabilize or destabilize the system via the disappearance or appearance of chaos. The species represented in the model can coexist for moderate levels of light intensity and nutrient availability. When the amount of light or nutrients present is extremely high or low, coexistence of the species becomes impossible, potentially harming biodiversity. Interestingly, stoichiometry can facilitate the re-emergence of severely endangered species as light intensity increases. In a temporally changing environment, the system can jump between different unstable states following changes in light intensity, with the trajectory followed depending strongly on initial conditions.

The microbial communities (bacteria, algae, zooplankton, and fungi) improved biofloc technology including the nitrogen-related material cycle in Litopenaeus vannamei farms

Microbes are essential in biofloc technology for controlling nitrogen levels in water. The composition and function of microorganisms with biofloc systems were reported; however, data on microorganisms other than bacteria, such as algae (which are essential in the nitrogen cycle) and zooplankton (which are bacterial and algal predators), remain limited. The microbial communities (including bacteria, algae, zooplankton, and fungi) were investigated in shrimp farms using biofloc technology. Using Illumina MiSeq sequencing, the V4 region of 18S rRNA and the V3-V4 region of 16S rRNA were utilized for the analysis of the eukaryotic and prokaryotic microbial communities. As a result, it was found that the biofloc in the shrimp farm consisted of 48.73%-73.04% eukaryotic organisms and 26.96%-51.27% prokaryotic organisms. In these shrimp farms, prokaryotic microbial communities had higher specie richness and diversity than eukaryotic microbial communities. However, the eukaryotic microbial communities were more abundant than their prokaryotic counterparts, while algae and zooplankton dominated them. It was discovered that the structures of the microbial communities in the shrimp farms seemed to depend on the effects of predation by zooplankton and other related organisms. The results provided the nitrogen cycle in biofloc systems by the algal and bacterial groups in microbial communities.

Contrasting stoichiometric dynamics in terrestrial and aquatic grazer-producer systems

The turnover rate of producer biomass in aquatic ecosystems is generally faster than in terrestrial. That is, aquatic producer biomass grows, is consumed, and is replaced considerably faster than terrestrial. The WKL model describes the flow of phosphorus and carbon through a grazer-producer system, hence varying the model parameters allows for analysis of different ecosystems of this type. Here we explore the impacts of the intrinsic growth rate of the producer and the maximal ingestion rate of the grazer on these dynamics because these parameters determine turnover rate. Simulations show that for low intrinsic growth rate and maximal ingestion rate, the grazer goes extinct; for higher values of these parameters, coexistence occurs in oscillations. Sensitivity analysis reveals the relative importance of all parameters on asymptotic dynamics. Lastly, the impacts of changing these two parameters in the LKE model appears to be quantitatively similar to the impacts in the WKL model.

Stoichiometric Ecotoxicology for a Multisubstance World

Nutritional and contaminant stressors influence organismal physiology, trophic interactions, community structure, and ecosystem-level processes; however, the interactions between toxicity and elemental imbalance in food resources have been examined in only a few ecotoxicity studies. Integrating well-developed ecological theories that cross all levels of biological organization can enhance our understanding of ecotoxicology. In the present article, we underline the opportunity to couple concepts and approaches used in the theory of ecological stoichiometry (ES) to ask ecotoxicological questions and introduce stoichiometric ecotoxicology, a subfield in ecology that examines how contaminant stress, nutrient supply, and elemental constraints interact throughout all levels of biological organization. This conceptual framework unifying ecotoxicology with ES offers potential for both empirical and theoretical studies to deepen our mechanistic understanding of the adverse outcomes of chemicals across ecological scales and improve the predictive powers of ecotoxicology.

Stoichiometric Modeling of Aboveground-Belowground Interaction of Herbaceous Plant and Two Herbivores

In a grassland ecosystem, the dynamics and coexistence mechanisms of two herbivores competing for one herbaceous plant have been widely studied, while the chemical heterogeneity of herbaceous plant's aboveground and belowground parts is usually ignored in dynamic modeling. Based on the traditional two herbivore-one herbaceous plant competition model, a new stoichiometric competition model, which incorporates the chemical heterogeneity of herbaceous plants, is formulated to investigate effects of the aboveground-belowground interactions and the chemical heterogeneity on the dynamics of the two herbivore-one herbaceous plant system. We perform theoretical analysis for the stability of boundary equilibria and show that a stable coexistent equilibrium is possible with two herbivores on one herbaceous plant. Moreover, numerical simulations reveal that various light intensity and nitrogen input can also allow all populations to coexist in periodic oscillations or irregularly cyclic oscillations. Our findings further indicate that when the nitrogen input is fixed, higher light intensity leads to a dominance of the lower N-demand herbivore, while the light intensity is fixed, higher nitrogen input leads to a dominance of the higher N-demand herbivore. Moderate levels of light and nutrient could promote the coexistence of two herbivores and herbaceous plant. This study also explains the functional mechanism for the decline of species diversity in response to nitrogen enrichment.

Coexistence and extinction in a data-based ratio-dependent model of an insect community

In theory, pure competition often leads to competitive exclusion of species. However, what we often see in nature is a large number of distinct predator or consumer species coexist in a community consisting a smaller number of prey or plant species. In an effort of dissecting how indirect competition and selective predation may have contributed to the coexistence of species in an insect community, according to the replicated cage experiments (two aphid species and a specialist parasitoid that attacks only one of the aphids) and proposed mathematical models, van Veen et. al. [5] conclude that the coexistence of the three species is due to a combination of density-mediated and trait-mediated indirect interactions. In this paper, we formulate an alternative model that observes the conventional law of mass conservation and provides a better fitting to their experimental data sets. Moreover, we present an initial attempt in studying the stabilities of the nonnegative steady states of this model.

The effects of excess food nutrient content on a tritrophic food chain model in the aquatic ecosystem

Ecological stoichiometry is an approach that focuses on the balance of energy and elements in environmental interactions, and it leads to new insights and a better understanding of ecological processes and outcomes. Modeling under this framework enables us to investigate the effects of nutrient content (i.e., food quality) on organisms, whether the imbalance involves insufficient or excess nutrient content. In this paper, we develop and analyze a tritrophic food chain model that captures the phenomenon known as the "stoichiometric knife-edge", where consumer growth is limited under conditions of excess nutrients. The model tracks two essential elements, carbon and phosphorus, in each species. The dynamics of the system such as boundedness and positivity of the solutions, existence and stability conditions of boundary and internal equilibria are analyzed. Through numerical simulations and bifurcation analyses, we observe the dynamics of the system switching between periodic oscillations and chaos. Our findings also show that nutrient-rich food consumption can cause adverse effects on species.

Environmental seasonality on predator-prey systems under nutrient and toxicant constraints

Bioaccumulation of toxicants in aquatic food webs can pose risks to ecosystem function and human health. Toxicant models of aquatic ecosystems can be improved by incorporating realistic environmental impacts such as nutrient availability and seasonality. It is well known that the carrying capacity of predator-prey systems can vary seasonally due to environmental cycles resulting from natural and human activities. As such, incorporating seasonal variation in the carrying capacity of a predator-prey system provides a better understanding of the underlying population dynamics bioaccumulation of toxicants. Here, we develop a seasonally varied predator-prey model subject to concurrent nutrient and toxicant stressors. We investigate the effects of seasonality on population dynamics to increase understanding of the complex governing processes of the trophic transfer of nutrients, energy, and toxicants. We observe that the strength of seasonality can shift solutions from periodic to quasi-periodic and models that neglect environmental seasonality may be under-predicting adverse effects of toxicity.

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.

Genotypic Selection in Spatially Heterogeneous Producer-Grazer Systems Subject to Stoichiometric Constraints

Various environmental conditions may exert selection pressures leading to adaptation of stoichiometrically important traits, such as organismal nutritional content or growth rate. We use theoretical approaches to explore the connections between genotypic selection and ecological stoichiometry in spatially heterogeneous environments. We present models of a producer and two grazing genotypes with different stoichiometric phosphorus/carbon ratios under spatially homogenous and heterogeneous conditions. Numerical experiments predict that selection of a single genotype, co-persistence of both genotypes, and extinction are possible outcomes depending on environmental conditions. Our results indicated that in spatially homogenous settings, co-persistence of both genotypes can occur when population dynamics oscillate on limit cycles near a key stoichiometric threshold on food quality. Under spatially heterogeneous settings, dynamics are more complex, where co-persistence is observed on limit cycles, as well as stable equilibria.

Eutrophication, harmful algae and biodiversity - Challenging paradigms in a world of complex nutrient changes

Eutrophication is a complex process and often associated with not only a change in overall algal biomass but also with a change in biodiversity. Common metrics of eutrophication (e.g., chlorophyll a), total nitrogen (TN) and phosphorus (TP) are not adequate for understanding biodiversity changes, especially those associated with harmful algal bloom (HAB) proliferations. Harmful algae can increase disproportionately with eutrophication, depending on which nutrients change and in what proportion. This paper challenges several classic paradigms in our understanding of eutrophication and associated biodiversity changes. The underlying message is that nutrient proportions and forms can alter biodiversity, even when nutrients are at concentrations in excess of those considered limiting. The global HAB problem is on a trajectory for more blooms, more toxins, more often, in more places. Our approach to management of HABs and eutrophication must consider the broader complexity of nutrient effects at scales ranging from physiological to ecological.