Food Chain Model Research Articles

Exploring quasi-periodic shrimps in the parameter space of a discrete-time food chain model.

Shrimps are islands of regularity within chaotic regimes in bi-parameter spaces of nonlinear dynamical systems. While the presence of periodic shrimps has been extensively reported, recent research has uncovered the existence of quasi-periodic shrimps. Compared to their periodic counterparts, quasi-periodic shrimps require a relatively higher-dimensional phase-space to come into existence and are also quite uncommon to observe. This Focus Issue contribution delves into the existence and intricate dynamics of quasi-periodic shrimps within the parameter space of a discrete-time, three-species food chain model. Through high-resolution stability charts, we unveil the prevalence of quasi-periodic shrimps in the system's unsteady regime. We extensively study the bifurcation characteristics along the two borders of the quasi-periodic shrimp. Our analysis reveals that along the outer border, the system exhibits transition to chaos via intermittency, whereas along the inner border, torus-doubling and torus-bubbling phenomena, accompanied by finite doubling and bubbling cascades, are observed. Another salient aspect of this work is the identification of quasi-periodic accumulation horizon and different quasi-periodic (torus) adding sequences for the self-distribution of infinite cascades of self-similar quasi-periodic shrimps along the horizon in certain parameter space of the system.

Dynamics of Fractional-Order Three-Species Food Chain Model with Vigilance Effect

Dynamics of Fractional-Order Three-Species Food Chain Model with Vigilance Effect

Population oscillations in a three-species food chain model and their possible control: a Z-type control approach

Population oscillations in a three-species food chain model and their possible control: a Z-type control approach

Global Boundedness of Solutions to a Food Chain Model with Nonlinear Taxis Sensitivity

Global Boundedness of Solutions to a Food Chain Model with Nonlinear Taxis Sensitivity

Evidence for low bioavailability of dietary nanoparticulate cerium in a freshwater food chain.

Radioactive 141Ce in ionic (I-Ce), nano (N-Ce, 11 ± 9 nm mean primary particle size ± standard deviation) and micron-sized (M-Ce, 530 ± 440 µm) forms associated with natural and artificial diets in natural river water and synthetic freshwater were used to measure the real-time biokinetics of dietary 141Ce assimilation in a freshwater food chain. The model food chain consisted of microalgae (Raphidocelis subcapitata), snails (Potamopyrgus antipodarum) and prawns (Macrobrachium australiense). Pulse-chase experiments showed that 91-100 % of all forms of cerium associated with all diets and water types were eliminated from the digestive system of the snail and prawn within 24 h, with no detectable cerium assimilation. The prawn and snail median elimination times (ET50) and elimination rates (Ke) for all cerium forms ranged from 0.05 to 1.7 d, and 30 to >100 % per d, respectively. The pulse-chase results were supported by the autoradiographic evidence for N-Ce and M-Ce that confirmed no detectable assimilation and translocation within the tissue of the prawn over time. In contrast, the more soluble I-Ce was found to be associated in low quantities with the hepatopancreas in the prawn confirming that the lack of dissolution by N-Ce and M-Ce in the digestive environment of these organisms makes these forms less bioavailable. In addition, hetero-agglomeration of N-Ce and M-Ce resulted in particles that did not dissociate in digestive fluids and were too large to be assimilated thereby making them non-bioavailable. Based on the results from this study and from the literature review, the risk of N-Ce biomagnification and chronic dietary toxicity in freshwater ecosystems is no greater than the risk associated with M-Ce or I-Ce.

B-tipping points in plankton dynamics: Stochasticity and early warning signals.

Near a tipping point, a critical transition occurs when small changes in input conditions lead to abrupt, often irreversible shifts in a dynamical system's state. This phenomenon is observed in various biological and physical systems, including the collapse of species in ecosystems. Several statistical indicators, known as early warning signals (EWSs), have been developed to anticipate such collapses, garnering significant attention for their broad applicability. This paper investigates the stochastic versions of a bistable algae-zooplankton food-chain model under demographic and environmental noise. Our findings show that an increase in the predatory fish population, which consumes zooplankton, triggers a collapse in zooplankton abundance through a saddle-node bifurcation. Basin stability measure reveals that the resilience of the underexploited steady state significantly diminishes as the system approaches the collapse point. We evaluate the efficacy of various generic EWSs in predicting sudden collapses under both types of noise through statistical analysis. The robustness of AR(1) and variance are assessed through a comprehensive sensitivity analysis of processing parameters. We also calculate conditional heteroskedasticity, which minimizes false positive signals in the time series. Our results indicate that the prediction accuracy of variance and conditional heteroskedasticity remains independent of the noise type. However, AR(1) and skewness perform better in the presence of environmental noise.

Energy transfer efficiency rather than productivity determines the strength of aquatic trophic cascades.

Trophic cascades are important determinants of food web dynamics and functioning, yet mechanisms accounting for variation in trophic cascade strength remain elusive. Here, we used food chain models and a mesocosm experiment (phytoplankton-zooplankton-shrimp) to disentangle the relative importance of two energetic processes driving trophic cascades: primary productivity and energy transfer efficiency. Food chain models predicted that the strength of trophic cascades was increased as the energy transfer efficiency between herbivore and predator (predator efficiency) increased, while its relationship with primary productivity was relatively weak. These model predictions were confirmed by a mesocosm experiment, which showed that the strength of trophic cascade increased with predator efficiency but remained unaffected by nutrient supply rate or primary productivity. Combined, our results indicate that the efficiency of energy transfer along the food chain, rather than the total amount of energy fixed by primary producers, determines the strength of trophic cascades. Our study provides an integrative perspective to reconcile energetic and population dynamics in food webs, which has implications for both ecological research and ecosystem management.

Uncertainty Analysis in Ecological Risk Assessment: Sensitivity Analysis of Different Exposure Models

Ecological risk assessments are used for decision making regarding releasing new products to market as well as remedial action required to address contamination at legacy sites. Uncertainty analysis is considered a key part of the process (USEPA 1997) but assessments seldom address uncertainty quantitatively. Key sources of uncertainty include: Sampling and analytical variability (soil/ sediment matrix, lab error), Choice of “Indicator” Species as Target Receptors for Different Exposure Pathways, Sample size. Bioaccumulation and Food Chain Modeling involve assumptions and uncertainty including: Linear assumption of bioaccumulation and selection of bioaccumulation factors; Other model inputs such as: Home range size, Dietary percentages, Body Weights, Ingestion Rate; Literature data on effects, Use of No Observed Adverse Effects Level (NOAEL) or Lowest Observed Adverse Effects Level (LOAEL) as a decision point. Toxicity testing where employed brings its own set of assumptions and interpretation that are subject to uncertainty. These include: Use of single organism, Correlation (or lack thereof) with Contaminants of Concern, Contaminant mixtures, and Reference area selection. Keywords: Ecotoxicology, Risk Assessment, Ecology

The human health risks and corresponding body thresholds of OCPs and PCBs in the Tibetan Plateau

Although organochlorine pesticides (OCPs) and polychlorinated biphenyls (PCBs) have been banned or restricted for years, they remain significant contaminants posing health risks in some places where are favorable for their enrichment, such as areas with high latitude or altitudes. However, the elucidation of environmental risks and the establishment of a clear relationship between these risks and human health are areas that remain significantly underdeveloped. Literatures on the concentrations of OCPs and PCBs in soil of the Tibetan Plateau were reviewed to calculate local health risk indexes and create risk maps. A food chain model (ACC-Human) was applied to predict human body concentrations of these chemicals. Higher risk areas are located in the southeastern Tibetan Plateau, suggesting influences from potential emissions in adjacent regions and long-range transport. Based on the concentration-risk relationships, the most toxic chemicals identified were PCBs, with a body concentration threshold as low as 13 ng/g lipid. Given the ongoing emissions from industrial production or historical legacy, it is hightime to lay a particular focus on PCBs in mountainous areas like the Tibetan Plateau.

Bifurcation Analysis and Multiobjective Nonlinear Model Predictive Control of Sustainable Ecosystems

<i>Objective: </i>All optimal control work involving ecological models involves single objective optimization. In this work, we perform multiobjective nonlinear model predictive control (MNLMPC) in conjunction with bifurcation analysis on an ecosystem model. <i>Methods: </i>Bifurcation analysis was performed using the MATLAB software MATCONT while the multi-objective nonlinear model predictive control was performed by using the optimization language PYOMO. <i>Results: </i>Rigorous proof showing the existence of bifurcation (branch) points is presented along with computational validation. It is also demonstrated (both numerically and analytically) that the presence of the branch points was instrumental in obtaining the Utopia solution when the multiobjective nonlinear model prediction calculations were performed. <i>Conclusions: </i>The main conclusions of this work are that one can attain the utopia point in MNLMPC calculations because of the branch points that occur in the ecosystem model and the presence of the branch point can be proved analytically. The use of rigorous mathematics to enhance sustainability will be a significant step in encouraging sustainable development. The main practical implication of this work is that the strategies developed here can be used by all researchers involved in maximizing sustainability The future work will involve using these mathematical strategies to other ecosystem models and food chain models which will be a huge step in developing strategies to address problems involving nutrition.

Differences in Tri-Trophic Community Responses to Temperature-Dependent Vital Rates, Thermal Niche Mismatches and Temperature-Size Rule.

Warming climate impacts aquatic ectotherms by changes in individual vital rates and declines in body size, a phenomenon known as the temperature-size rule (TSR), and indirectly through altered species interactions and environmental feedbacks. The relative importance of these effects in shaping community responses to environmental change is incompletely understood. We employ a tri-trophic food chain model with size- and temperature-dependent vital rates and species interaction strengths to explore the role of direct kinetic effects of temperature and TSR on community structure along resource productivity and temperature gradients. We find that community structure, including the propensity for sudden collapse along resource productivity and temperature gradients, is primarily driven by the direct kinetic effects of temperature on vital rates and thermal mismatches between the consumer and predator species, overshadowing the TSR-mediated effects. Overall, our study enhances the understanding of the complex interplay between temperature, species traits and community dynamics in aquatic ecosystems.

Persistence and zero-Hopf equilibrium in the tritrophic food chain model with Holling functional response

In this paper, we analyze the persistence of three species in a three-level food chain model. We characterize when such a model exhibits a zero-Hopf equilibrium point and show that it is possible only if the functional responses in the model are of type Holling III or IV.

Zero‐Hopf bifurcation in a family of tritrophic food chain model with Holling III–III functional response

In this paper, we apply the second‐order averaging theory for obtaining an explicit expression of the small amplitude periodic solution that bifurcates from a zero‐Hopf equilibrium point of a tritrophic food chain model. This model considers logistic growth rate for the lowest trophic level, Holling functional responses type III for the middle and for the highest level. We first prove that this model has a zero‐Hopf equilibrium point, after we show that from this equilibrium bifurcates a small limit cycle, and finally, we provide the explicit expression of the first two terms in the power series of this limit cycle. These differential systems for which the equilibrium point is non hyperbolic are not easy to study, in particular, if the equilibrium is zero‐Hopf. As far as we know, this is the first time that the averaging theory has been used to exhibit the first and second terms of the power series expansion of the analytical expression of a limit cycle that bifurcates from a zero‐Hopf equilibrium in the food chain models.

Global boundedness and asymptotic stability for a food chain model with nonlinear diffusion

This paper is concerned with a food chain model with nonlinear diffusion ut = Δu + u(1 − u − b1v), vt=∇⋅((v+1)m∇v)−∇⋅(ξv∇u)+vu−b2w1+v+w−θ1−α1v,wt=∇⋅((w+1)l∇w)−∇⋅(χw∇v)+wv1+v+w−θ2−α2w in a smooth bounded domain Ω ⊂ Rn(n ≥ 2) with homogeneous Neumann boundary conditions, where the parameters ξ, χ, α1, bi, θi (i = 1, 2) > 0 and α2 ≥ 0 as well as m, l∈R. We study the global boundedness of classical solutions to the problem if either n = 2 and m ≥ 0, l > − 1 or n ≥ 3 and m>1−2n, l > − 1. Moreover, we prove the global stability of the prey-only steady state and semi-coexistence steady as well as coexistence steady states under certain conditions on parameters.

Bistability and bifurcations for a food chain model with nonlinear harvesting of top predator

In this paper, we study a prey–predator–top predator food chain model with nonlinear harvesting of top predator. We have derived two important thresholds: the top predator extinction threshold and the coexistence threshold. We found that the top predator will die out if the nonlinear harvesting from predator to top predator is larger than the top predator extinction threshold. On the other hand, the prey, predator and top predator coexist if the nonlinear harvesting from predator to top predator is less than the coexistence threshold. While the parameter value of nonlinear harvesting from predator to top predator is between two critical thresholds, the system displays bistability phenomena, implying that the top predator species either die out or exist with the prey and predator species, which largely depend on the initial condition. Thus, a bistable interval exists between two critical thresholds, which is a significant phenomenon for the model. Meanwhile, we performed bifurcation analysis for the model, showing that the system would arise backward/forward bifurcation and saddle-node bifurcation and Hopf bifurcation. Finally, we performed numerical simulations to verify the theoretical analysis.

Bifurcation analysis and exploration of noise-induced transitions of a food chain model with Allee effect

Seeking shelter from threats is a widespread instinct across species, specially employed by prey to avoid direct confrontations with predators. The present investigation centers on a three-species food chain model wherein the basal prey, characterized by logistic growth, seeks refuge to evade the intermediate predator, while the intermediate predator, in turn, seeks refuge to avoid encounters with the top predator. Additionally, our model assumes that the presence of the top predator induces a mate-finding Allee effect among the intermediate predator. We investigate how varying levels of refuge, along with other critical parameters such as the reproduction rate of the basal prey and the natural mortality rate of the top predator, influence system’s dynamics within the biparametric planes. Our model displays multistability and undergoes transcritical, saddle–node, Bogdanov–Takens and cusp bifurcations across different parameters. Moreover, the external environmental noise can induce interesting dynamics in the predator–prey system, resulting in noise-induced frequent transitions between distinct interior attractors or from interior to axial attractors. This phenomenon is particularly notable in scenarios where the deterministic model exhibits tristability. In summary, our findings offer potential new avenues for developing control strategies within the realm of community ecology in constant as well as fluctuating environments.

Temperature variability regulates the interactive effects of warming and pharmaceutical on aquatic ecosystem dynamics

Climate warming and pharmaceutical contaminants have profound impacts on population dynamics and ecological community structure, yet the consequences of their interactive effects remain poorly understood. Here, we explore how climate warming interacts with pharmaceutical-induced boldness change to affect aquatic ecosystems, built on an empirically well-informed food-chain model, consisting of a size-structured fish consumer, a zooplankton prey, and a fish predator. Climate warming is characterized by both daily mean temperature (DMT) and diurnal temperature range (DTR) in our model. Results show that DMT and high levels of species’ boldness are the primary drivers of community instability. However, their interactive effects can lead to diverse outcomes: from predator collapse to coexistence with seasonality-driven cycles and coexistence with population interaction-driven cycles. The interactive effects are significantly modulated by daily temperature variability, where moderate DTR counteracts the destabilizing interactive effects by increasing consumer reproduction, while large temperature variability considerably reduces consumer biomass, destabilizing the community at high mean temperatures. Our analyses disentangle the respective roles of DMT, DTR and boldness in mediating the response of aquatic ecosystems to the impacts from pharmaceutical contaminants in the context of climate warming. The interactive effects of the environmental stressors reported here underscore the pressing need for studies aimed at quantifying the cumulative impacts of multiple environmental stressors on aquatic ecosystems.

Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations.

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.

Global existence and uniform boundedness in a diffusive food chain model with direct and indirect prey‐taxis

In this paper, we propose a food chain model in which the primary predator moves directly toward areas of high prey density. Simultaneously, the primary predator, which serves as the prey for the secondary predator, indirectly influences the directional movements of the secondary predator through cues such as chemical signals, scents, or excretions. We investigate whether the distinct influences of direct taxis and indirect taxis, as observed in prey–predator dynamics, are also manifested in the proposed food chain model. Our study demonstrates that the model, which incorporates both direct and indirect prey‐taxis, possesses bounded and global solutions up to three‐dimensional space.

Qualitative Properties and Optimal Control Strategy on a Novel Fractional Three-Species Food Chain Model

Qualitative Properties and Optimal Control Strategy on a Novel Fractional Three-Species Food Chain Model