Food Chain Model Research Articles

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

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

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

Martingale solutions and asymptotic behaviors for a stochastic cross-diffusion three-species food chain model with prey-taxis.

The stochastic food chain model is an important model within the field of ecological research. Since existing models are difficult to describe the influence of cross-diffusion and random factors on the evolution of species populations, this work is concerned with a stochastic cross-diffusion three-species food chain model with prey-taxis, in which the direction of predators' movement is opposite to the gradient of prey, i.e., a higher density of prey. The existence and uniqueness of martingale solutions are established in a Hilbert space by using the stochastic Galerkin approximation method, the tightness criterion, Jakubowski's generalization of the Skorokhod theorem, and the Vitali convergence theorem. Furthermore, asymptotic behaviors around the steady states of the stochastic cross-diffusion three-species food chain model in the time mean sense are investigated. Finally, numerical simulations are carried out to illustrate the results of our analysis.

Partial tipping in bistable ecological systems under periodic environmental variability.

Periodic environmental variability is a common source affecting ecosystems and regulating their dynamics. This paper investigates the effects of periodic variation in species growth rate on the population dynamics of three bistable ecological systems. The first is a one-dimensional insect population model with coexisting outbreak and refuge equilibrium states, the second one describes two-species predator-prey interactions with extinction and coexistence states, and the third one is a three-species food chain model where chaotic and limit cycle states may coexist. We demonstrate with numerical simulations that a periodic variation in species growth rate may cause switching between two coexisting attractors without crossing any bifurcation point. Such a switchover occurs only for a specific initial population density close to the basin boundary, leading to partial tipping if the frozen system is non-chaotic. Partial tipping may also occur for some initial points far from the basin boundary if the frozen system is chaotic. Interestingly, the probability of tipping shows a frequency response with a maximum for a specific frequency of periodic forcing, as noticed for equilibrium and non-equilibrium limit cycle systems. The findings suggest that unexpected outbreaks or abrupt declines in population density may occur due to time-dependent variations in species growth parameters. Depending on the selective frequency of the periodic environmental variation, this may lead to species extinction or help the species to survive.

Dynamics of a Stochastic Regime-Switching Four-Species Food Chain Model with Distributed Delays and Harvesting

Dynamics of a Stochastic Regime-Switching Four-Species Food Chain Model with Distributed Delays and Harvesting

Effect of fear with saturated fear cost and harvesting on aquatic food chain model (plankton–fish model) in the presence of nanoparticles

Studying the interplay of phytoplankton–zooplankton–fish (PP–ZP–F) in an aquatic system is crucial for better understanding of nutrient cycling, assessing ecosystem health, predicting and mitigating harmful algal blooms, and managing fisheries in the water bodies. In order to investigate the effectiveness of nanoparticles (NPs), fear, and harvesting, this paper focuses on exploring the dynamics of a food chain model among PP–ZP–F species. We consider the fear of fish on zooplankton species (which reduces the reproduction rate of ZPs) with saturated fear cost in the presence of nanoparticles (NPs) and harvesting in fish. The system dynamics are studied from the viewpoint of proving positivity, boundedness, and uniqueness, followed by analysing the existence and local stability of biologically feasible equilibria. Conditions for the global stability of the interior equilibrium point are also found. Furthermore, we established the transversality conditions for the occurrence of Hopf, transcritical, and saddle–node bifurcations. To validate our theoretical results, we made numerous phase portraits, time-series graphs, tables showing the extinction of species, and bifurcation diagrams. It is numerically observed that increasing the contact rate of NPs with PPs makes the system stable from chaos, and further increase of contact rate may lead to extinction. Chaos at a low contact rate can also be managed by increasing the fear level, and the chaotic behaviour at a low fear level can again be controlled by enhancing the harvesting of fish species. Over-exploitation may result in the extinction of fish, whereas fear may promote coexistence, stability, and long-term survival of the species. Increased saturated fear cost can make the system chaotic from stable dynamics. Therefore, the theoretical as well as numerical findings of our paper may be of great interest in estimating the behaviour of aquatic systems biologically and practically.

Using algae and brine shrimp as food chain model for bioaccumulation and biomagnification of lead and cadmium

Using algae and brine shrimp as food chain model for bioaccumulation and biomagnification of lead and cadmium

Chaos and extinction risks of sexually reproductive generalist top predator in a seasonally forced food chain system with Allee effect.

This paper investigates the dynamics of a tritrophic food chain model incorporating an Allee effect, sexually reproductive generalist top predators, and Holling type IV and Beddington-DeAngelis functional responses for interactions across different trophic levels. Analytically, we explore the feasible equilibria, their local stability, and various bifurcations, including Hopf, saddle-node, transcritical, and Bogdanov-Takens bifurcations. Numerical findings suggest that higher Allee intensity in prey growth leads to the inability of species coexistence, resulting in a decline in species density. Likewise, a lower reproduction rate and a higher strength of intraspecific competition among top predators also prevent the coexistence of species. Conversely, a rapid increase in the reproduction rate and a decrease in the strength of intraspecific competition among top predators enhance the densities of prey and top predators while decreasing intermediate predator density. We also reveal the presence of bistability and tristability phenomena within the system. Furthermore, we extend our autonomous model to its nonautonomous counterpart by introducing seasonally perturbed parameters. Numerical analysis of the nonautonomous model reveals that higher seasonal strength in the reproduction rate and intraspecific competition of top predators induce chaotic behavior, which is also confirmed by the maximum Lyapunov exponent. Additionally, we observe that seasonality may lead to the extinction of species from the ecosystem. Factors such as the Allee effect and growth rate of prey can cause periodicity in population densities. Understanding these trends is critical for controlling changes in population density within the ecosystem. Ecologists, environmentalists, and policymakers stand to benefit significantly from the invaluable insights garnered from this study. Specifically, our findings offer pivotal guidance for shaping future strategies aimed at safeguarding biodiversity and maintaining ecological stability amidst changing environmental conditions. By contributing to the existing body of knowledge, our study advances the field of ecological science, enhancing the comprehension of predator-prey dynamics across diverse ecological conditions.

Periodic solutions of an NPZ model with periodic delay and space heterogeneity

In this paper, we investigate a nutrient-phytoplankton-zooplankton (NPZ) model with seasonality and spatial heterogeneity, in which the immature zooplankton with size structure is described by the quasi-linear first-order partial differential equation. A three-dimensional periodic delay food chain model with initial value is derived from the original model. To establish the threshold dynamics of NPZ model, we first obtain the global attractivity of the positive periodic solution of the nutrient-phytoplankton (NP) subsystem by using the method of sub-super solutions and associated iterations. We then give the integral expression of threshold for the corresponding system, which is convenient for numerical simulations. Based on this, the threshold dynamics is discussed by appealing to the comparison arguments and the persistence theory. Finally, the analytic results are in good consistence with our numerical simulations.

Killing the predator: impacts of highest-predator mortality on the global-ocean ecosystem structure

Abstract. Recent meta-analyses suggest that microzooplankton biomass density scales linearly with phytoplankton biomass density, suggesting a simple, general rule may underpin trophic structure in the global ocean. Here, we use a set of highly simplified food web models, solved within a global general circulation model, to examine the core drivers of linear predator–prey scaling. We examine a parallel food chain model which assumes microzooplankton grazers feed on distinct size classes of phytoplankton and contrast this with a diamond food web model allowing shared microzooplankton predation on a range of phytoplankton size classes. Within these two contrasting model structures, we also evaluate the impact of fixed vs. density-dependent microzooplankton mortality. We find that the observed relationship between microzooplankton predators and prey can be reproduced with density-dependent mortality on the highest predator, regardless of choices made about plankton food web structure. Our findings point to the importance of parameterizing mortality of the highest predator for simple food web models to recapitulate trophic structure in the global ocean.

Chaos in a seasonal food-chain model with migration and variable carrying capacity

Chaos in a seasonal food-chain model with migration and variable carrying capacity

Complex Dynamics of a Discrete-Time Food Chain Model

In a food chain, the role of intake patterns of predators is very influential on the survival and extinction of the interacting species as well as the entire dynamics of the ecological system. In this study, we investigate the affluent and intricate dynamics of a simple three-species food chain model in a discrete-time framework by analyzing the parameter plane of the system with simultaneous changes of two crucial parameters, the predation rates of middle and top predators. From the theoretical viewpoint, we study the model by determining the fixed points’ biological feasibility and local asymptotic stability criteria, and performing some analyses of local bifurcations, namely, transcritical, flip, and Neimark–Sacker bifurcations. Here, we initiate the numerical simulation by plotting the changes of the prey population density in terms of a vital parameter of the system, and we observe the switching among different dynamical behaviors of the system. We also draw some phase portraits and plot the time series solutions to show the diverse characteristics of the system dynamics. Further, we move one step ahead to explore the intricate dynamical scenarios appearing in the parameter plane by forming Lyapunov exponent and isoperiodic diagrams. In the parameter plane of the system, we see the emergence of innumerable Arnold tongues. All these Arnold tongues are organized along a particular direction, and the beautiful arrangement of these tongues forms several kinds of period-adding sequences. The study sheds more light on various types of multistability occurring in the model system. We see the coexistence of three periodic attractors in the parameter plane. In this study, the most striking observation is the coexistence of four periodic attractors, which occurs infrequently in ecological systems. We draw the basins of attraction for the tristable and tetrastable attractors, which are complex Wada basins. A system with Wada basin is very sensitive to initial conditions and more erratic in nature than a system with fractal basin. Also, we plot the density of all interacting species in terms of the predation rates of middle and top predators and observe the variation in the population densities of all species with the variability of these two key parameters. In the parameter plane created by the simultaneous changes of two parameters, the system exhibits a variety of intricate and subtle dynamics, which cannot be found by changing only a single parameter.

Chaotic dynamics of a three-species food chain model

An analysis of a discrete-time food chain model is presented in this paper to investigate the effects of strong prey pressure on the chain. A bifurcation and stability theory is used to analyze equilibrium formation in system. It is shown by numerical simulations that chaos occurs as a result of the bifurcation of invariant curves following Neimark-Sacker (NS) bifurcation rules. The OGY method on stable periodic orbits of period 1 further controls chaos. Stability is achieved through iterations for chaotic motion controlled by different regulator poles. Numerical simulations are performed to examine the effects of the control method along with the results of our theoretical analysis.