Three-species Food Chain Model Research Articles

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.

Stability and Bifurcation Control for a Generalized Delayed Fractional Food Chain Model

In this paper, a generalized fractional three-species food chain model with delay is investigated. First, the existence of a positive equilibrium is discussed, and the sufficient conditions for global asymptotic stability are given. Second, through selecting the delay as the bifurcation parameter, we obtain the sufficient condition for this non-control system to generate Hopf bifurcation. Then, a nonlinear delayed feedback controller is skillfully applied to govern the system’s Hopf bifurcation. The results indicate that adjusting the control intensity or the control target’s age can effectively govern the bifurcation dynamics behavior of this system. Last, through application examples and numerical simulations, we confirm the validity and feasibility of the theoretical results, and find that the control strategy is also applicable to eco-epidemiological systems.

Dynamics of a reaction–diffusion three-species food chain model: Effect of space-time white noise

This paper is devoted to analyzing the influence of space-time white noise on the dynamics of biological mathematical models in spatiotemporal scenarios based on stochastic partial differential equations (SPDEs). Here, we propose a stochastic reaction–diffusion three-species food chain model with various functional response functions. The motivation of the SPDEs model construction is mainly twofold: (1) when species migrate in space, space-time white noise is a factor that cannot be ignored; (2) functional response function is an essential indicator for directly measuring the relationship between species in biological mathematical models. To discuss the long-term behaviors of the SPDEs model in spatial heterogeneous environment, we first analyze the well-posedness of mild and strong solutions, respectively. Then, the critical conditions for determining the permanence and extinction of populations are given, which reveal that high-intensity white noise can lead to the extinction of predator populations. The validity of the theoretical results is also performed by numerical simulation. In particular, this new theoretical method may be extended to the study of other stochastic biological mathematical models in spatiotemporal scenarios, such as eco-epidemiological models, high-dimensional population models and epidemic models.

An investigation of the parameter space in a tri-trophic food chain model with refuge

Aside from humans, the instinct to run and hide from dangerous situations is ingrained in individuals of many other species. Taking refuge is a very common precautionary strategy among prey species to avoid direct contact with their predators. In this article, we investigate a three-species food chain model where the logistically growing basal prey seeks refuge to avoid the middle predator. To assess the impact of varying levels of refuge, we explore the system dynamics in the biparametric space of refuge and another important system parameter related to the half-saturation constant of the basal prey. The isospike diagrams along with the largest Lyapunov exponent diagrams show a stabilizing role of refuge. Apart from this, we observe an abundance of organized periodic structures embedded inside the chaotic zone of the biparametric space. The most notable ones include the shrimp-shaped structures and an “elliptic disc” shaped structure. The article also explores the coexistence between multiple pairs of coexisting attractors and one instance of spike-bubbling phenomena. The basins of attraction for the coexisting attractors also have a self-similar pattern. We also modify the present model by replacing the density-dependent prey refuge with constant proportion of prey refuge, and then explore the dynamics of the modified model within the same biparametric space and find similar complexity in its dynamics. We compare the effects of both types of refuge on the density variations of all three species when they show stable coexistence and observe that density-dependent refuge is more beneficial to the system from the standpoint of species conservation.

Chaos and Bistabilities in a Food-Chain Model with Allee Effect and Additional Food

In this research article, a three-species food chain model with Allee effect and additional food is proposed and analyzed. The Allee effect and additional food are introduced to the top predator population. The dynamical behavior of the system is studied analytically and numerically. We have performed equilibrium analysis and local stability analysis around the non-negative equilibria. We have also explored different bifurcations in the system. We have drawn several one- and two-parameter bifurcation diagrams to explore complex dynamical behaviors. We observe that top predator goes to extinction when Allee parameter crosses a threshold value, whereas additional food enhances the stability and persistence of the system.

Exploring the Evolution of the Food Chain under Environmental Pollution with Mathematical Modeling and Numerical Simulation

Environmental pollution has led to many ecological issues, including air, water, and soil contamination. Developing appropriate pollution control measures to mitigate these hazards and protect our environment is critical. In that respect, we developed a mathematical model to study the evolution of ecosystems containing food chains under environmental pollution. We integrate environmental pollution into a three-species food chain model, which includes a prey population, an intermediate predator population, and an apex predator population. The equilibrium points of the model are obtained and we analyze their stability. Numerical simulations are carried out to explore the dynamics of the model. The simulation results show that the model presents complex, chaotic, dynamic behaviors. Our study demonstrates that the interactions of individual populations in the food chain and the effects of environmental pollution can result in complex dynamics. The investigation provides insights into the evolution of the food chain in a polluted environment. Our research shows that pollution can disturb the equilibrium in nature, leading to complex and chaotic effects. Reducing environmental pollution can restore the food chain to an orderly state. Environmental pollution will harm the healthy development of each species in the ecosystem. Reducing pollution and restoring each species’ habitats are effective strategies for restoring a healthy ecosystem. Natural ecosystems are often polluted by domestic and industrial sources. The environmental protection department should allocate more resources to address domestic pollution and enhance domestic wastewater treatment methods. Industrial pollution can be reduced by encouraging companies to invest in treating wastewater and waste gases. It is also vital to prevent the establishment of highly polluting industries in environmentally sensitive environments.

Structure of parameter space of a three-species food chain model with immigration and emigration

Structure of parameter space of a three-species food chain model with immigration and emigration

A remark on “Dynamical behavior of a fractional three-species food chain model” [Nonlinear Dynamics, 95, February 2019

A remark on “Dynamical behavior of a fractional three-species food chain model” [Nonlinear Dynamics, 95, February 2019

Effect of Environmental Fluctuation in the Dynamics of a Three-Species Food Chain Model with Sexually Reproductive Generalized Type Top Predator and Crowley-Martin Type Functional Response Between Predators

Effect of Environmental Fluctuation in the Dynamics of a Three-Species Food Chain Model with Sexually Reproductive Generalized Type Top Predator and Crowley-Martin Type Functional Response Between Predators

Designing Meyer wavelet neural networks for the three-species food chain model

<abstract> <p>The current research work is related to present the numerical solutions of three-species food chain model (TS-FCM) by exploiting the strength of Meyer wavelet neural networks (MWNNs) along with the global and local search competencies. The particle swarm optimization technique works as a global operator, while the sequential quadratic programming scheme is applied as a local operator for the TS-FCM. The nonlinear TS-FCM is dependent upon three categories, called consistent of prey populations, specialist predator and top predator. The optimization of an error-based fitness function is presented by using the hybrid computing efficiency of the global and local search schemes, which is designed through the differential form of the designed ordinary differential model and its initial conditions. The proposed results of the TS-FCM are calculated through the stochastic numerical techniques and further comparison is performed by the Adams method to check the exactness of the scheme. The absolute error in good ranges is performed, which shows the competency of the proposed solver. Moreover, different statistical procedures have also been used to check the reliability of the proposed stochastic procedure along with forty numbers of independent trials and 10 numbers of neurons.</p> </abstract>

Fear effect in a three-species food chain model with generalist predator.

Within the framework of a food web, the foraging behavior of meso-carnivorous species is influenced by fear responses elicited by higher trophic level species, consequently diminishing the fecundity of these species. In this study, we investigate a three-species food chain model comprising of prey, an intermediate predator, and a top predator. We assume that both the birth rate and intraspecies competition of prey are impacted by fear induced by the intermediate predator. Additionally, the foraging behavior of the intermediate predator is constrained due to the presence of the top predator. It is essential to note that the top predators exhibit a generalist feeding behavior, encompassing food sources beyond the intermediate predators. The study systematically determines all feasible equilibria of the proposed model and conducts a comprehensive stability analysis of these equilibria. The investigation reveals that the system undergoes Hopf bifurcation concerning various model parameters. Notably, when other food sources significantly contribute to the growth of the top predators, the system exhibits stable behavior around the interior equilibrium. Our findings indicate that the dynamic influence of fear plays a robust role in stabilizing the system. Furthermore, a cascading effect within the system, stemming from the fear instigated by top predators, is observed and analyzed. Overall, this research sheds light on the intricate dynamics of fear-induced responses in shaping the stability and behavior of multi-species food web systems, highlighting the profound cascading effects triggered by fear mechanisms in the ecosystem.

Three-species food chain model with cannibalism in the second level

Three-species food chain model with cannibalism in the second level

Qualitative properties for a three-species food chain model with cross-diffusion and intra-specific competition

In this paper, we consider a three-species spatial food chain system as follows$ \begin{equation} \nonumber \begin{cases} u_t = d_1\Delta u + u(1-u)-b_1 uv, \ \quad\quad\ \ x\in \Omega, \; \; t>0, \\ v_t = \nabla \cdot(\gamma_1 (u) \nabla v)- \nabla \cdot(\chi_1(u) v \nabla u)+uv -b_2 vw -\theta_1 v - \alpha_1 v^2, \\ \quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\ x\in \Omega, \; \; t>0, \\ w_t = \nabla \cdot(\gamma_2 (v) \nabla w)- \nabla \cdot(\chi_2(v) w \nabla v)+vw -\theta_2 w - \alpha_2 w^2, \\ \quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\quad\quad \quad\ x\in \Omega, \; \; t>0, \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = \frac{\partial w}{\partial \nu} = 0, \ \quad\quad \quad\quad \quad\quad \ \ \ x\in\partial\Omega, \; \; t>0, \\ (u, v, w)(x, 0) = (u_0, v_0, w_0)(x), \ \ \ \ \quad x\in\Omega, \end{cases} \end{equation} $where $ \Omega\subset{\mathbb{R}}^2 $ is a bounded domain with smooth boundary. For $ i = 1, 2 $, all the parameters $ b_i, \alpha_i, \theta_i $ are positive and the functions $ \gamma_i>0 $ and $ \chi_i>0 $ satisfy $ (\gamma_i, \chi_i)\in [C^2([0, \infty))]^2. $ We first establish the global existence of classical solutions with uniform-in-time bound by using the coupling energy estimates and Moser iteration. Moreover, by constructing Lyapunov functionals, the global stability and convergence rate of steady states are established under certain conditions.

Controlling of periodicity and chaos in a three dimensional prey predator model introducing the memory effect

In an ecological system, omnivores are often significantly important as they feed upon more than one trophic level in a food chain model. Intraguild predation is a special kind of omnivory that is ubiquitous in many ecological communities. A lot of field experiments suggest that the experience over a time duration in the recent past affects the growth rate of all species at the present time. Here, we have considered the Caputo type fractional ordered three-species food chain model. The memory effect is explained in terms of the order of the fractional derivative. Intermediate, top predators feed upon prey by Holling type I functional response and the top predator consumes intermediate predator by Holling type II functional response. Here, we established the existence and uniqueness of the solution of the considered system. Also, we prove the positivity and boundedness of the system solution. All possible feasible equilibrium points with required parametric conditions are discussed in the text. The local stability of all feasible equilibrium points and the parametric conditions of global stability of all non-trivial equilibrium points are discussed. The memory effect can stabilize the system from the periodic oscillatory situation which has been proved through the Hopf bifurcation analysis. Finally, some numerical simulation has been carried out to make significant comments on the dynamics of the considered system about the memory effect and other system parameters. It is clear from the numerical simulations that the order of fractional derivative can be taken as the indicator of chaos control. The biological interpretations make the article most significant from the ecological point of view.

Dynamics in the biparametric spaces of a three-species food chain model with vigilance

In nature, vigilance is a common anti-predator strategy prey individuals employ to protect themselves from a possible attack. It gives them sufficient time to take cover or flee. Yet this seemingly profitable strategy, which lowers the number of successful predatory attacks, has a significant impact on the prey's growth rate. Researchers attribute this to reduced foraging time. In this article, we study a three-species food chain model in which both the basal prey and middle predator show active vigilance against their respective predators. Apart from the positivity, boundedness, and local stability analysis of the equilibrium points of the model, we find the conditions for global stability and the occurrence of Hopf bifurcation around the coexisting equilibrium point. Bifurcation diagrams and their corresponding Lyapunov exponent diagrams (largest and second-largest) illustrate diverse dynamical scenarios ranging from stable coexistence to periodic to chaotic oscillations. The isospike diagrams and their associated largest Lyapunov exponent diagrams in the biparametric space of basal prey's and middle predator's vigilance reveal a broader picture of the effects of different levels of vigilance on the dynamics of the system. A great deal of attention has been paid to unwrapping the complex structural beauties like shrimp-shaped periodic structures inside the chaotic sea and multistability between several sets of attractors in the overlapping zones of periodic windows. We observe that multiple coexisting attractors possess fractal basin boundaries. We also explore the dynamics of the system in two other biparametric spaces. The density variation maps at the end show the effect of vigilance on the equilibrium density of all three species. Our results suggest that vigilance promotes regularity and coexistence, but too high levels of vigilance can cause species extinction. For better visualization of the system dynamics, we provide a few high-resolution animations of the phase portraits and the basins of attraction in the supplementary material.

Complex Dynamics of a Three-Species Food Chain Model with Fear and Allee Effect

In ecology, predator–prey interactions are very complex in nature. Apart from direct killing, predator induces fear among their prey which affects the life-history, behavioral changes, and reproduction potential of the prey population. On the other hand, the Allee effect has a great impact on regulating the population size, community structure and population dynamics. In the present investigation, we modify the Hastings–Powell (HP) [1991] model by considering the cost of fear in middle predator and the Allee effect in top predator. The stability conditions for the biologically feasible equilibria are derived using linear stability analysis. Considering the cost of fear and the Allee effect as key parameters, the Hopf bifurcation analysis is carried out around the interior equilibrium. The direction of Hopf bifurcation and the stability of the bifurcating periodic solution are determined by applying the normal form theory and center manifold theorem. Our numerical results suggest that the fear effect can stabilize the system. It is observed that high levels of fear among middle predator decrease the population density of top predator. We also observe that if the Allee parameter is increased, then the system becomes stable from chaotic oscillations. However, further increase in the Allee parameter leads to population extinction. We have also drawn several one- and two-parameter bifurcation diagrams which explore rich dynamical behaviors.

Stability and Hopf bifurcation analysis for a three-species food chain model with fear and two different delays

Stability and Hopf bifurcation analysis for a three-species food chain model with fear and two different delays

Impact of vigilance on the density variations in a food chain model

Anti-predator behavior in the form of vigilance greatly influences the dynamics of a predator-prey system. In the present work, we investigate the impact of prey vigilance in a three-species food chain model where both prey and middle predator use vigilance as a survival strategy in the presence of their respective predators. We present basic mathematical results such as local and global stability, bifurcation behavior of the system. We explore the variation of the densities of the populations in different bi-parameter spaces and observe that vigilance plays a crucial role in the survival and extinction of the populations.

Bifurcation Analysis of a Discrete Food Chain Model with Harvesting

We explore existence of fixed points, topological classifications around fixed points, existence of periodic points and prime period, and bifurcation analysis of a three-species discrete food chain model with harvesting. Finally, theoretical results are numerically verified.