Three-species Food Chain Model Research Articles

Global Stability for a Three-Species Food Chain Model in a Patchy Environment

We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse amongndifferent patches(n≥2). By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive equilibrium of this model is unique and globally asymptotically stable if it exists.

Analysis of Dynamics in a Complex Food Chain with Ratio-Dependent Functional Response

In this paper, we study a new model obtained as an extension of a three-species food chain model with ratio-dependent functional response. We provide non-persistence and permanence results and investigate the stability of all possible equilibria in relation to the ecological parameters. Results are obtained for the trivial and prey-only equilibria where the singularity of the model prevents linearization, and the remaining semi-trivial equilibria are studied using linearization. We provide a detailed analysis of conditions for existence, uniqueness, and multiplicity of coexistence equilibria, as well as permanent eect for all species. The complexity of the dynamics in this model is theoretically discussed and graphically demonstrated through various examples and numerical simulations.

Bifurcation behaviours in a delayed three-species food-chain model with Holling type-II functional response

This article is concerned with the local stability of a positive equilibrium and the Hopf bifurcation of a delayed three-species food-chain system with the Holling type-II functional response. Some new sufficient conditions ensuring the local stability of a positive equilibrium and the existence of Hopf bifurcation for the system are established. Some explicit formulae are derived to determine the direction of bifurcations and the stability of bifurcating periodic solutions using the normal form theory and the centre manifold theory. Numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are included.

Finite Time Blowup in a Realistic Food-Chain Model

We investigate a realistic three-species food-chain model, with generalist top predator. The model based on a modified version of the Leslie-Gower scheme incorporates mutual interference in all the three populations and generalizes several other known models in the ecological literature. We show that the model exhibits finite time blowup in certain parameter range and for large enough initial data. This result implies that finite time blowup is possible in a large class of such three-species food-chain models. We propose a modification to the model and prove that the modified model has globally existing classical solutions, as well as a global attractor. We reconstruct the attractor using nonlinear time series analysis and show that it pssesses rich dynamics, including chaos in certain parameter regime, whilst avoiding blowup in any parameter regime. We also provide estimates on its fractal dimension as well as provide numerical simulations to visualise the spatiotemporal chaos.

Stability and Hopf bifurcation for a three-species food chain model with time delay and spatial diffusion

This paper deals with a delayed three-species Lotka–Volterra food chain model with diffusion effects and homogeneous Neumann boundary conditions. By taking the sum of delays as the bifurcation parameter, spatially homogeneous and nonhomogeneous Hopf bifurcations at the positive constant steady state are proved to occur for a sequence of critical values of the delay parameter. In addition, sufficient conditions for global asymptotic stability of the positive constant steady state are derived by using an iteration technique. Furthermore, in order to determine the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcated periodic solutions, the formulas are given by using the normal form theory and the center manifold reduction for PFDEs. Finally, to verify our theoretical predictions, some numerical simulations are also included.

TRAVELING WAVE SOLUTIONS IN A THREE-SPECIES FOOD-CHAIN MODEL WITH DIFFUSION AND DELAYS

This paper deals with the existence of traveling wave solutions in a three-species food-chain model with spatial diffusion and time delays due to gestation and negative feedback. By using a cross iteration scheme and Schauder's fixed point theorem, we reduce the existence of traveling wave solutions to the existence of a pair of upper-lower solutions. By constructing a pair of upper-lower solutions, we derive the existence of a traveling wave solution connecting the trivial steady state and the positive steady state. Numerical simulations are carried out to illustrate the main results. In particular, our results extend and improve some known results.

Stability and Bifurcation Analysis in a Food Chain Model with Delay

In this paper, we investigate a delayed three-species food chain model. The existence and stability of equilibria are obtained. A explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by using the normal form and the center manifold theory.

A numeric-analytic method for approximating three-species food chain models

This paper investigates the accuracy of the differential transformation method (DTM) for solving the three-species food chain models which is described as three-dimensional system of ODES with quadratic and rational nonlinearities. Numerical results are compared to those obtained by the fourth-order Runge-Kutta method to illustrate the preciseness and effectiveness of the proposed method. The direct symbolic-numeric scheme is indicated to be efficient and accurate. Key words: Three-species food chain models, multi-step differential transformation method, attractors.

Chaos in a Hybrid Three-Species Food Chain with Beddington-Deangelis Functional Response

In this paper, a hybrid three-species food-chain model with Beddington-DeAngelis functional response has been studied. Local stability of the equilibria described by this model have been proved. By means of computer simulations, the complex dynamics of the model, especially chaos, have been identified. Furthermore, the largest Lyapunov exponen has been used to demonstrate the chaotic dynamics.

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincaré return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing δ1. However, for higher values of δ1 the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter ζ) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.

Chaos control and synchronization of a three-species food chain model via Holling functional response

Chaos control and synchronization problem of a three-species food chain model with Holling I-type functional response is investigated. By the linearization approach, a feedback control law is developed to stabilize the model system to equilibrium points. Based on the bidirectional coupled method, chaos synchronization of the model system is studied. Numerical simulations show that the feedback control law can suppress chaos to unstable equilibrium points successfully. Further, the numeric results show that when the coupling strength 0<k<0.012, synchronization of two model systems cannot be achieved; when the coupling strength 0.012≤k<2, synchronization of two model systems is always achieved, furthermore, the larger is the coupling strength k, the smaller are the synchronization errors, the better is the effect of chaos synchronization; when the coupling strength k≥2, synchronization of two model systems can also be achieved, but the synchronization errors are bigger than that of 0.012≤k<2.

Complex dynamics in a food chain with slow and fast processes

This paper is devoted to the analysis of the dynamic behavior of a three-species food chain model, in which two predators compete for the same prey while one of the predators feeds on the other. Under the assumption that the time responses of the three trophic levels are extremely diversified, the model is proved to have homoclinic orbit. We firstly use geometric singular perturbation method to detect singular homoclinic orbits as well as parameter combinations for which these orbits exist. Then, we show, numerically, that there exist also nonsingular homoclinic orbits that tend toward the singular ones for slightly different parameter values. This analysis is particularly helpful to understanding the chaotic behavior of the food chains.

Dynamics of a three-species food chain model with adaptive traits

A three-species food chain model is proposed with dynamically variable adaptive traits in the intermediate consumer. We prove that its solutions are non-negative and bounded, and we analyze the existence and stability of its equilibria. By applying Li and Muldowney’s [Li MY, Muldowney J. On Bendixson’s criterion. J Differ Equ 1993;106:27–39] high-dimensional Bendixson criterion, we show that the positive equilibrium is globally stable under specific conditions. We support our analytical findings with numerical simulations.

Chaos in a three-species food chain model with a Beddington–DeAngelis functional response

This paper investigates a three-species food chain model with a Beddington–DeAngelis functional response, both analytically and through numerical simulations. First the equilibrium states of the system are identified and their stability analyzed analytically. The results of simulations demonstrate chaotic long-term behavior over a broad range of parameters. The existence of a strange attractor and computation of the largest Lyapunov exponent also demonstrate the presence of chaotic dynamics in the model.

INFLUENCE OF ENVIRONMENTAL NOISE ON THE DYNAMICS OF A REALISTIC ECOLOGICAL MODEL

The present paper investigates the influence of environmental noise on a fairly realistic three-species food chain model based on the Leslie-Gower scheme. The self- growth parameter for the prey species is assumed to be perturbed by white noise characterized by a Gaussian distribution with mean zero and unit spectral density. Using tools borrowed from the nonlinear dynamical system theory, we study the dynamical behavior of the model system. The behavior of the stochastic system (perturbed one) is studied and the fluctuations in the populations are measured both analytically (for the linearized system) and numerically by computer simulation. Varying one of the control parameters in its range, while keeping all the others constant, we monitor the changes in the dynamical behavior of the model system, thereby fixing the regimes in which the system exhibits chaotic dynamics. Our study suggests that the trophic level (top, middle or bottom) at which a population is positioned, the amplitude of environmental noise and the population's susceptibility to environmental noise play key roles in how noise affects the population dynamics.

Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion

In the paper, we investigate a three-species food chain model with diffusion and ratio-dependent predation functional response. We mainly focus on the coexistence of the three species. For this coupled reaction-diffusion system, we study the persistent property of the solution, the stability of the constant positive steady state solution, and the existence and nonexistence of nonconstant positive steady state solutions. Both the general stationary pattern and Turing pattern are observed as a result of diffusion. Our results also exhibit some interesting effects of diffusion and functional responses on pattern formation. (An erratum to this article has been appended at the end of the pdf file.)

The dynamic complexity of a three species food chain model

In this paper, a three-species food chain model is analytically investigated on theories of ecology and using numerical simulation. Bifurcation diagrams are obtained for biologically feasible parameters. The results show that the system exhibits rich complexity features such as stable, periodic and chaotic dynamics.

Blowup estimates for a parabolic system in a three-species cooperating model

Blowup estimates for solutions of a parabolic system, which arises in a cooperating three-species food chain model, with homogeneous Dirichlet boundary conditions are investigated. The upper bound of blowup rate for any N and the lower bound of blowup rate for N=1 are obtained.

Blow-up in a three-species cooperating model

The properties of solutions for a parabolic system with homogeneous Dirichlet boundary conditions, which arises in a cooperating three-species food chain model, are investigated. It is shown that global solutions exist if the intraspecific competitions are strong whereas blowup solutions exist under certain conditions if the intra-specific competitions are weak.

Dynamical and numerical analyses of a generalized food-chain model

This paper focuses on the qualitative and quantitative analysis of a generalized three-species food-chain model. The conditions for the existence and stability of the boundary and positive equilibria of the model are established. By constructing an appropriate “bifurcation function”, it is shown that the model undergoes a Hopf bifurcation from the positive equilibrium for certain parameter values. The root of this bifurcation function gives the bifurcation value. A robust non-standard numerical method is constructed and used to obtain the solution of the model. Numerical simulations using this method show that the Hopf bifurcation is supercritical with a stable limit cycle in the positive octant.