On the dynamical behavior of three species food web model

The present article deals with a constant proportion of prey refuge in presence of both the inter specific competition and the intra-specific competition among predator populations of a prey-dependent three component food chain model consisting of two competitive predators sharing one prey species as their food. The behaviour of the system near the biologically feasible equilibria is thoroughly analyzed. Boundedness and dissipativeness of the system are established. The stability analysis including local and global stability of the equilibria has been carried out in order to examine the behaviour of the system. The present model system experiences Hopf-Andronov bifurcation for suitable choice of the parameter values. The influences of the prey refuge parameters on the dynamical behaviour of the system are exhibited through several plots and discussed at some equilibrium positions. It is worthnoting that prey refuge has stabilization effect in some selected situations which may be of some use for biological control. Numerical simulations are performed to illustrate and to support the analytical findings so as to validate the applicability of the model under consideration.

Nonlinear Dynamics, 2014

This paper discusses a prey-predator model with reserved area. The feeding rate of consumers (predators) per consumer (i.e., functional response) is considered to be Beddington-DeAngelis type. The Beddington-DeAngelis functional response is similar to the Holling type II functional response but contains an extra term describing mutual interference by predators. We investigate the role of reserved region and degree of mutual interference among predators in the dynamics of system. We obtain different conditions that affect the persistence of the system. We also discuss local and global asymptotic stability behavior of various equilibrium solutions to understand the dynamics of the model system. The global asymptotic stability of positive interior equilibrium solution is established using suitable Lyapunov functional. It is found that the Hopf bifurcation occurs when the parameter corresponding to reserved region (i.e., m) crosses some critical value. Our result indicates that the predator species exist so long as prey reserve value (m) does not cross a threshold value and after this value the predator species extinct. To mimic the real-world scenario, we also solve the inverse problem of estimation of model parameter (m) using the sampled data of the system. The results can also be interpreted in different contexts such as resource conservation, pest management and

2021

In this paper, a mathematical model is proposed to study the effect of prey refuge on the dynamics of three species food web system. The food web comprises of a single prey and two competing predators. The two predators predate their prey following Holling type II functional response. In this work we discussed boundedness of the system, existence condition of the equilibrium points and the Jacobean matrix is obtained by linearization techniques. The local stability of the equilibrium points was discussed by using Routh-Hurwitz criteria and the global stability of the equilibrium points by constructing suitable Lyapunov function. Numerical simulation is conducted to support the analytical result. Finally, the effect of prey refuge on the dynamics of one prey two predator was discussed based on the analytical and numerical simulation results. From the numerical simulations, it is found that the dynamical system is persistent for a small value of the refuge constant. However, an increa...

Journal of Biological Dynamics, 2012

A four-dimensional food-web system consisting of a bottom prey, two middle predators and a generalist predator has been developed with modified functional response. The system is well posed and dissipative. Some results on uniform persistence have been developed. The dynamics of the system is found to be chaotic for certain choice of parameters. The coexistence of all four species is possible in the form of periodic orbits/strange attractors for suitably chosen set of parameters.

Communications in Nonlinear Science and Numerical …, 2005

A food web consisting of two independent preys and a predator is modeled incorporating modified Holling type-II functional response. The mathematical model has a unique and bounded solution. The necessary and sufficient conditions for persistence of the food web are obtained. Bifurcation diagram has been obtained for selected range of different parameters. The system exhibits chaos for a range of parametric values when long time behavior is studied. The computation of Lyapunov exponents and the existence of strange attractor also indicate the chaotic behavior of the system.

Mathematical Biosciences, 2013

The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf-Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.

Iraqi Journal of Science, 2021

This paper aims to study the role of a prey refuge that depends on both prey and predator species on the dynamics of a food web model. It is assumed that the food transfer among the web levels occurs according to Lotka-Volterra functional response. The solution properties, such as existence, uniqueness, and uniform boundedness, are discussed. The local, as well as the global, stabilities of the solution of the system are investigated. The persistence of the system is studied with the assistance of average Lyapunov function. The local bifurcation conditions that may occur near the equilibrium points are established. Finally, numerical simulation is used to confirm our obtained results. It is observed that the system has only one type of attractors that is a stable point, while periodic dynamics do not exist even on the boundary planes.

BAREKENG: Jurnal Ilmu Matematika dan Terapan

This research develops a mathematical model of three species of food chains between prey, predator, and top predator by adding intraspecific competition and harvesting factors. Interaction between prey with predator and interaction between predator with top predator uses the functional response type II. Model formation begins with creating a diagram food chain of three species compartments. Then a nonlinear differential equation system is formed based on the compartment diagram. Based on this system four equilibrium points are obtained. Analysis of local stability at the equilibrium points by linearization shows that there is one unstable equilibrium point and three asymptotic stable local equilibrium points. Numerical simulations at equilibrium points show the same results as the results of the analysis. Then numerical simulations on several parameter variations show that intraspecific competition has little effect on population changes in predator and top predator. While the harve...

Chaos, Solitons & Fractals, 2007

This paper investigates the dynamical behavior of an exploited system consisting of two preys and a predator which is being harvested. The existence of biological, economic and optimum equilibrium of the system is examined. The local and global stability analysis of the model has been carried out. The optimal harvesting policy for harvesting the predator species is studied. The bifurcation diagram is drawn for biologically feasible choice of parameters and the harvest parameter is chosen in the range for which optimum equilibrium also exist. It is observed that harvesting can control the chaos.

2020

We study of dynamics of a three species food chain model, where the growth rate of middle predator is reduced due to prey population is also consumed by top predator and growth rate of prey is also suppressed due to the same reason. The goal of our study is to demonstrate the presence of chaos in the class of ecological models. The model is analyzed in term of stability like as local and global stability at each equilibrium point including interior equilibrium point. To check the the validate of theoretical formulation by numerical simulation by considering the required set of parameters. Subject class :30C45.

Applied Mathematics and Computation, 2005

The dynamics of a predator-prey model with a Beddington-DeAngelis functional response and linear intrinsic growth rate of the prey population is fully analyzed. Conditions on local and global stability of the interior equilibrium are established. The equilibria type are determined. All possible global asymptotic behaviors of the system are considered, including the determination of the extinction conditions and existence of periodic orbits. It is shown that mutual interference between predators can alone stabilize predator-prey interactions even when only a linear intrinsic growth rate of the prey population is considered in the mathematical model. Additional biological implications and a set of numerical simulations supporting the analysis are also presented.

International Journal of Modern Physics: Conference Series, 2012

In this paper, we study mathematical model of ecology with a tritrophic food chain composed of a classical Lotka-Volterra functional response for prey and predator, and a Holling type-III functional response for predator and super predator. There are two equilibrium points of the system. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. Furthermore the dynamical behaviors of this model are investigated. Models for biologically reasonable parameter values, exhibits stable, unstable periodic and limit cycles. The dynamical behavior is found to be very sensitive to parameter values as well as the parameters of the practical life. Computer simulations are carried out to explain the analytical findings.

We characterize the dynamics of a three-species food chain model with a Beddington-DeAngelis functional response in two-parameter space by using Lyapunov exponents. We identify periodic and chaotic behaviors. In this model, the periodic windows are associated with shrimp-shaped structures.

In this paper, a spatial predator-prey system with Beddington-DeAngelis functional response and the modified Leslie-Gower type dynamics under homogeneous Neumann boundary condition is considered. The local and global asymptotic stability of the unique positive homogeneous steady state of the corresponding temporal model are discussed. Moreover, the local stability of the unique constant steady state of the spatiotemporal model is investigated and it is pointed out that spatial patterns cannot occur in the vicinity of this stable steady state.

Journal of Tropical Life Science, 2015

A modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and Michaelis-Menten type prey harvesting is studied. The equilibrium points of the system are investigated. To see the stability of each equilibrium point, we perform some numerical simulations. Our numerical simulations show that the extinction of prey or survival of both prey and predator are conditionally stable.