Beddington-DeAngelis Functional Response Research Articles

An example of chaotic dynamics in 3D systems via stretching along paths

We extend the concept of linked twist maps to a 3D setting and develop a global geometrical method to detect the presence of complex dynamics. Our approach, which is based on a recent variant of the theory of topological horseshoes, provides an analytical proof of “chaos” which does not involve small/large parameter techniques and is robust with respect to small perturbations. An application is given to a predator-prey system in $${{\mathbb R}^3}$$ with a Beddington-DeAngelis functional response.

Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response

In this paper spatial dynamics of the Beddington-DeAngelis predator-prey model is investigated. We analyze the linear stability and obtain the condition of Turing instability of this model. Moreover, we deduce the amplitude equations and determine the stability of different patterns. In Turing space, we found that this model has coexistence of H(0) hexagon patterns and stripe patterns, H(π) hexagon patterns, and H(0) hexagon patterns. To better describe the real ecosystem, we consider the ecosystem as an open system and take the environmental noise into account. It is found that noise can decrease the number of the patterns and make the patterns more regular. What is more, noise can induce two kinds of typical pattern transitions. One is from the H(π) hexagon patterns to the regular stripe patterns, and the other is from the coexistence of H(0) hexagon patterns and stripe patterns to the regular stripe patterns. The obtained results enrich the finding in the Beddington-DeAngelis predator-prey model well.

Persistence of pollination mutualisms in plant–pollinator–robber systems

Interactions between pollinators, nectar robbers, defensive plants and non-defensive plants are characterized by evolutionary games, where payoffs for the four species are represented by population densities at steady states in the corresponding dynamical systems. The plant–robber system is described by a predator–prey model with the Holling II functional response, while the plant–pollinator system is described by a cooperative model with the Beddington–DeAngelis functional response. By combining dynamics of the models with properties of the evolutionary games, we show mechanisms by which pollination mutualisms could persist in the presence of nectar robbers. The analysis leads to an explanation for persistence of plant–pollinator–robber systems in real situations.

Seasonally Perturbed Prey‐Predator Ecological System with the Beddington‐DeAngelis Functional Response

On the basis of the theories and methods of ecology and ordinary differential equation, a seasonally perturbed prey‐predator system with the Beddington‐DeAngelis functional response is studied analytically and numerically. Mathematical theoretical works have been pursuing the investigation of uniformly persistent, which depicts the threshold expression of some critical parameters. Numerical analysis indicates that the seasonality has a strong effect on the dynamical complexity and species biomass using bifurcation diagrams and Poincaré sections. The results show that the seasonality in three different parameters can give rise to rich and complex dynamical behaviors. In addition, the largest Lyapunov exponents are computed. This computation further confirms the existence of chaotic behavior and the accuracy of numerical simulation. All these results are expected to be of use in the study of the dynamic complexity of ecosystems.

Dynamics of Mutualism‐Competition‐Predator System with Beddington‐DeAngelis Functional Responses and Impulsive Perturbations

We explore the dynamics of a class of mutualism‐competition‐predator interaction models with Beddington‐DeAngelis functional responses and impulsive perturbations. Sufficient conditions for existence of positive periodic solution are established by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are given for the global stability and the globally exponential stability of system by employing comparison principle and Lyapunov method.

Non-Turing pattern formation in a modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response

In this paper a modified Leslie-Gower predator-prey system with Beddington-DeAngelis functional response under homogeneous Neumann boundary condition is considered. The prey and predator species are assumed to have equal diffusion coefficient. The possibility of non-Turing spatio-temporal patterns is explored. The ratio of two intrinsic growth rates is considered as a control parameter. Regular spatio-temporal oscillatory patterns as well as non-stationary irregular patterns are obtained. The numerical simulations reveal the dependence of the patterns on the choice of initial distribution and reaction parametric values.

THE EFFECTS OF HARVESTING AND TIME DELAY ON PREDATOR-PREY SYSTEMS WITH BEDDINGTON–DEANGELIS FUNCTIONAL RESPONSE

The combined effects of harvesting and time delay on predator-prey systems with Beddington–DeAngelis functional response are studied. The region of stability in model with harvesting of the predator, local stability of equilibria and the existence of Hopf bifurcation are obtained by analyzing the associated characteristic equation due to the two-parameter geometric criteria developed by Ma, Feng and Lu [Discrete Contin. Dyn. Syst. Ser B9 (2008) 397–413]. The global stability of the positive equilibrium is investigated by the comparison theorem. Furthermore, local stability of steady states and the existence of Hopf bifurcation for prey harvesting are also considered. Numerical simulations are given to illustrate our theoretical findings.

A With-In Host Dengue Infection Model with Immune Response and Beddington-DeAngelis Incidence Rate

A model of viral infection of monocytes population by dengue virus is formulated in a system of four ordinary differenttial equations. The model takes into account the immune response and the incidence rate of susceptible and free virus particle as Beddington-DeAngelis functional response. By constructing a block, the global stability of the unin-fected steady state is investigated. This steady state always exists. If this is the only steady state, then it is globally asymptotically stable. If any infected steady state exists, then uninfected steady state is unstable and one of the infected steady states is locally asymptotically stable. These different cases depend on the values of the basic reproduction ratio and the other parameters.

Global Behavior for a Strongly Coupled Predator‐Prey Model with One Resource and Two Consumers

We consider a strongly coupled predator‐prey model with one resource and two consumers, in which the first consumer species feeds on the resource according to the Holling II functional response, while the second consumer species feeds on the resource following the Beddington‐DeAngelis functional response, and they compete for the common resource. Using the energy estimates and Gagliardo‐Nirenberg‐type inequalities, the existence and uniform boundedness of global solutions for the model are proved. Meanwhile, the sufficient conditions for global asymptotic stability of the positive equilibrium for this model are given by constructing a Lyapunov function.

The effect of mutual interference between predators on a predator–prey model with diffusion

We consider a diffusive predator–prey model with Beddington–DeAngelis functional response under homogeneous Dirichlet boundary conditions. The effect of large k which represents the extent of mutual interference between predators is extensively studied. By making use of the fixed point index theory, we obtain a complete understanding of the existence, uniqueness and stability of positive steady-states when k is sufficiently large. Moreover, we present some numerical simulations that supplement the analytic results in one dimension.

Coexistence states for a diffusive one-prey and two-predators model with B–D functional response

In this paper, we study a diffusive one-prey and two-predators system with Beddington–DeAngelis functional response. The sufficient and necessary conditions for the existence of coexistence states are obtained by means of the fixed point index theory. In addition, the stability and uniqueness of coexistence states are investigated. Finally, we give the sufficient conditions for extinction and permanence of the time-dependent system.

Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator–prey models

This paper is concerned with two-species spatial homogeneous and inhomogeneous predator–prey models with Beddington–DeAngelis functional response. For the spatial homogeneous model, the asymptotic behavior of the interior equilibrium and the existence of Hopf bifurcation of nonconstant periodic solutions surrounding the interior equilibrium are considered. Furthermore, the direction of Hopf bifurcation and the stability of bifurcated periodic solutions are investigated. For the model with no-flux boundary conditions, Turing instability of the interior equilibrium solution is studied. In particular, Turing instability region regarding the parameters is established. Finally, to verify our theoretical results, some numerical simulations are also included.

Dynamics of a stochastic predator-prey model with the Beddington-DeAngelis functional response

Dynamics of a stochastic predator-prey model with the Beddington-DeAngelis functional response

Dynamics of species in a model with two predators and one prey

In this paper, we study a predator–prey model which has one prey and two predators with Beddington–DeAngelis functional responses. Firstly, we establish a set of sufficient conditions for the permanence and extinction of species. Secondly, the periodicity of positive solutions is studied. Thirdly, by using Liapunov functions and the continuation theorem in coincidence degree theory, we show the global asymptotic stability of such solutions. Finally, we give some numerical examples to illustrate the behavior of the model.

Periodic solutions for a delayed predator–prey system with stage-structured predator on time scales

In this paper, a modified delayed predator–prey system with stage structure for predator is proposed and studied, which generalizes and incorporates as special cases some known models. With the help of continuation theorem based on the Gaines and Mawhin coincidence degree theory, we investigate the existence of periodic solutions for the proposed delayed predator–prey system with stage-structured predator and two kinds of functional responses (called Holling III functional response and Beddington–DeAngelis functional response) on time scales. In particular, when the time scale is chosen as the set of the real numbers or the integers, the existence of the periodic solutions of the corresponding continuous-time and discrete-time models follows.

ON THE DYNAMICS OF PREDATOR-PREY SYSTEMS WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE

This paper studies a predator-prey system with Beddington-DeAngelis functional response. We establish sufficient criteria posed on the coefficients for the permanence of the system, globally asymptotic stability of solutions and the non-existence of periodic orbits.

Global analysis for delay virus dynamics model with Beddington–DeAngelis functional response

A class of virus dynamics model with intracellular delay and nonlinear infection rate of Beddington–DeAngelis functional response is analysed in this paper. By constructing suitable Lyapunov functionals and using LaSalle-type theorem for delay differential equations, we show that the global stability of the infection-free equilibrium and the infected equilibrium depends on the basic reproductive ratio R 0 , that is, the former is globally stable if R 0 ≤ 1 and so is the latter if R 0 > 1 . Our results extend the known results on delay virus dynamics considered in the other papers and suggest useful methods to control virus infection.

Analysis of a Delayed Predator-Prey Reaction-Diffusion System with Beddington-DeAngelis Functional Response and Two Stage Structure

In this paper, we study a delayed predator-prey reaction-diffusion system with Beddington-DeAngelis functional response and two stage structure on prey, an immature and a mature stage. The time delays are the time lengths between the immature’s birth and maturity prey species. Under some simple assumptions, using upper and lower solutions method and a comparison principle we prove that the unique constant positive steady states solution is global asymptotic stability. The results shows that the dynamic behavior of solutions to this model is similar to that of solutions to corresponding model without diffusion.

Dynamics of a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response

In this paper, we discuss a stochastic density dependent predator–prey system with Beddington–DeAngelis functional response. First, we show that this system has a unique positive solution as this is essential in any population dynamics model. Then, we investigate the asymptotic behavior of this system. When the white noise is small, the stochastic system imitates the corresponding deterministic system. Either there is a stationary distribution, or the predator population will die out. While if the white noise is large, besides the extinction of the predator population, both species in the system may also die out, which does not happen in the deterministic system. Finally, simulations are carried out to conform to our results.

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.