Beddington-DeAngelis Functional Response Research Articles

Dynamical Analysis of a Discrete Amensalism System with the Beddington–DeAngelis Functional Response and Allee Effect for the Unaffected Species

Dynamical Analysis of a Discrete Amensalism System with the Beddington–DeAngelis Functional Response and Allee Effect for the Unaffected Species

Analysis of a stochastic predator–prey system with fear effect and Lévy noise

This paper studies a stochastic predator–prey model with Beddington–DeAngelis functional response, fear effect, and Lévy noise, where the fear is of prey induced by predator. First, we use Itô’s formula to prove the existence and uniqueness of a global positive solution and its moment boundedness. Next, sufficient conditions for the persistence and extinction of both species are given. We further investigate the stability in distribution of our system. Finally, we verify our analytical results by exhaustive numerical simulations.

Fractional order system dynamical behaviors with Beddington-DeAngelis functional response

The present study proposes a fractional order prey-predator model with Beddington-DeAngelis functional response, that the Caputo fractional derivative is applied. There is exploration of the solutions' existence, uniqueness, non-negativity, and boundedness. Stability of all feasible equilibrium points is determined locally by the use of Matignon's condition. Moreover, we also provide sufficient conditions to assure global asymptotic stability for both the predator-extinction equilibrium point and the positive equilibrium point, with selecting a relevant Lyapunov function and the incidence of Hopf-bifurcation is also displayed. Finally, the fractional order effect on the stability behavior of systems is investigated theoretically and also illustrated numerically to support theoretical results.

Dynamics of the Predator-Prey Model with Beddington-DeAngelis Functional Response Perturbed by Lévy Noise

We study the non-autonomous stochastic predator-prey model with Beddington-DeAngelies functional response driven by the system of stochastic differential equations with white noise, centered and non-centered Poisson noises. It is proved the existence and uniqueness of the global positive solution of considered system. We obtain sufficient conditions of stochastic ultimate boundedness, stochastic permanence, non-persistence in the mean, weak and strong persistence in the mean and extinction of the population densities in the considered stochastic predator-prey model.

Spatiotemporal Dynamics in a Diffusive Predator–Prey System with Beddington–DeAngelis Functional Response

Spatiotemporal Dynamics in a Diffusive Predator–Prey System with Beddington–DeAngelis Functional Response

Persistence and periodic measure of a stochastic predator–prey model with Beddington–DeAngelis functional response

In this paper, we study a stochastic predator–prey model with Beddington–DeAngelis functional response and time-periodic coefficients. By analyzing the stability of the solution on the boundary and some stochastic estimates, the threshold conditions for the time-average persistence in probability and extinction of each population are established. Furthermore, the existence of a unique periodic measure of the model is also presented under the condition of the time-average persistence in probability of the model. Several numerical simulations are given to verify the effectiveness of the theoretical results and to illustrate the effects of the white noises on the persistence and periodic measure of the model.

Dynamics of a predator–prey model with nonlinear growth rate and B–D functional response

In this paper, a predator–prey diffusive model, subject to homogeneous Dirichlet boundary conditions, with Beddington–DeAngelis functional response and nonlinear growth rate on the predator is proposed and the well posedness of its solution is systematically studied. Taking the capture rate m as the main parameter, we mainly investigate the existence, stability and exact number of positive solutions when m is large and other parameters meet a certain range of conditions. Meanwhile, some numerical simulations are applied to illustrate the analytical results. The main techniques used in this paper include the fixed point index theory, the super-sub solution method, the Lyapunov-Schmidt reduction procedure and the perturbation principle.

A Predator–Prey Model with Beddington–DeAngelis Functional Response and Multiple Delays in Deterministic and Stochastic Environments

In view of prey’s delayed fear due to predators, delayed predator gestation, and the significance of intra-specific competition between predators when their populations are sufficiently large, a prey–predator population model with a density-dependent functional response is established in a deterministic environment. We research the existence and asymptotic stability of the equilibrium statuses. Then, taking into consideration environmental disturbances, we extend the deterministic model to a stochastic model and research the existence and stationary distributions of stochastic solutions. Finally, we perform some numerical simulations to verify the theoretical results. Numerical examples indicate that fear, delays and environmental disturbance play crucial roles in the system stability of the equilibrium status.

Positive periodic solution of a discrete commensal symbiosis model with Beddington-DeAngelis functional response

A non-autonomous discrete commensal symbiosis model with Beddington-DeAngelis functional response is proposed and studied in this paper. Sufficient conditions are obtained for the existence of positive periodic solution of the system.

On the Existence of Positive Periodic Solution of an Amensalism Model with Beddington-DeAngelis Functional Response

A non-autonomous discrete amensalism model with Beddington-DeAngelis functional response is proposed and studied in this paper. Sufficient conditions are obtained for the existence of positive periodic solution of the system.

Global bifurcation and pattern formation for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses

Of concern in this paper is to propose an accurate description for the global bifurcation structure of the nonconstant steady states for a reaction–diffusion predator–prey model with prey-taxis and double Beddington–DeAngelis functional responses systematically. By treating the coefficient of nonlinear prey-taxis as a bifurcation parameter and utilizing the user-friendly version of Crandall–Rabinowitz bifurcation theory, we study the global bifurcation theory of the system. Meanwhile, the existence of nonconstant steady states will be offered by the exported global bifurcation theorem under a rather natural condition. In the proof, a priori estimate of steady states will play an important role. The local stability analysis with a numerical simulation and bifurcation analysis are given.

Dynamics of a stochastic phytoplankton–toxic phytoplankton–zooplankton system under regime switching

In this paper, a stochastic phytoplankton–toxic phytoplankton–zooplankton system with Beddington–DeAngelis functional response, where both the white noise and regime switching are taken into account, is studied analytically and numerically. The aim of this research is to study the combined effects of the white noise, regime switching, and toxin‐producing phytoplankton (TPP) on the dynamics of the system. Firstly, the existence and uniqueness of global positive solution of the system is investigated. Then some sufficient conditions for the extinction, persistence in the mean, and the existence of a unique ergodic stationary distribution of the system are derived. Significantly, some numerical simulations are carried to verify our analytical results and show that high intensity of white noise is harmful to the survival of plankton populations, but regime switching can balance the different survival states of plankton populations and decrease the risk of extinction. Additionally, it is found that an increase in the toxin liberation rate produced by TPP will increase the survival chance of phytoplankton, while it will reduce the biomass of zooplankton. All these results may provide some insightful understanding on the dynamics of phytoplankton–zooplankton systems in randomly disturbed aquatic environments.

Bifurcation analysis of a predator–prey model with Beddington–DeAngelis functional response and predator competition

In this paper, we consider a predator–prey model with Beddington–DeAngelis functional response and predator competition, which is a five‐parameter family of planar vector field. It is shown that the model can undergo a sequence of bifurcations including focus type degenerate Bogdanov–Takens bifurcation of codimension 3 and Hopf bifurcation of codimension at least 2 as the parameters vary. Our theoretical results indicate that predator competition can cause richer dynamics such as two limit cycles enclosing one or three hyperbolic positive equilibria and three kinds of homoclinic orbits (homoclinic to hyperbolic saddle, saddle‐node, or neutral saddle). Moreover, there exists a threshold value for predator capturing rate , below or equal to which the predators always tend to extinction, above which the predators and preys will coexist in the form of multiple steady states or periodic oscillations for all positive initial populations. Numerical simulations are presented to illustrate the theoretical results.

On optimal harvesting policy for two economically beneficial species mysida and herring: a clue for conservation biologist through mathematical model

ABSTRACT In this paper, we have studied the dynamics of two species, ecological model, considering two economically beneficial marine species herring (Clupea harengus) as predator and mysida (Hemimysis anomala) as its prey with Beddington–DeAngelis functional response, the effect of prey refuge and linear harvesting in both populations. The local and global stability of the equilibrium points has been established here. The axial equilibrium point goes through transcritical bifurcation depending on the harvesting effort of both species. The interior equilibrium point goes through super critical Hopf bifurcation depending on prey intrinsic growth rate. Additionally, we have determined the suitable value of the MSY policy for the harvesting and find suitable harvesting level using the optimal control theory, such that both the species persist simultaneously. Finally, numerical simulations have been performed to justify the theoretical findings. It is clear from the numerical simulations that the prey refuge and mutual interference between predators stabilized the unstable system to a stable one. This finding could give some clue to the experimental biologist and business personnel for making management decisions in controlling the harvesting strategies of the two economic beneficial fish populations, mysida and herring.

Impact of fear effect and prey refuge on a fractional order prey-predator system with Beddington-DeAngelis functional response.

In this article, a fractional-order prey-predator system with Beddington-DeAngelis functional response incorporating two significant factors, namely, dread of predators and prey shelter are proposed and studied. Because the life cycle of prey species is memory, the fractional calculus equation is considered to study the dynamic behavior of the proposed system. The sufficient conditions to ensure the existence and uniqueness of the system solution are found, and the legitimacy and well posedness in the biological sense of the system solution, such as nonnegativity and boundedness, are proved. The stability of all equilibrium points of the system is analyzed by an eigenvalue analysis method, and it is proved that the system generates Hopf bifurcation nearby the coexistence equilibrium with regard to three parameters: the fear coefficient k, the rate of prey shelters p, and the order of fractional derivative q. Compared with the integer derivative, the system dynamics in the situation of fractional derivative is more stable. We observe an interesting phenomenon through the simulation: with the increase in the level of the fear effect, the stability of the positive equilibrium point changes from stable to unstable and then to stable. At this time, there are two Hopf branches nearby the positive equilibrium point with respect to the fear coefficient k, and the system can be in a stable state at very low or high level of the fear effect. In addition, when the order of the fractional differential equation of the system decreases continuously, the stability of the system will change from unstable to stable, especially in the case of low-level fear caused by predators and low rate of prey shelters. Therefore, our findings support the view that the strong memory can promote the stable coexistence of two species in the prey-predator system, while fading memory of species will worsen the stable coexistence of two species in the proposed system.

An epidemic model with Beddington–DeAngelis functional response and environmental fluctuations

An epidemic model with the saturated incidence rate and the environmental fluctuations is investigated in this paper. We study the extinction and the persistence in the mean, and stationary distribution of the solution as well. By contradiction, we firstly show that stochastic epidemic model admits a unique global positive solution for any given positive initial value. Further, when R˜0>1 is valid, we derive the persistence in the mean, and also prove the existence of an ergodic stationary distribution by constructing moderate functions. By comparison theorem of stochastic differential equations and properties of inequalities, the extinction of the solution is finally derived when R0<1 and ν<0 hold, where ν indicates the exponential rate for decline. Meanwhile, the distribution for the density of the susceptible is estimated. As a consequence, numerical simulations and illustrative examples are separately carried out to support the main results of this paper.

A delayed plant disease model with Caputo fractional derivatives

We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington–DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams–Bashforth–Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.

Dynamics of stage-structured prey–predator model with prey refuge and harvesting

ABSTRACT A stage-structured prey–predator model with prey refuge is assumed, where predators have two stages as immature and mature. We consider here the harvesting on prey only and analyze the model using Beddington-DeAngelis functional response. Local stability as well as global stability at an interior equilibrium point are investigated in the proposed model. We choose different types of bifurcation like Hopf bifurcation, saddle-node bifurcation, and transcritical bifurcation in the formulated model. Moreover, in this paper, we check the optimality of the harvesting function with the help of optimal control. We provide the numerical simulation of the proposed model for the confirmation of analytical results.

Dynamical analysis of a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps

&lt;abstract&gt;&lt;p&gt;In this paper, a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps is studied. Firstly, it is proved that the model has a unique global solution. Secondly, sufficient conditions for weak persistence in the mean and extinction of prey and predator populations are established. Finally, sufficient conditions for the existence and uniqueness of ergodic stationary distribution are established. Moreover, several numerical simulations are presented to illustrate the main results.&lt;/p&gt;&lt;/abstract&gt;

Global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis

&lt;abstract&gt;&lt;p&gt;In this paper, our purpose is to discuss the global dynamics of a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response and prey-taxis under homogeneous Neumann boundary conditions. First, we derive that the global classical solutions of the system are globally bounded by taking advantage of the Morse's iteration of the parabolic equation, which further arrives at the global existence of classical solutions with a uniform-in-time bound. In addition, we establish the global stability of the spatially homogeneous coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals.&lt;/p&gt;&lt;/abstract&gt;