Beddington-DeAngelis Type Functional Response Research Articles

Qualitative analysis and traveling wave solutions of a predator-prey model with time delay and stage structure

<abstract><p>In this paper, we considered a delayed predator-prey model with stage structure and Beddington-DeAngelis type functional response. First, we analyzed the stability of the constant equilibrium points of the model by the linear stability method. Furthermore, we considered the existence of traveling wave solutions connecting the zero equilibrium point and the unique positive equilibrium point. Second, we transformed the existence of traveling wave solutions into the existence of fixed points of an operator by constructing suitable upper and lower solutions, and combined with the Schauder fixed point theorem, we gave the existence of fixed points and obtained the existence of traveling wave solutions of the model.</p></abstract>

Extinction and strong persistence in the Beddington–DeAngelis predator–prey random model

This paper studies a predator–prey model with a Beddington–DeAngelis type functional response under the stationary Ornstein–Uhlenback process. First, we prove the existence and uniqueness of the global positive solution for the given system for any initial datum. Second, we turn to its internal structures and establish the existence of a compact forward‐absorbing set as well as a forward attracting one. Finally, we carry out some numerical simulations to support the theoretical results.

Study on Dynamic Behavior of a Stochastic Predator–Prey System with Beddington–DeAngelis Functional Response and Regime Switching

In this paper, by introducing environmental white noise and telegraph noise, we proposed a stochastic predator–prey model with the Beddington–DeAngelis type functional response and investigated its dynamical behavior. Persistence and extinction are two core contents of population model research, so we analyzed these two properties. The sufficient conditions of the strong persistence in the mean and extinction were established and the threshold between them was obtained. Moreover, we took stability into account and, by means of structuring a suitable Lyapunov function with regime switching, we proved that the stochastic system has a unique stationary distribution. Finally, numerical simulations were used to illustrate our theoretical results.

Qualitative Analysis in a Beddington–DeAngelis Type Predator–Prey Model with Two Time Delays

In this paper, we investigate a delayed predator–prey model with a prey refuge where the predator population eats the prey according to the Beddington–DeAngelis type functional response. Firstly, we consider the existence of equilibrium points. By analyzing the corresponding characteristic equations, the local stability of the trivial equilibrium, the predator–extinction balance, and the coexistence equilibrium of the system are discussed, and the existence of Hopf bifurcations concerning both delays at the coexistence equilibrium are established. Then, in accordance with the standard form method and center manifold theorem, the explicit formulas which determine the direction of Hopf bifurcation and stability of bifurcating period solutions are derived. Finally, representative numerical simulations are performed to validate the theoretical analysis.

Fear effect in a predator-prey model with additional food, prey refuge and harvesting on super predator

The shape of the interaction between three species: prey, predator, and super predator has been considered in this work as a new predator-prey model. In this article, the fear of predator in prey species has been introduced. Also, we have considered that the predation of both prey and predator occurs according to the Beddington-DeAngelis type functional response. The predator is thought to be acting as a shelter for the super predator species. It is anticipated that a constant amount of increased food will be delivered to super predators due to their dread of predators and their refuge behavior, and that predators will benefit from this excess food. The non-negativity and boundedness of solutions of the prosed model have been examined. The system's local and global stability at the interior equilibrium points has been explored. The existence condition of Hopf bifurcation of the system with respect to fear f of predator has been studied. From the analysis of the model, it is found that the fear f of predator may be responsible for the system's stability. It's also been discovered that providing extra food to the super predator may help to keep the system in check. Further, it is found that the refuge of predator to the super predator has the ability to stabilize the predator-prey system. Finally, some intriguing numerical simulation findings were provided in order to study the model's dynamics.

On the Dynamics of Impulsive Predator-Prey Systems with Beddington–Deangelis-Type Functional Response

On the Dynamics of Impulsive Predator-Prey Systems with Beddington–Deangelis-Type Functional Response

On the dynamics of the impulsive predator-prey systems with Beddington – DeAngelis type functional response

UDC 517.9 In this study, the two-dimensional predator-prey system with Beddington–DeAngelis type functional response with impulses is considered in a periodic environment. For this special case, necessary and sufficient conditions are found for the considered system when it has at least one -periodic solution. This result is mainly based on the continuation theorem in the coincidence degree theory and to get the globally attractive -periodic solution of the given system, an inequality is given as the necessary and sufficient condition by using the analytic structure of the system.

Pelican–Tilapia interaction in Salton sea: an eco-epidemiological model with strong Allee effect and additional food

The effect of diseases in ecological system is an important issue from mathematical and experimental points of view. We have analysed the dynamic nature of an eco-epidemiological model consisting of susceptible Tilapia fish, infected Tilapia fish and their predator the Pelican birds. Again, the susceptible prey population is subjected to a strong Allee effect. Moreover, the provision of additional food to predator is one of the momentous concepts in the field of biological control. Beddington–DeAngelis-type functional response has been taken to represent the interaction between susceptible prey and predator, whereas predator consumes the infected prey as Volterra-type (Holling type I) functional response. Feeding on infected prey increases the death rate of predator and is regarded to contribute negative growth, whereas consumption of susceptible prey enhances predator growth rate and is regarded to contribute positive growth. It is assumed that the predator population has sufficient resistance power to overcome the effect of environmental toxicants and so they are free from transmission of infection. In this work, we have studied some basic properties, such as non-negativity, uniform boundedness, and stability criteria of all feasible equilibria. The existence of transcritical bifurcations has been observed around the planer equilibrium points taking the mortality rate of predator and transmission rate as bifurcation parameters. Also, we have incorporated gestation delay in the system to examine the changing nature of the delay parameter via Hopf bifurcation. Further, the formulae for direction, period, and the stability of bifurcating periodic solutions have been established. Several numerical simulations are illustrated using MATLAB to validate all the theoretical results.

Dynamical study of infochemical influences on tropic interaction of diffusive plankton system

We propose a model for tropic interaction among the infochemical-producing phytoplankton and non-info chemical-producing phytoplankton and microzooplankton. Volatile information-conveying chemicals (infochemicals) released by phytoplankton play an important role in the food webs of marine ecosystems. Microzooplankton is an ecologically important grazer of phytoplankton for coexistence of a large number of phytoplankton species. Here, we discuss how information transferred by dimethyl sulfide shapes the interaction of phytoplankton. Phytoplankton deterrents may lead to propagation of IPP bloom. The interaction between IPP and microzooplankton follows the Beddington–DeAngelis-type functional response. Analytically, we discuss boundedness, stability and Turing instability of the model system. We perform numerical simulation for temporal (ODE model) as well as a spatial model system. Our numerical investigation shows that microzooplankton grazing refuse of IPP leads to oscillatory dynamics. Increasing diffusion coefficient of microzooplankton shows Turing instability. Time evolution also plays an important role in the stability of system dynamics. The results obtained in this paper are useful to understand the dominance of algal bloom in coastal and estuarine ecosystem.

Bifurcation Analysis of a Prey–Predator Model with Beddington–DeAngelis Type Functional Response and Allee Effect in Prey

The article aims to study a prey–predator model which includes the Allee effect phenomena in prey growth function, density dependent death rate for predators and Beddington–DeAngelis type functional response. We notice the changes in the existence and stability of the equilibrium points due to the Allee effect. To investigate the complete global dynamics of the Allee model, we present here a two-parametric bifurcation diagram which describes the effect of density dependent death rate parameter of predator on dynamical changes of the system. We have also analyzed all possible local and global bifurcations that the system could go through, namely transcritical bifurcation, saddle-node bifurcation, Hopf-bifurcation, cusp bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Finally, the impact of the Allee effect in the considered system is investigated by comparing the dynamics of both the systems with and without Allee effect.

Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition

Incorporating Prey Refuge in a Prey-Predator Model with Beddington-DeAngelis Type Functional Response: A Comparative Study on Intra-Specific Competition

Modeling the plankton–fish dynamics with top predator interference and multiple gestation delays

Many prey organisms have developed defense mechanisms against predation. If the predator can encounter in a limited spatial domain, the feeding rates reflect interference between predators and the corresponding functional response depends on both predator and prey densities. In this manuscript, an attempt has been made to understand the role of top predator interference and gestation delays on the dynamics of a three-species food chain model involving intermediate and top predators population. Interaction between the prey and an intermediate predator follows the Monod–Haldane functional response, while that between the top predator and its prey depends on Beddington–DeAngelis-type functional response. Analytically, we study the essential mathematical features such as boundedness, stability and direction of bifurcating periodic solution around the coexisting equilibrium for the model system. Numerically, we study the Hopf and transcritical bifurcations scenarios with respect to inhibitory effect of phytoplankton against zooplankton and death rate of fish population for the non-delayed system. Further, we study the stability behavior of the delayed model system. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Further, the system shows extinction of predators with the increase in inhibitory effect. Stability domain plots with respect to different important system parameters for non-delayed system and delay parameters for delayed system give a significant impact to study the stability of the different equilibrium points and bifurcation scenarios of both non-delayed and delayed systems. Our results point out the complexity of three-way interactions between phytoplankton, zooplankton and fish population in marine environment and highlight the role of predator interference and gestation delays in exhibiting the chaotic dynamics and extinction of predator population.

Pattern formation by fractional cross-diffusion in a predator–prey model with Beddington–DeAngelis type functional response

In this paper, we explore the emergence of patterns in a fractional cross-diffusion model with Beddington–DeAngelis type functional response. First, we explore the stability of the equilibrium points with or without fractional cross-diffusion. Instability of equilibria can be induced by cross-diffusion. We perform the linear stability analysis to obtain the constraints for the Turing instability. It is found by theoretical analysis that cross-diffusion is an important mechanism for the appearance of Turing patterns. For the dynamics of pattern, the weakly nonlinear multi-scaling analysis has been performed to obtain the amplitude equations. Finally, we ensure the existence of Turing patterns such as squares, spots and stripes by using the stability analysis of the amplitude equations. Moreover, with the assistance of numerical simulations, we verify the theoretical results.

A Beddington–DeAngelis type one-predator two-prey competitive system with help

In this paper, we present and study a two competing prey and one predator system where during predation both the teams of prey help each other and the rate of predation on both the teams are different. The prey consumption rate per individual of predator is considered as Beddington–DeAngelis type. Beddington–DeAngelis type functional response describes predator feeding rate depending on both the prey and predator density. Ecological significance of considering this type of functional response depends on how frequently a predator encounters another predator while feeding on the same type of prey, resulting in strong intraspecific competition and thereby reducing the predator feeding rate. Firstly we establish persistence and permanence of the system and examine the role of cooperation coefficient. Different conditions for the coexistence of equilibrium solutions are discussed. Then the local asymptotic stability of various equilibrium points and the existence of Hopf bifurcation at the positive equilibrium are carried out to understand the dynamics of the model system. The Kolmogorov analysis of subsystems suggests for the existence of transcritical bifurcation. Sufficient conditions for the nonexistence of periodic orbits of the considered model system have been presented. The analytical results found in this paper are illustrated with the help of numerical examples.

A fractional-order predator–prey model with Beddington–DeAngelis functional response and time-delay

Herein, we propose a fractional-order prey–predator dynamical system with Beddington–DeAngelis type functional response and time-delay. We study the existence of various equilibrium points, and sufficient conditions that ensure the local asymptotic stability of the steady states of such system. The system shows a Hopf-bifurcation which depends on the time-delay. The presence of fractional-order and time-delay in the differential model improves the stability of the solutions and enriches the dynamics of the model. Some numerical examples and simulations are provided to validate the derived theoretical results.

Complex dynamics of diffusive predator–prey system with Beddington–DeAngelis functional response: The role of prey-taxis

An attempt has been made to understand the complex dynamics of a spatial predator–prey system with Beddington–DeAngelis type functional response in the presence of prey-taxis and subjected to homogenous Neumann boundary condition. To describe the active movement of predators to the regions of high prey density or if the predator is following some sort of odor to find the prey, the prey-taxis phenomenon is included in a general reaction–diffusion equation. We have studied the linear stability analysis of both spatial and non-spatial models. We have performed extensive simulations to identify the conditions to generate spatiotemporal patterns in the presence of prey-taxis. It has been observed that the increasing predator active movement from the bifurcation value, the system shows chaotic behavior whereas increasing value of random movement brings the system back to order from the disordered state.

Hopf bifurcation of a delayed diffusive predator-prey model with strong Allee effect

The paper is concerned with a delayed diffusive predator-prey system where the growth of prey population is governed by Allee effect and the predator population consumes the prey according to Beddington-DeAngelis type functional response. The situation of bi-stability and the existence of two coexisting equilibria for the proposed model system are addressed. The stability of the steady state together with its dependence on the magnitude of time delay has been obtained. The conditions that guarantee the occurrence of the Hopf bifurcation in presence of delay are demonstrated. Furthermore, the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem. Finally, some numerical simulations have been carried out in order to validate the assumptions of the model.

Impacts of additional food on diffusion induced instabilities in a predator-prey system with mutually interfering predator

In this paper, diffusion driven pattern forming instabilities in a predator-prey system with mutually interfering predators described by the Beddington-DeAngelis type functional response, are investigated in the presence of additional food for predators. Conditions for Hopf, Turing and wave instabilities are investigate around the coexisting equilibrium point analytically. Numerical simulation results are presented to show different types of spot, stripe and their mixture patterns. Different spatial domains in the parameter space are plotted. The existence and non-existence of positive, non-constant, steady states of the reaction-diffusion model are established. It is observed that spatio-temporal pattern of a predator prey system can change significantly depending upon the parameters related to additional food. We can conclude from our study, that the reasons of appearance of different spatio-temporal patterns in the real life ecological systems may be due to variation of additional food.

Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response.

In any reaction-diffusion system of predator-prey models, the population densities of species are determined by the interactions between them, together with the influences from the spatial environments surrounding them. Generally, the prey species would die out when their birth rate is too low, the habitat size is too small, the predator grows too fast, or the predation pressure is too high. To save the endangered prey species, some human interference is useful, such as creating a protection zone where the prey could cross the boundary freely but the predator is prohibited from entering. This paper studies the existence of positive steady states to a predator-prey model with reaction-diffusion terms, Beddington-DeAngelis type functional response and non-flux boundary conditions. It is shown that there is a threshold value [Formula: see text] which characterizes the refuge ability of prey such that the positivity of prey population can be ensured if either the prey's birth rate satisfies [Formula: see text] (no matter how large the predator's growth rate is) or the predator's growth rate satisfies [Formula: see text], while a protection zone [Formula: see text] is necessary for such positive solutions if [Formula: see text] with [Formula: see text] properly large. The more interesting finding is that there is another threshold value [Formula: see text], such that the positive solutions do exist for all [Formula: see text]. Letting [Formula: see text], we get the third threshold value [Formula: see text] such that if [Formula: see text], prey species could survive no matter how large the predator's growth rate is. In addition, we get the fourth threshold value [Formula: see text] for negative [Formula: see text] such that the system admits positive steady states if and only if [Formula: see text]. All these results match well with the mechanistic derivation for the B-D type functional response recently given by Geritz and Gyllenberg (J Theoret Biol 314:106-108, 2012). Finally, we obtain the uniqueness of positive steady states for [Formula: see text] properly large, as well as the asymptotic behavior of the unique positive steady state as [Formula: see text].

Stationary patterns induced by self- and cross-diffusion in a Beddington–DeAngelis predator–prey model

The study of spatial pattern formation through diffusion-driven instability of reaction–diffusion models of interacting species has long been one of the fundamental problems in mathematical ecology. The present article is concerned with interacting predator–prey reaction–diffusion model with Beddington–DeAngelis type functional response. The essential conditions for Hopf and Turing bifurcations are derived on the spatial domain. The parameter space for Turing spatial structure is established. Based on the bifurcation analysis, the spatial pattern formation in Turing space through numerical simulations is carried out in order to study the evolution procedure of the proposed model system in the vicinity of coexistence equilibrium point. The consequences of the results obtained reveal that the effects of self- and cross-diffusion play significant role on the steady state spatiotemporal pattern formation of the reaction–diffusion predator–prey model system which concerns the influence of intra-species competition among predators. Finally, ecological implications of the present results obtained are discussed at length towards the end in order to validate the applicability of the model under consideration.