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This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results. [ABSTRACT FROM AUTHOR]

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Abstract This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin’s coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified.

This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations.

This paper investigates a delayed predator-prey model with discontinuous harvesting and Beddington–DeAngelis functional response. Using the theory of differential inclusion theory, the existence of positive solutions in the sense of Filippov is discussed. Under reasonable assumptions and periodic disturbances, the existence of positive periodic solutions of the model is studied based on the theory of Mawhin's coincidence degree. Finally, through numerical simulation, the correctness and feasibility of the conclusions are verified. [ABSTRACT FROM AUTHOR]

This study considers two time delays applied in a commensalism system with a Beddington–DeAngelis functional response. In contrast with existing literature on commensalism systems, the system considered in the present study has two time delays in one species. The local stability of the positive equilibrium and Hopf bifurcation are investigated. The linearized stability is thoroughly examined. Furthermore, the characteristic equations are investigated, and the time delays are applied as the bifurcation parameter. Eventually, the presence of Hopf bifurcation is demonstrated. The Lyapunov functional is constructed, and the system is shown to have uniform persistence. The consistent persistent domain of the system is obtained by constructing a persistent function. Numerical simulations are conducted, demonstrating the reliability of the derived results. [ABSTRACT FROM AUTHOR]

The dynamics of prey–predator system, when one or both the species are harvested non-linearly, has become a topic of intense study because of its wide applications in biological control and species conservation. In this paper we have discuss different bifurcation analysis of a two dimensional prey–predator model with Beddington–DeAngelis type functional response in the presence of prey refuge and non-linear harvesting of both species. We have studied the positivity and boundedness of the model system. All the biologically feasible equilibrium points are investigated and their local stability is analyzed in terms of model parameters. The global stability of coexistence equilibrium point has been discussed. Depending on the prey harvesting effort ( E 1 ) and degree of competition among the boats, fishermen and other technology ( l 1 ) used for prey harvesting, the number of axial and interior equilibrium points may change. The system experiences different type of co-dimension one bifurcations such as transcritical, Hopf, saddle-node bifurcation and co-dimension two Bogdanov–Takens bifurcation. The parameter values at the Bogdanov–Takens bifurcation point are highly sensitive in the sense that the nature of coexistence equilibrium point changes dramatically in the neighbourhood of this point. The feasible region of the bifurcation diagram in the l 1 - E 1 parametric plane divides into nine distinct sub-regions depending on the number and nature of equilibrium points. We carried out some numerical simulations using the Maple and MATLAB software to justify our theoretical findings and finally some conclusions are given. [ABSTRACT FROM AUTHOR]

In this paper, a fractional density-dependent prey-predator model has been considered. Certain reading of local and global stabilities of an equilibrium point of a system was extracted and conducted by applying fractional systems' stability theorems along with Lyapunov functions. Meanwhile, the persistence of the aforementioned system has been discussed and claimed to imply a local asymptotic stability for the given positive equilibrium point. Moreover, the presented model was extended to a periodic impulsive model for the prey population. Such an expansion was implemented through the periodic catching of the prey species and the periodic releasing of the predator population. By studying the effect of changing some of the system's parameters and drawing their bifurcation diagram, it was observed that different periodic solutions appear in the system. However, the effect of an impulse on the system subjects the system to various dynamic changes and makes it experience behaviors including cycles, period-doubling bifurcation, chaos and coexistence as well. Finally, by comparing the fractional system with the classic one, it has been concluded that the fractional system is more stable than its classical one. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.

This paper introduces diffusion into an immunosuppressive infection model with virus stimulation delay and Beddington-DeAngelis functional response. First, we study the stability of positive constant steady state solution and show that the Hopf bifurcation will exist under certain conditions. Second, we derive the normal form of the Hopf bifurcation for the model reduced on the center manifold by using the multiple time scales (MTS) method. Moreover, the direction and stability of the bifurcating periodic solution are investigated. Finally, we present numerical simulations to verify the results of theoretical analysis and provide corresponding biological explanations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion [Math Processing Error] d 2 or cross-diffusion [Math Processing Error] d 4 by using the Leray-Schauder degree theory, and the results reveal that the diffusion [Math Processing Error] d 2 and the fractional type cross-diffusion [Math Processing Error] d 4 can create spatial patterns. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This paper studies the existence and multiplicity of periodic orbits of a stage-structured predator-prey model with Beddington-DeAngelis functional response. We derive an existence criterion of periodic orbit in terms of inequalities of system's nine parameters and prove that the system admits at least two limit cycles or three limit cycles via subcritical Hopf bifurcation or generalized Hopf bifurcation theory and hypersurface theory. We also prove that each one-parameter system possesses at least one limit cycle when it is larger or smaller than its Hopf bifurcating value, or between its Hopf bifurcating values. Combining theoretic analysis and numerical simulations, we investigate global bifurcations of limit cycles for a varying parameter system, which provides a plenty of bifurcating properties and again suggests that the system has at least three limit cycles. Similar results are obtained for the system without mutual interference. [ABSTRACT FROM AUTHOR]

This research involves a discrete amensalism system with the Beddington–DeAngelis functional response and Allee effect for the unaffected species. We begin by investigating the presence and local stability of fixed points. Then, utilizing the central manifold theorem and bifurcation theory, we analyze a variety of codimension one and codimension two bifurcations, which include transcritical, pitchfork, fold, flip, fold-flip and 1 : 2 strong resonance bifurcations. These theoretical findings suggest that Allee effect serves a crucial role in stabilizing the population sizes of both species. In addition, Allee effect would make the system spend more time to achieve its stable steady-state solution. They are illustrated via numerical simulations. [ABSTRACT FROM AUTHOR]

This manuscript examines the dynamics of a predator-prey model of the Beddington-DeAngelis type with strong Allee effect on prey growth function. Conditions for the existence and equilibria types are established. By taking Allee effect, predation rate of the prey and growth rate of the predator as bifurcation parameters, different potential bifurcations are explored, including codimension one bifurcations: fold bifurcation, transcritical bifurcation, Hopf bifurcation, and codimension two bifurcations: cusp bifurcation, Bogdanov-Takens bifurcation, and Bautin bifurcation. In addition, to confirm the dynamic behavior of the system, bifurcation diagrams are given in different parameter spaces and phase portraits are also presented to provide corresponding interpretation. The findings indicate that the dynamics of our system is much richer than the system with no strong Allee effect. [ABSTRACT FROM AUTHOR]

Abstract In this paper, a modified cross-diffusion Leslie–Gower predator–prey model with the Beddington–DeAngelis functional response is studied. We use the linear stability analysis on constant steady states to obtain sufficient conditions for the occurrence of Turing instability and Hopf bifurcation. We show that the Turing instability and associated patterns are induced by the variation of parameters in the cross-diffusion term. Some numerical simulations are given to illustrate our results.

We propose four predator–prey models: RM (Rosenzweig–MacArthur) model, BD model (RM type model with Beddington–DeAngelis functional response), RMI model (i.e., RM model with intraspecific competition among predators) and BDI model (BD model with intraspecific competition among predators). Each model incorporates time delay in the predators' numerical response. We first analyse the delay-induced stability for all the models. We show that increasing delay always destabilizes a coexisting stable equilibrium in RM and BD models. However, increasing delay does not always destabilize a stable equilibrium in RMI and BDI models. Indeed, the stable equilibrium, in the latter two models, may also maintain its stability due to varying delay. Thus, one of the major conclusions is that the invariance property of the local stability in RMI and BDI models is due to the influence of intraspecific competition. Analytically, we prove that stability switching is impossible to occur in all the models. Later, we implement harvesting of the prey and predator separately, which may generate stability switching. If populations oscillate in the unharvested system, extensive effort has a potential to stabilize the equilibrium. Under the same natural condition (unharvested situation), prey harvesting and predator harvesting may produce opposite dynamic modes. [ABSTRACT FROM AUTHOR]

This paper is concerned with a reaction–diffusion predator–prey system with a Beddington–DeAngelis functional response and subject to Neumann boundary conditions. The stability criteria and Turing instability conditions of the constant positive steady state of the system are provided. Some prior estimates, conditions to the nonexistence and the existence of non-constant positive steady-state solutions to the system are established and Turing patterns are also explored. The existence of spatially homogeneous and nonhomogeneous Hopf bifurcations of the constant positive equilibrium are discussed. Their achievements provide that the presence of the Beddington–DeAngelis functional response essentially increases the spatiotemporal complexity of the model and also leads to the appearance of Turing instability. Finally, some numerical tests are given to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper formulates and explores a nonautonomous impulsive stochastic predator-prey system with Beddington-DeAngelis (BD) functional response, where only the prey has a disease, which incorporates modified saturated incidence. The sufficient criteria of extinction and non-persistence in the mean of the target model are established, revealing that different intensities of stochastic perturbations contribute to dynamics of the system mentioned above. Stochastically ultimate boundedness is examined, and we further establish sufficient conditions for global attractivity. Our analytical findings are verified through numerical simulations. [ABSTRACT FROM AUTHOR]

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 m0$$ {m}_0 $$ for predator capturing rate m$$ m $$, 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. [ABSTRACT FROM AUTHOR]

In this study, a 2D age-dependent predation equations characterizing Beddington − − DeAngelis type schemes are established to investigate the evolutionary dynamics of population, in which the predator is selected to be depicted with an age structure and its fertility function is assumed to be a step function. The dynamic behaviors of the equations are derived from the integrated semigroup method, the Hopf bifurcation theorem, the center manifold reduction and normal form theory of semilinear equations with non-dense domain. It turns out that the equations appear the oscillation phenomenon via Hopf bifurcation (positive equilibrium age distribution lose its stability and give rise to periodic solutions), as the bifurcation parameter moves across certain threshold values. Additionally, the explicit expressions are offered to determine the properties of Hopf bifurcation (the direction the Hopf bifurcation and the stability of the bifurcating periodic solutions). This technique can also be employed to other epidemic and ecological equations. Eventually, some numerical simulations and conclusions are executed to validating the major results of this work. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the existence and nonexistence of traveling wave solution for a nonlocal dispersal delayed predator-prey system with the Beddington-DeAngelis functional response and harvesting. By constructing the suitable upper-lower solutions and applying Schauder's fixed point theorem, we show that there exists a positive constant $ c^* $ such that the system possesses a traveling wave solution for any given $ c > c^* $. Moreover, the asymptotic behavior of traveling wave solution at infinity is obtained by the contracting rectangles method. The existence of traveling wave solution for $ c = c^* $ is established by means of Corduneanu's theorem. The nonexistence of traveling wave solution in the case of $ c < c^* $ is also discussed.

This paper is concerned about a diffusive degenerate predator–prey model with Beddington–DeAngelis functional response subject to homogeneous Neumann boundary condition. First, the global bifurcation branches of positive stationary solutions are studied, which are quite different from those with different degeneracy or functional response. Second, the multiplicity and stability of positive stationary solutions are obtained as the parameter k or m in the Beddington–DeAngelis functional response is large enough, from which the effects of the functional response on the coexistence region are revealed. In particular, the global stability of the positive stationary solution is derived as it exists uniquely. [ABSTRACT FROM AUTHOR]

In this paper, a modified cross-diffusion Leslie–Gower predator–prey model with the Beddington–DeAngelis functional response is studied. We use the linear stability analysis on constant steady states to obtain sufficient conditions for the occurrence of Turing instability and Hopf bifurcation. We show that the Turing instability and associated patterns are induced by the variation of parameters in the cross-diffusion term. Some numerical simulations are given to illustrate our results. [ABSTRACT FROM AUTHOR]

This paper is designed based on the combined bioeconomic harvesting of Holling-Tanner prey-predator competition model with Beddington-DeAngelis functional response with two different delays. The situation and existence of the steady state points for proposed model are discussed. The conditions for local stability of the interior steady state points together with the existence of Hopf bifurcation at the interior steady state point are addressed. The situations for stability and the direction of bifurcating periodic solutions are obtained by using center manifold theory. Also total revenue is calculated using bionomic equilibrium, and it is maximized by using Pontryagin's maximum principle. Some numerical simulations together with pictorial representations are placed in the paper for showing the effects of various parameters in the considered model. Conclusions with the future study are described at last. [ABSTRACT FROM AUTHOR]

In this paper, a Beddington-DeAngelis amensalism system with weak Allee effect on the first species are introduced and investigated. The existence and stability of all possible trivial, semi-trivial and interior equilibria of the model are studied. By utilizing Sotomayor's theorem, bifurcation analysis has been proposed and obtain one saddle-node bifurcation. Furthermore, in view of Poincaré transformation, the behaviors near infinity and the nonexistence of close orbits are obtained and lead to the presentation of all possible global phase portraits. The global phase portrait in R 5 with two stable node E 2 and E 1 ∗ is a new case due to the appearance of bistable structure. Finally, some numerical examples are offered to verify and extend the analytical results and visualize the interesting phenomenon. [ABSTRACT FROM AUTHOR]

A HIV virus-to-cell dynamical model with distributed delay and Beddington-DeAngelis functional response is proposed in this paper. Using the characteristic equations and analytical means, the principle reproduction number R0 on the local stability of infection-free and chronic-infection equilibria is established. Furthermore, by constructing suitable Lyapunov functionals and using LaSalle invariance principle, we show that if R0 ≤ 1 the infection-free equilibrium is globally asymptotically stable, while if R0 > 1 the chronic-infection equilibrium is globally asymptotically stable. Numerical simulations are presented to illustrate the theoretical results. Comparing the effects between discrete and distributed delays on the stability of HIV virus-to-cell dynamical models, we can see that they could be same and different even opposite.

The most important fact in the field of theoretical ecology and evolutionary biology is the strategy of predation for predators and avoidance of prey from predator attack. A lot of experimental works suggest that the reduction of prey depends on both direct predation and fear of predation. We explore the impact of fear effect and mutual interference among predators into a three-species food chain model. In this manuscript, we have considered a tri-topic food chain model with Beddington–DeAngelis functional response between interacting species, incorporating the reduction of prey and intermediate predator growth due to the fear of intermediate and top predator, respectively. We have provided parametric conditions for existence of biologically feasible equilibria as well as their local and global stability. We have established conditions of transcritical, saddle-node and Hopf bifurcation about different equilibria. Finally, we have performed some numerical investigations to justify analytical findings. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we explore a stochastic non-autonomous one-predator–two-prey system with Beddington–DeAngelis functional response and impulsive perturbations. First, by using Itô’s formula, exponential martingale inequality, Chebyshev’s inequality and other mathematical skills, we establish some sufficient conditions for extinction, non-persistence in the mean, weak persistence, persistence in the mean and stochastic permanence of the solution of the stochastic system. Then the limit of the average in time of the sample path of the solution is estimated by two constants. Afterwards, the lower-growth rate and the upper-growth rate of the positive solution are estimated. In addition, sufficient conditions for global attractivity of the system are established. Finally, we carry out some simulations to verify our main results and explain the biological implications: the large stochastic interference is disadvantageous for the persistence of the population and the strong impulsive harvesting can lead to extinct of the population.

In this paper, a differential algebraic predator-prey model including two delays, Beddington-DeAngelis functional response and nonlinear predator harvesting is proposed. Without considering time delay, the existence of singularity induced bifurcation is analyzed by regarding economic interest as bifurcation parameter. In order to remove singularity induced bifurcation and stabilize the proposed system, state feedback controllers are designed in the case of zero and positive economic interest respectively. By the corresponding characteristic transcendental equation, the local stability of interior equilibrium and existence of Hopf bifurcation are discussed in the different case of two delays. By using normal form theory and center manifold theorem, properties of Hopf bifurcation are investigated. Numerical simulations are given to demonstrate our theoretical results.

In this paper, a predator–prey model with age structure, Beddington–DeAngelis functional response and time delays is considered. Using a geometric method for studying transcendental equation with two delays, we conduct detailed analysis on the distribution of the roots for the characteristic equation of the model. Then, applying the integrated semigroup theory and the Hopf bifurcation theorem for an abstract Cauchy problem within a nondense domain, we proved the existence of Hopf bifurcation for the model. Stability switches can also occur, as the two time delays pass through a continuous curve in the parameter plane. To illustrate the theoretical results, numerical simulations are presented. [ABSTRACT FROM AUTHOR]

A diffusive intraguild predation model with delay and Beddington-DeAngelis functional response is considered. Dynamics including stability and Hopf bifurcation near the spatially homogeneous steady states are investigated in detail. Further, it is numerically demonstrated that delay can trigger the emergence of irregular spatial patterns including chaos. The impacts of diffusion and functional response on the model's dynamics are also numerically explored.

This work investigates a prey–predator model with Beddington–DeAngelis functional response and discrete time delay in both theoretical and numerical ways. Firstly, we incorporate into the system a discrete time delay between the capture of the prey by the predator and its conversion to predator biomass. Moreover, by taking the delay as a bifurcation parameter, we analyze the stability of the positive equilibrium in the delayed system. We analytically prove that the local Hopf bifurcation critical values are neatly paired, and each pair is joined by a bounded global Hopf branch. Also, we show that the predator becomes extinct with an increase of the time delay. Finally, before the extinction of the predator, we find the abundance of dynamical complexity, such as supercritical Hopf bifurcation, using the numerical continuation package DDE-BIFTOOL. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

• We construct a Leslie-Gower type prey-predator system with feedback. • We systematically analyze the effects of feedback controls on the dynamics of ecosystems. • Pattern transition emerges as feedback intensity varies. In this study, we investigate the global dynamics of non-autonomous and autonomous systems based on the Leslie–Gower type model using the Beddington–DeAngelis functional response (BDFR) with time-independent and time-dependent model parameters. Unpredictable disturbances are introduced in the forms of feedback control variables. BDFR explains the feeding rate of the predator as functions of both the predator and prey densities. The global stability of the unique positive equilibrium solution of the autonomous model is determined by defining an appropriate Lyapunov function. The condition obtained for the global stability of the interior equilibrium ensures that the global stability is free from control variables, which is also a significant issue in the ecological balance control procedure. The autonomous system exhibits complex dynamics via bifurcation scenarios, such as period doubling bifurcation. We prove the existence of a globally stable almost periodic solution of the associated non-autonomous model. The different coefficients of the system are taken as almost periodic functions by generalizing periodic assumptions. The permanence of the non-autonomous system is established by defining upper and lower averages of a function. Our results also explain how the important hypothesis in ecology known as the "intermediate disturbance hypothesis" applies in predator–prey interactions. We show that moderate feedback intensity can make both the ordinary differential equation system and partial differential equation system more robust. The results obtained provide new insights into the protection of populations, where moderate feedback intensity can promote the coexistence of species and adjusting the intensity of the feedback in appropriate regions can control the population biomass while maintaining the stability of the system. Finally, the results obtained from extensive numerical simulations support the analytical results as well as the usefulness of the present study in terms of ecological balance and bio-control problems in agro-ecosystems. [ABSTRACT FROM AUTHOR]

In this paper, we present the theoretical results on the pattern formation of a modified Leslie–Gower diffusive predator–prey system with Beddington–DeAngelis functional response and nonlocal prey competition under Neumann boundary conditions. First, we investigate the local stability of homogeneous steady-state solutions and describe the effect of the nonlocal term on the stability of the positive homogeneous steady-state solution. Lyapunov–Schmidt method is applied to the study of steady-state bifurcation and Hopf bifurcation at the interior of constant steady state. In particular, we investigate the existence, stability and multiplicity of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous periodic solutions. Furthermore, we present a simple description of the dynamical behaviors of the system around the interaction of steady-state bifurcation curve and Hopf bifurcation curve. Finally, a numerical simulation is provided to show that the nonlocal competition term can destabilize the constant positive steady-state solution and lead to the occurrence of spatially nonhomogeneous steady-state solutions and spatially nonhomogeneous time-periodic solutions. [ABSTRACT FROM AUTHOR]

In this study, we depicted the spread of pests in tea plants and their control by biological enemies in the frame of a fractional-order model, and its dynamics are surveyed in terms of boundedness, uniqueness, and the existence of the solutions. To reduce the harm to the tea plant, a harvesting term is introduced into the equation that estimates the growth of tea leaves. We analyzed various points of equilibrium of the projected model and derived the conditions for the stability of these equilibrium points. The complex nature is examined by changing the values of various parameters and fractional derivatives. Numerical computations are conducted to strengthen the theoretical findings.

Taking impulsive effects into account, an impulsive stochastic predator–prey system with the Beddington–DeAngelis functional response is proposed in this paper. First, the impulsive system is transformed into an equivalent system without pulses. Then, by constructing suitable functionals and applying the extreme-value theory of quadratic functions, sufficient conditions on the existence of periodic Markovian processes are provided. The uniform continuity and global attractivity of solutions are also investigated. Additionally, we investigate the extinction and permanence in the mean of all species with the help of comparison methods and inequality techniques. Sufficient conditions on the existence and ergodicity of the stationary distribution of solutions for the autonomous and non-impulsive case are given. Finally, numerical simulations are performed to illustrate the main results.

In this paper, we investigate the stability of equilibrium in the stage-structured and density-dependent predator–prey system with Beddington–DeAngelis functional response. First, by checking the sign of the real part for eigenvalue, local stability of origin equilibrium and boundary equilibrium are studied. Second, we explore the local stability of the positive equilibrium for τ=0 and τ≠0 (time delay τ is the time taken from immaturity to maturity predator), which shows that local stability of the positive equilibrium is dependent on parameter τ. Third, we qualitatively analyze global asymptotical stability of the positive equilibrium. Based on stability theory of periodic solutions, global asymptotical stability of the positive equilibrium is obtained when τ=0; by constructing Lyapunov functions, we conclude that the positive equilibrium is also globally asymptotically stable when τ≠0. Finally, examples with numerical simulations are given to illustrate the obtained results.

Functional responses are non-linear functions commonly used to describe the variation in the rate of consumption of resources by a consumer. They have been widely used in both theoretical and empirical studies, but a comprehensive understanding of their parameters at different levels of description remains elusive. Here, by depicting consumers and resources as stochastic systems of interacting particles, we present a minimal set of reactions for consumer resource dynamics. We rigorously derived the corresponding system of ODEs, from which we obtained via asymptotic expansions classical 2D consumer-resource dynamics, characterized by different functional responses. We also derived functional responses by focusing on the subset of reactions describing only the feeding process. This involves fixing the total number of consumers and resources, which we call chemostatic conditions. By comparing these two ways of deriving functional responses, we showed that classical functional response parameters in effective 2D consumer-resource dynamics differ from the same parameters obtained by measuring (or deriving) functional responses for typical feeding experiments under chemostatic conditions, which points to potential errors in interpreting empirical data. We finally discuss possible generalizations of our models to systems with multiple consumers and more complex population structures, including spatial dynamics. Our stochastic approach builds on fundamental ecological processes and has natural connections to basic ecological theory.

In this paper, we investigate the influence of anti-predator behavior in prey due to the fear of predators with a Beddington–DeAngelis prey–predator model analytically and numerically. We give the existence and stability of equilibria of the model, and provide the existence of Hopf bifurcation. In addition, we investigate the influence of the fear effect on the population dynamics of the model and find that the fear effect can not only reduce the population density of both predator and prey, but also prevent the occurrence of limit cycle oscillation and increase the stability of the system. [ABSTRACT FROM AUTHOR]

In this paper, a three species food chain model has been developed among the interaction of prey, predator and super predator. It is assumed that the predator shows refuge behavior to the super predator. It is also assumed that a certain amount of additional food will be supplied to the super predator. It is considered that the predator population is benefiting partially from the additional food. To get optimal harvesting of super predator the Pontryagin's maximum principle has been used. It is found that super predator may be extinct if harvesting rate increase. It is observed that as the refuge rate increases, predator population gradually increases, but super predator population decreases. Also, it is found that our proposed system undergoes oscillatory or periodic behavior as the value of refuge rate (m1), harvesting rate (E), the intrinsic growth rate of prey (r), carrying capacity of prey (k) and conservation rate of prey (c1) varies for some certain range of these parameters. It is found that this study may be useful for the increase of harvesting of a super predator by supplying the additional food to our proposed system. [ABSTRACT FROM AUTHOR]

A Holling-Tanner system with Beddington- DeAngelis functional response and prey refuge takes the form... is investigated in this paper, where a1, b, c1, β, and ε are all positive constants, m is a nonnegative constant which satisfies 0 = m < 1. For the system without prey refuge, i.e., m = 0 case, by developing the new analysis technique, we show that c1 ≥ 2 is enough to ensure the global attractivity of the positive equilibrium of the system, such a result seems amazing since it is independent of the parameter a1, b, ε and β. Consequently, we can draw the conclusion that for the most of the parameters, system admits a unique globally attractive positive equilibrium. For 0 < c1 < 2, we also investigate the stability property of the positive equilibrium. Two examples together with their numerical simulations show the feasibility of the main results. For the system with prey refuge, we show that there exists a m*, such that for all m > m*, the system always admits a unique positive equilibrium, which means that enough large prey refuge can improve the coexistence of the species. Refuge plays important role on the persistent property of the system. [ABSTRACT FROM AUTHOR]

Modeling rodent populations has been always a challenge for population ecologists. These populations have oscillations that are dynamically complex. In this paper, we consider the population dynamics of rodents under the effect of the "specialist" and "generalist" predators with Beddington–DeAngelis and sigmoidal functional responses. We discover that the ODE system has one axial state and two boundary states. If the rate of predation by the generalist predator is more than the critical value (c 2 > c 2 ∗) , then the system has a unique internal equilibrium which is stable if the predator's intrinsic growth rate of population is more than the critical value s ∗ . We show that the predation rates of the both predators ( c 1 , c 2 ) play an important role on rodent population dynamic. Then, we have considered a delay differential equation (DDE) model to account for the time delays in the transient dynamics. By treating time delays as the bifurcation parameter, we show that a Hopf bifurcation about the equilibria could happen for critical time delays. Finally, we gave an biological interpretation of our analytical results. [ABSTRACT FROM AUTHOR]

Abstract A two species amensalism model with Beddington–DeAngelis functional response is proposed and studied in this paper. The existence and stability of possible equilibria are investigated. Under some additional assumptions, there are two stable equilibria which implies this system is not asymptotically stable. Based on the stability analysis of equilibria, closed orbits and the saddle connection, we give some comprehensive bifurcation and global dynamics of the system. Next, we further incorporate the Allee effect into the second species and provide a complete qualitative and bifurcation analysis of the system with Allee effect. Numerical simulations show that the system with an Allee effect must take a longer time to reach its stable steady-state solution than that without Allee effect. There is a good agreement between the present results and the numeric simulations. [ABSTRACT FROM AUTHOR]

In the present article, we investigate the impact of fear effect in a predator–prey model, where predator–prey interaction follows Beddington–DeAngelis functional response. We consider that due to fear of predator the birth rate of prey population reduces. Mathematical properties, such as persistence, equilibria analysis, local and global stability analysis, and bifurcation analysis, have been investigated. We observe that an increase in the cost of fear destabilizes the system and produces periodic solutions via supercritical Hopf bifurcation. However, with further increase in the strength of fear, system undergoes another Hopf bifurcation and becomes stable. The stability of the Hopf-bifurcating periodic solutions is obtained by computing the first Lyapunov coefficient. Our results suggest that fear of predation risk can have both stabilizing and destabilizing effects. [ABSTRACT FROM AUTHOR]

In this paper, a planar system of ordinary differential equations is considered, which is a modified Leslie‐Gower model, considering a Beddington‐DeAngelis functional response. It generates a complex dynamics of the predator‐prey interactions according to the associated parameters. From the system obtained, we characterize all the equilibria and its local behavior, and the existence of a trapping set is proved. We describe different types of bifurcations (such as Hopf, Bogdanov‐Takens, and homoclinic bifurcation), and the existence of limit cycles is shown. Analytic proofs are provided for all results. Ecological implications and a set of numerical simulations supporting the mathematical results are also presented. [ABSTRACT FROM AUTHOR]