Beddington-DeAngelis Functional Response Research Articles

Bifurcation branch and stability of stationary solutions of a predator–prey model

ABSTRACT 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.

Global Stability of An Eco-epidemiological Model with Beddington-DeAngelis Functional Response and Delay

Global Stability of An Eco-epidemiological Model with Beddington-DeAngelis Functional Response and Delay

Significance of additional food to mutually interfering predator under herd behavior of prey on the stability of a spatio-temporal system

In this paper we represent a predator-prey model in temporal as well as spatio-temporal contexts which follows Beddington-DeAngelis (simply written as BD) functional response while prey forming a square shaped herd as a form of group defense and the predator who is provided additional food with the combination of primary food to improve the absolute growth of the predator to restrain the prey population. Boundedness, local stability criterion of equilibrium points, persistence conditions, global stability conditions of interior equilibrium point, the extinction criterion for the prey and predator population, bifurcation theory at the equilibrium points of the system are elaborately investigated. The Turing instability of two species prey-predator dynamical model system is thoroughly discussed. The structural interpretation of Turing patterns are derived by weakly non-linear analysis using amplitude equations. Lastly, the diffusion of dynamical system in the spatial domain is depicted by different pattern formations. The results of theoretical analysis are well justified with numerical simulations. Thus we conduct our results in the ecosystem and yield the concept of group defence of prey and additional food to predator.

Bifurcation and patterns induced by flow in a prey-predator system with Beddington-DeAngelis functional response.

In this paper, a prey-predator system described by a couple of advection-reaction-diffusion equations is studied theoretically and numerically, where the migrations of both prey and predator are considered and depicted by the unidirectional flow (advection term). To investigate the effect of population migration, especially the relative migration between prey and predator, on the population dynamics and spatial distribution of population, we systematically study the bifurcation and pattern dynamics of a prey-predator system. Theoretically, we derive the conditions for instability induced by flow, where neither Turing instability nor Hopf instability occurs. Most importantly, linear analysis indicates the instability induced by flow depends only on the relative flow velocity. Specifically, when the relative flow velocity is zero, the instability induced by flow does not occur. Moreover, the diffusion-driven patterns at the same flow velocity may not be stationary because of the contribution of flow. Numerical bifurcation analyses are consistent with the analytical results and show that the patterns induced by flow may be traveling waves with different wavelengths, amplitudes, and speeds, which are illustrated by numerical simulations.

Chaos in Beddington–DeAngelis food chain model with fear

In the current paper, the effect of fear in three species Beddington–DeAngelis food chain model is investigated. A three species food chain model incorporating Beddington-DeAngelis functional response is proposed, where the growth rate in the first and second level decreases due to existence of predator in the upper level. The existence, uniqueness and boundedness of the solution of the model are studied. All the possible equilibrium points are determined. The local as well as global stability of the system are investigated. The persistence conditions of the system are established. The local bifurcation analysis of the system is carried out. Finally, numerical simulations are used to investigate the existence of chaos and understand the effect of varying the system parameters. It is observed that the existence of fear up to a critical value has a stabilizing effect on the system; otherwise it works as an extinction factor in the system.

Predator–prey interaction system with mutually interfering predator: role of feedback control

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.

Stationary distribution of a stochastic chemostat model with Beddington–DeAngelis functional response

In this paper, we study a stochastic chemostat model with Beddington–DeAngelis functional response. By constructing a suitable stochastic Lyapunov function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution to the model. The existence of a stationary distribution implies stochastic weak stability.

Patterns in a Modified Leslie–Gower Model with Beddington–DeAngelis Functional Response and Nonlocal Prey Competition

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.

Global stability of the boundary solution of a nonautonomous predator-prey system with Beddington-DeAngelis functional response

ABSTRACT In this paper, we consider a nonautonomous predator-prey system with Beddington-DeAngelis functional response and explore the global stability of boundary solution. Based on the dynamics of logistic equation, some new sufficient conditions on the global asymptotic stability of boundary solution are presented for general time-dependence case. Our main results indicate that (i) the long-term ineffective predation behaviour or high mortality of predator species will lead the predator species to extinction, even if the intraspecies competition of predator species is weak or no intraspecies competition; (ii) the long-term intense intraspecific competition may lead the predator species to extinction, even though the long-term accumulative predation benefit is higher than the death lose. When all parameters are periodic functions with common period, a necessary and sufficient condition on the global stability of boundary periodic solution is obtained. In addition, some numerical simulations are performed to illustrate the theoretical results.

Bifurcation analysis and dynamic behavior to a predator-prey model with Beddington-DeAngelis functional response and protection zone

<p style='text-indent:20px;'>In this paper we study the protection zone problem to a predator-prey model subject to Beddington-DeAngelis functional responses and small prey growth rate. This is a successive work to a previous paper of the authors [X. He, S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol. 75 (2017) 239-257], where the model with large prey growth rate was considered. At first we establish the existence and multiplicity of positive steady state solutions, and then give the dynamic behavior of the evolution problem. It is proved that there may be no positive steady state, or may have at leat one, two, or even three positive steady states, depending on the parameters involved such as the growth rate, the predation rate, and the food handling time of the predators, the growth rate and the refuge ability of the preys, and the sizes of the habitat with protection zone. In addition, it is shown that the dynamics of the solutions rely on the initial state as well, e.g., though there could be multiple positive steady states, the prey will go to extinction as time tends to infinity if its initial value is small.

Global stability for a class of HIV virus-to-cell dynamical model with Beddington-DeAngelis functional response and distributed time delay.

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.

具反馈控制和Beddington-DeAngelis功能反应的离散竞争系统的持久性

具反馈控制和Beddington-DeAngelis功能反应的离散竞争系统的持久性

具反馈控制和Beddington-DeAngelis功能反应的离散竞争系统的概周期解

具反馈控制和Beddington-DeAngelis功能反应的离散竞争系统的概周期解

GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number $R_0$ is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if $R_0 $ is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when $R_0 $ is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.

Modeling the fear effect and stability of non-equilibrium patterns in mutually interfering predator–prey systems

Recent demographic experiments have demonstrated that both birth and survival in free-living animals are essentially affected due to having sufficient exposure to predators and further leaving physiological stress effects. In this paper, we have proposed and analyzed a predator–prey interaction model with Beddington–DeAngelis functional response (BDFR) and incorporating the cost of fear into prey reproduction. Stability analysis and the existence of transcritical bifurcation are studied. For the spatial system, the Hopf-bifurcation around the interior equilibrium, stability of homogeneous steady state, direction and stability of spatially homogeneous periodic orbits have been established. Using Normal form of the steady state bifurcation, the possibility of pitchfork bifurcation has been established. The impact of the level of fear and mutual interference on the stability and Turing patterns of the spatiotemporal system have been discussed in detail. Simulation results ensure that the fear of predator stabilizes the system dynamics and cost the overall population size of the species.

Dynamics of One-Consumer-Two-Resources Ecological System with Beddington-Deangelis Functional Response

In this paper we study a one-consumer-two-resources ecological system with simple mass action and Beddington-DeAngelis functional responses. The essential mathematical features of the present model have been analyzed thoroughly in terms of the local and the global stability and the bifurcations arising in some selected situations as well. The ranges of the significant parameters under which the system admits a Hopf bifurcation are investigated. The explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions are also derived with the use of both the normal form and the central manifold theory (cf. Carr [1], Hassard et al. [2]). Numerical illustrations are performed finally in order to validate the applicability of the model under consideration.

Shape Effects on Herd Behavior in Prey–Predator Interaction with Multiple Delays and Alternative Food Source to Predator in Non-autonomous Environment

Prey population gathers together in a herd, thus possesses a social behavior. We consider the fact that the individuals of prey and predator interact mostly along the perimeter (boundary) of the pack (herd) formed in a 2D space or along the total surface area of the herd in a 3D space. Herding in prey implies the prey–predator interaction would be limited to the prey exposed on the outside of the herd. Thus prey on the inside of the herd has a diluted risk of predation. We model the three-dimensional shaped herds of hilsa (Tenualosa ilisha) fishes and their association with eel (Macrognathus pancalus) fishes (predator to hilsa) in a non-autonomous environment. Eel’s reliance on additional food sources during the absence of the hilsa has been modeled using Beddington DeAngelis functional response. We consider the delay in hilsa attaining sexual maturity and also the much-needed delay in collecting the harvest. We discuss various dynamical aspects such as persistence, the existence of a globally stable solution, and periodic solution. Through numerical simulation, we prove the significance of the presence of two-time delays for hilsa and the importance of the alternative food source for the predator. We do not see any difference in the dynamical aspects of time-variant hilsa–eel models with the cube and sphere-shaped herding in hilsa.

Long-time behavior of a diffusive prey–predator system with Beddington–DeAngelis functional response in heterogeneous environment

In this paper, we study a diffusive prey–predator model with Beddington–DeAngelis functional response when the intraspecific crowding effect for the prey population disappears in some subdomain $$\Omega _0$$ of their whole habitat. We are concerned about the global dynamics of the system and discuss it based on two coefficients: the growth rate of prey $$\lambda $$ and that of predator $$\mu $$. In particular, the results show that, with the degeneracy of prey population’s intraspecific crowding effect in the subdomain $$\Omega _0$$, the density of prey may tend to infinity in $$\Omega _0$$. On the other hand, the unboundedness of prey means that the solutions of the system lose compactness which may bring difficulty to investigate the long-time behavior of the solutions. Finally, some numerical simulations are presented to support and strengthen our theoretical analysis.

Spatial Patterns of a Predator–Prey Model with Beddington–DeAngelis Functional Response

This paper is concerned with the spatial patterns of a predator–prey system with Beddington–DeAngelis functional response, in which the parameter [Formula: see text] measuring the mutual interference between predators can play an essential role. By using the bifurcation theory and implicit function theorem we first consider the positive steady state solution bifurcating from the semitrivial steady state solution set of the system and prove that the positive steady state solution is constant. Then we show that nonconstant positive steady state solution may bifurcate from the constant positive steady state solution when [Formula: see text] is neither small nor large. Finally, we show that spatially nonhomogeneous periodic orbits may also bifurcate from the constant positive steady state solution as [Formula: see text] is not large.

Persistence of pollination mutualisms under pesticides

A plant–pollinator system with pesticides is considered, in which the plant or plant’s seeds are treated by pesticides and the food (e.g., pollen and nectar) produced by plant is contaminated. The effect of pesticides is characterized by a toxin-determined functional response, which is a modification of the traditional Beddington–DeAngelis functional response through explicitly including a reduction in the consumption of food by the pollinator due to pesticides. Dynamics of the model demonstrate basic mechanisms by which the toxicity results in reduction of pollinators and leads to extinction of pollination mutualisms. It is also shown that varying the toxicity level could make the dynamics transition between uniform persistence, bi-stability, tri-stability and species’ extinction in a smooth fashion. Moreover, a detailed bifurcation analysis of the system reveals a rich array of novel behaviors including double saddle-node bifurcation and double transcritical bifurcation, which exhibits important biological implications in the complex dynamics of pollination mutualisms under pesticides.