Global stability of a nonlinear stochastic predator–prey system with Beddington–DeAngelis functional response

Stochastically asymptotic stability in the large of a predator–prey system with Beddington–DeAngelis functional response with stochastic perturbation is considered. The result shows that if the positive equilibrium of the deterministic system is globally stable, then the stochastic model will preserve this nice property provided the noise is sufficiently small. Some simulation figures are introduced to support the analytical findings.     

Persistence and global stability of Bazykin predator–prey model with Beddington–DeAngelis response function

In this article, a predator–prey model of Beddington–DeAngelis type with discrete delay is proposed and analyzed. The essential mathematical features of the proposed model are investigated in terms of local, global analysis and bifurcation theory. By analyzing the associated characteristic equation, it is found that the Hopf bifurcation occurs when the delay parameter τ crosses some critical values. In this article, the classical Bazykin’s model is modified with Beddington–DeAngelis functional response. The parametric space under which the system enters into Hopf bifurcation for both delay and non-delay cases are investigated. Global stability results are obtained by constructing suitable Lyapunov functions for both the cases. We also derive the explicit formulae for determining the stability, direction and other properties of bifurcating periodic solutions by using normal form and central manifold theory. Our analytical findings are supported by numerical simulations. Biological implication of the analytical findings are discussed in the conclusion section.     

Dynamics of cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response

In this paper, cooperative predator–prey system with impulsive effects and Beddington–DeAngelis functional response is studied. By using comparison theorem and some analysis techniques as well as the coincidence degree theory, sufficient conditions are obtained for the permanence, extinction and the existence of positive periodic solution.     

Global properties for virus dynamics model with Beddington–DeAngelis functional response

This paper investigates the global stability of virus dynamics model with Beddington–DeAngelis infection rate. By constructing Lyapunov functions, the global properties have been analysed. If the basic reproductive ratio of the virus is less than or equal to one, the uninfected steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is more than one, the infected steady state is globally asymptotically stable. The conditions imply that the steady states are always globally asymptotically stable for Holling type II functional response or for a saturation response.     

Global asymptotic stability for predator–prey models with environmental time-variations

This paper considers a Lotka–Volterra predator–prey model with predators receiving an environmental time-variation. For such a system, a unique interior equilibrium is shown to be globally asymptotically stable if the time-variation is bounded and weakly integrally positive. Our result tells that the equilibrium can be stabilized even by nonnegative functions that make the limiting system structurally unstable. Numerical simulations are also shown to illustrate the result and to suggest that cases with time-variation acting on predators have larger-scale convergence to the equilibrium than population dynamics with time-variation acting on prey.     

Stability and traveling waves of a stage-structured predator–prey model with Holling type-II functional response and harvesting

In this paper, we consider a reaction–diffusion predator–prey model with stage-structure, Holling type-II functional response, nonlocal spatial impact and harvesting. The stability of the equilibria is investigated. Furthermore, by the cross-iteration scheme companied with a pair of admissible upper and lower solutions and Schauder fixed point theorem, we deduce the existence of traveling wave solution which connects the zero solution and the positive constant equilibrium.     

Stability of the virus dynamics model with Beddington–DeAngelis functional response and delays

A virus dynamics model with Beddington–DeAngelis functional response and delays is introduced. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and the LaSalle invariance principle, we show that the infection-free equilibrium is globally asymptotically stable if _R0?1$R_0?1$ and the chronic-infection equilibrium is globally asymptotically stable if _R0>1$R_0>1$. Numerical simulations are also given to explain our results.     

A qualitative study on general Gause-type predator–prey models with non-monotonic functional response

In this paper, we study a diffusive predator–prey model with general growth rates and non-monotonic functional response under homogeneous Neumann boundary condition. A local existence of periodic solutions and the asymptotic behavior of spatially inhomogeneous solutions are investigated. Moreover, we show the existence and non-existence of non-constant positive steady-state solutions. Especially, to show the existence of non-constant positive steady-states, the fixed point index theory is used without estimating the lower bounds of positive solutions. More precisely, calculating the indexes at the trivial, semi-trivial and positive constant solutions, some sufficient conditions for the existence of non-constant positive steady-state solutions are studied. This is in contrast to the works in previous papers. Furthermore, on obtaining these results, we can observe that the monotonicity of a prey isocline at the positive constant solution plays an important role.     

Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response

A diffusive predator–prey system with Ivlev-type functional response subject to Neumann boundary conditions is considered. Hopf and steady-state bifurcation analysis are carried out in detail. First, the stability of the positive equilibrium and the existence of spatially homogeneous and inhomogeneous periodic solutions are investigated by analysing the distribution of the eigenvalues. The direction and stability of Hopf bifurcation are determined by the normal form theory and the centre manifold reduction for partial functional differential equations and then steady-state bifurcation is studied. Finally, some numerical simulations are carried out for illustrating the theoretical results.     

Global asymptotical stability and persistent property for a diffusive predator–prey system with modified Leslie–Gower functional response

A diffusive predator–prey system with modified Holling–Tanner functional response and no-flux boundary condition is considered in this work. A sufficient condition which ensures persistence of the system is obtained. Furthermore, sufficient conditions for the global asymptotical stability of the unique positive equilibrium of the system are derived by using a comparison method. It is shown that our result supplements and complements one of the main results of Shi et al. [H.B. Shi, W.T. Li, G. Lin, Positive steady states of a diffusive predator–prey system with modified Holling–Tanner functional response, Nonlinear Analysis: Real World Applications 11 (2010) 3711–3721].     

Global stability of a Leslie–Gower predator–prey model with feedback controls

In this work the global stability of a unique interior equilibrium for a Leslie–Gower predator–prey model with feedback controls is investigated. The main result together with its numerical simulations shows that feedback control variables have no influence on the global stability of the Leslie–Gower model, which means that feedback control variables only change the position of the unique interior equilibrium and retain its global stability.     

Discrete-time analogues of predator–prey models with monotonic or nonmonotonic functional responses

Discrete-time analogues of predator–prey models with monotonic and nonmonotonic functional responses are introduced, respectively. The discrete-time analogues are considered to be numerical discretizations of the continuous-time models and we study their dynamical characteristics. It is shown that the discrete-time analogues preserve the periodicity of the continuous-time models with monotonic functional responses. Moreover, it is the first time that multiplicity of periodic solutions are studied when modeled with nonmonotonic functional responses. Unlike other types of functional responses, nonmonotone functional response declines at high prey densities. Thus the existing arguments for obtaining bounds of solutions to the operator equation $\mathit{Lx}=\lambda \mathit{Nx}$ are inapplicable to our case and some new arguments are employed for the first time.     

Predator–prey model of Beddington–DeAngelis type with maturation and gestation delays

A predator–prey model of Beddington–DeAngelis type with maturation and gestation delays is formulated and analyzed. This two-delay model is similar to the stage-structured model by Liu and Beretta [S. Liu, E. Beretta, Stage-structured predator–prey Model with the Beddington–DeAngelis functional response, SIAM J. Appl. Math. 66 (2006) 1101–1129] but contains an extra gestation delay term. Criteria for permanence and for predator extinction as well as the global attractiveness of the interior equilibrium are derived. The combined effects of the two delays and the degree of predator interference on the dynamical behaviors of the coexistence equilibrium are also studied both analytically and numerically. It is shown that complicated behaviors including chaotic and multi-periodic solutions may occur with the introduction of gestation delay, and that the predator interference can stabilize the system by simplifying the dynamical behaviors and enlarging the stability parameter fields.     

A Lyapunov functional for a stage-structured predator–prey model with nonlinear predation rate

We consider the dynamics of a general stage-structured predator–prey model which generalizes several known predator–prey, SEIR, and virus dynamics models, assuming that the intrinsic growth rate of the prey, the predation rate, and the removal functions are given in an unspecified form. Using the Lyapunov method, we derive sufficient conditions for the local stability of the equilibria together with estimations of their respective domains of attraction, while observing that in several particular but important situations these conditions yield global stability results. The biological significance of these conditions is discussed and the existence of the positive steady state is also investigated.     

Asymptotic properties of a stochastic predator–prey system with Holling II functional response

A stochastic predator–prey system with Holling II functional response is proposed and investigated. We show that there is a unique positive solution to the model for any positive initial value. And we show that the positive solution to the stochastic system is stochastically bounded. Moreover, under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean.