A delayed HIV-1 infection model with Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV-1 infection with intracellular delay and Beddington–DeAngelis functional response is investigated. We obtain a necessary and sufficient condition for the global stability of the infection-free equilibrium and give some sufficient conditions for the local stability of the infected equilibrium.     

Dynamical behavior of a class of prey-predator system with impulsive state feedback control and Beddington–DeAngelis functional response

This paper studies systematically a Bedd-ington?CDeAngelis prey?Cpredator system with harvesting and impulsive state feedback control. Conditions for existence and stability of predator-free periodic solution are obtained. When the predator-free periodic solution loses its stability, the existence and stability of nontrivial period solution are also established. Furthermore, computer simulations show that this impulsive system displays a series of complex phenomena, including period-doubling bifurcation and cascade, period window, and chaotic bands. Through numerical simulation, it is also observed that capture capability can influence the amount of predator released and the interval of the stability for nontrivial period-1 solution. Moreover, the superiority of impulsive state feedback control strategy is also exhibited over the impulsive fixed-time control.     

Asymptotic Properties of a Stochastic SIR Epidemic Model with Beddington–DeAngelis Incidence Rate

In this paper, the stochastic SIR epidemic model with Beddington–DeAngelis incidence rate is investigated. We classify the model by introducing a threshold value \(\lambda \). To be more specific, we show that if \(\lambda <0\) then the disease-free is globally asymptotic stable i.e., the disease will eventually disappear while the epidemic is persistence provided that \(\lambda >0\). In this case, we derive that the model under consideration has a unique invariant probability measure. We also depict the support of invariant probability measure and prove the convergence in total variation norm of transition probabilities to the invariant measure. Some of numerical examples are given to illustrate our results.     

Analysis of Bogdanov–Takens bifurcations in a spatiotemporal harvested-predator and prey system with Beddington–DeAngelis-type response function

Nonlinear Dynamics - In this article, we consider a predator–prey system with constant rate of harvesting, which exhibits Hopf and Bogdanov–Takens bifurcations under certain parametric...     

A high-static–low-dynamic-stiffness vibration isolator with the auxiliary system

High-static–low-dynamic-stiffness (HSLDS) vibration isolators seek to widen the frequency range of isolation by decreasing the dynamic stiffness so as to reduce the natural frequency, while maintaining the same static displacement as equivalent linear isolators. However, in many cases especially under light damping or large excitations, the peak transmissibility is very large and the hardening nonlinearity causes the transmissibility curve to bend to the right seriously or in other words causes the jump phenomenon, resulting in a greatly reduced isolation region. In this paper, an auxiliary system is added to the HSLDS isolator to overcome these disadvantages, with the static displacement of the isolation object remaining unchanged. Coefficient-varying harmonic balance method is proposed in this paper to find the dynamic response and most importantly analyze the stability of the steady-state response. The isolation performance of the HSLDS-AS isolator, which is evaluated by displacement transmissibility, is compared with that of the equivalent HSLDS isolator. The effects of system parameters on the isolation performance are investigated. It is shown that the auxiliary system can lower the peak transmissibility and eliminate the jump phenomenon, resulting in a wide isolation region, and the HSLDS-AS isolator has strong designability with many parameters tunable.     

Global stability of a virus dynamics model with Beddington–DeAngelis incidence rate and CTL immune response

In this paper, the global stability of virus dynamics model with Beddington–DeAngelis infection rate and CTL immune response is studied by constructing Lyapunov functions. We derive the basic reproduction number R ₀ and the immune response reproduction number R ⁰ for the virus infection model, and establish that the global dynamics are completely determined by the values of R ₀. We obtain the global stabilities of the disease-free equilibrium E ₀, immune-free equilibrium E ₁ and endemic equilibrium E ^∗ when R ₀≤1, R ₀>1, R ⁰>1, respectively.     

Bifurcations and chaotic dynamics in a 4-dimensional competitive Lotka–Volterra system

In this paper, we study the Hopf bifurcation for the four-dimensional competitive Lotka–Volterra system, and give an example which can display chaotic dynamics apparent like Rössler’s folded band attractor. This demonstrates that Smale’s conclusions in (J. Math. Biol., 3:5–7, 1976) are true even for the simplest competitive Lotka–Volterra systems when the dimension n is four. We explore the mechanism of occurrence of chaotic behavior for the four-dimensional competitive Lotka–Volterra system. The numerical study indicates that a periodic solution by Hopf bifurcation can undergo successive period-doubling cascades, and a homoclinic orbit can undergo homoclinic bifurcation by Shil’nikov theorem.     

Dynamics of a discrete-time stage-structured predator–prey system with Holling type II response function

Nonlinear Dynamics - A discrete-time model of predator–prey dynamics with Holling type II response function is proposed. Each species considered has its age structure which is typical for...     

Distributed multiple-bipartite consensus in networked Lagrangian systems with cooperative–competitive interactions

Nonlinear Dynamics - In combination with the collective behavior evolution of bipartite consensus and cluster/group consensus, this paper proposes the notion of multiple-bipartite consensus in...     

A chaotic system with Hölder continuity

This paper presents a new chaotic system with infinitely many equilibria. The new system contains two system parameters and a nonlinear term which does not satisfy Lipschitz continuity but does satisfy $\frac{1}{2}$ -Hölder continuity condition. The complicated dynamics are studied through theoretical analysis and numerical simulation. Synchronization for two identical systems by a piecewise linear feedback controller is investigated based on Lyapunov stability criteria.     

A delay-induced predator–prey model with Holling type functional response and habitat complexity

Nonlinear Dynamics - A delay-induced predator–prey system with the effect of habitat complexity and Holling type functional response is proposed. The dynamical behaviors of the presented...     

Weighted bipartite containment motion of Lagrangian systems with impulsive cooperative–competitive interactions

Nonlinear Dynamics - The present paper investigates weighted bipartite containment motion of networked Lagrangian systems with instantaneous cooperative–competitive interactions. A control...     

Dynamics of a mass–spring–pendulum system with vastly different frequencies

We investigate the dynamics of a simple pendulum coupled to a horizontal mass?Cspring system. The spring is assumed to have a very large stiffness value such that the natural frequency of the mass?Cspring oscillator, when uncoupled from the pendulum, is an order of magnitude larger than that of the oscillations of the pendulum. The leading order dynamics of the autonomous coupled system is studied using the method of Direct Partition of Motion (DPM), in conjunction with a rescaling of fast time in a manner that is inspired by the WKB method. We particularly study the motions in which the amplitude of the motion of the harmonic oscillator is an order of magnitude smaller than that of the pendulum. In this regime, a pitchfork bifurcation of periodic orbits is found to occur for energy values larger that a critical value. The bifurcation gives rise to nonlocal periodic and quasi-periodic orbits in which the pendulum oscillates about an angle between zero and ??/2 from the down right position. The bifurcating periodic orbits are nonlinear normal modes of the coupled system and correspond to fixed points of a Poincare map. An approximate expression for the value of the new fixed points of the map is obtained. These formal analytic results are confirmed by comparison with numerical integration.     

A delayed ratio-dependent predator–prey model of interacting populations with Holling type III functional response

In this article, we study a ratio-dependent predator–prey model described by a Holling type III functional response with time delay incorporated into the resource limitation of the prey logistic equation. This investigation includes the influence of intra-species competition among the predator species. All the equilibria are characterized. Qualitative behavior of the complicated singular point (0,0) in the interior of the first quadrant is investigated by means of a blow-up transformation. Uniform persistence, stability, and Hopf bifurcation at the positive equilibrium point of the system are examined. Global asymptotic stability analyses of the positive equilibrium point by the Bendixon–Dulac criterion for non-delayed model and by constructing a suitable Lyapunov functional for the delayed model are carried out separately. We perform a numerical simulation to validate the applicability of the proposed mathematical model and our analytical findings.