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The local dynamics with different topological classifications, bifurcation analysis, and chaos control for the phytoplankton–zooplankton model, which is a discrete analogue of the continuous-time model by a forward Euler scheme, are investigated. It is proved that the discrete-time phytoplankton–zooplankton model has trivial and semitrivial fixed points for all involved parameters, but it has an interior fixed point under the definite parametric condition. Then, by linear stability theory, local dynamics with different topological classifications are investigated around trivial, semitrivial, and interior fixed points. Further, for the discrete-time phytoplankton–zooplankton model, the existence of periodic points is also investigated. The existence of possible bifurcations around trivial, semitrivial, and interior fixed points is also investigated, and it is proved that there exists a transcritical bifurcation around a trivial fixed point. It is also proved that around trivial and semitrivial fixed points of the phytoplankton–zooplankton model there exists no flip bifurcation, but around an interior fixed point there exist both Neimark–Sacker and flip bifurcations. From the viewpoint of biology, the occurrence of Neimark–Sacker implies that there exist periodic or quasi-periodic oscillations between phytoplankton and zooplankton populations. Next, the feedback control method is utilized to stabilize chaos existing in the phytoplankton–zooplankton model. Finally, simulations are presented to validate not only obtained results but also the complex dynamics with orbits of period-8, 9, 10, 11, 14, 15 and chaotic behavior of the discrete-time phytoplankton–zooplankton model. [ABSTRACT FROM AUTHOR]

The study considers the problem of finite-time event-triggered H-infinity consensus for second-order multi-agent systems (MASs) with intrinsic nonlinear dynamics and external bounded disturbances. Based on the designed triggering function, a distributed event-triggered control strategy is presented on the basis of the designed triggering function to ensure consensus in the system, which effectively reduces the data transmission. Then, sufficient conditions for the finite-time consensus with H-infinity performance level of the resulting event-triggered MAS are derived by utilizing the Lyapunov function and finite-time stability theory. Furthermore, Zeno behavior is proven to be excluded under the proposed event-triggered scheme. Finally, the validity of the proposed results is verified by numerical simulation. [ABSTRACT FROM AUTHOR]

In this study, we use a compartmental nonlinear deterministic mathematical model to investigate the effect of different optimal control strategies in controlling Tuberculosis (TB) disease transmission in the community. We employ stability theory of differential equations to investigate the qualitative behavior of the model by obtaining the basic reproduction number and determining the local stability conditions for the disease-free and endemic equilibria. We consider three control strategies representing distancing, case finding, and treatment efforts and numerically compare the levels of exposed and infectious populations with and without control strategies. The results suggest that combination of all controls is the best strategy to eradicate TB disease from the community at an optimal level with minimum cost of interventions. [ABSTRACT FROM AUTHOR]

This paper considers the Clifford-valued recurrent neural network (RNN) models, as an augmentation of real-valued, complex-valued, and quaternion-valued neural network models, and investigates their global exponential stability in the Lagrange sense. In order to address the issue of non-commutative multiplication with respect to Clifford numbers, we divide the original n-dimensional Clifford-valued RNN model into 2 m n real-valued models. On the basis of Lyapunov stability theory and some analytical techniques, several sufficient conditions are obtained for the considered Clifford-valued RNN models to achieve global exponential stability according to the Lagrange sense. Two examples are presented to illustrate the applicability of the main results, along with a discussion on the implications. [ABSTRACT FROM AUTHOR]

In this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new fractional-order predator–prey model with Holling type-III functional response and stage structure. Based on the Lyapunov stability theory and by constructing a suitable Lyapunov functional, we obtain some sufficient conditions for the existence and uniqueness of positive solutions and the asymptotic stability of the positive equilibrium to the system. Finally, we give some numerical examples to illustrate the feasibility of our results. [ABSTRACT FROM AUTHOR]

This paper proposes a modified adaptive sliding-mode control technique and investigates the reduced-order and increased-order synchronization between two different fractional-order chaotic systems using the master and slave system synchronization arrangement. The parameters of the master and slave systems are different and uncertain. These systems exhibit different chaotic behavior and topological properties. The dynamic behavior of the proposed synchronization schemes is more complex and unpredictable. These attributes of the proposed synchronization schemes enhance the security of the information signal in digital communication systems. The proposed switching law ensures the convergence of the error vectors to the switching surface and the feedback control signals guarantee the fast convergence of the error vectors to the origin. Lyapunov stability theory proves the asymptotic stability of the closed-loop. The paper also designs suitable parameters update laws the estimate the unknown parameters. Computer-based simulation results verify the theoretical findings. [ABSTRACT FROM AUTHOR]

Some essential conditions for existence theory and stability analysis to a class of boundary value problems of fractional delay differential equations involving Atangana–Baleanu-Caputo derivative are established. The deserted results are derived by using the Banach contraction and Krasnoselskii's fixed point theorems. Moreover, different kinds of stability theory including Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam-Rassias and generalized Hyers–Ulam–Rassias stability are also developed for the problem under consideration. Appropriate examples are given for illustrative purposes. [ABSTRACT FROM AUTHOR]

Nonlinearities, such as dead-zone, backlash, hysteresis, and saturation, are common in the mechanical and mechatronic systems' components and actuators. Hence, an effective control strategy should take into account such nonlinearities which, if unaccounted for, may cause serious response problems and might even result in system failure. Input saturation is one of the most common nonlinearities in practical control systems. So, this article introduces a novel adaptive variable structure control strategy for nonlinear Caputo fractional-order systems despite the saturating inputs. Owing to the complex nature of the fractional-order systems and lack of proper identification strategies for such systems, this research focuses on the canonic systems with complete unknown dynamics and even those with model uncertainties and external noise. Using mathematical stability theory and adaptive control strategy, a simple stable integral sliding mode control is proposed. The controller will be shown to be effective against actuator saturation as well as unknown characteristics and system uncertainties. Finally, two case studies, including a mechatronic device, are considered to illustrate the effectiveness and practicality of the proposed controller in the applications. [ABSTRACT FROM AUTHOR]

This paper investigates a deterministic and stochastic SIQS epidemic model with vertical transmission and Beddington–DeAngelis incidence. Firstly, for the corresponding deterministic system, the global asymptotic stability of disease-free equilibrium and the endemic equilibrium is proved through the stability theory. Secondly, for the stochastic system, the threshold conditions which decide the extinction or permanence of the disease are derived. By constructing suitable Lyapunov functions, we investigate the oscillation behavior of the stochastic system solution near the endemic equilibrium. The results of this paper show that there exists a great difference between the deterministic and stochastic systems, which implies that the large stochastic noise contributes to inhibiting the spread of disease. Finally, in order to validate the theoretical results, a series of numerical simulations are presented. [ABSTRACT FROM AUTHOR]

The work reported in this paper deals with the study of a coupled system for fractional terminal value problems involving ψ-Hilfer fractional derivative. The existence and uniqueness theorems to the problem at hand are investigated. Besides, the stability analysis in the Ulam–Hyers sense of a given system is studied. Our discussion is based upon known fixed point theorems of Banach and Krasnoselskii. Examples are also provided to demonstrate the applicability of our results. [ABSTRACT FROM AUTHOR]

This article investigates a modified adaptive sliding-mode controller to achieve synchronisation between two different fractional-order chaotic systems with fully unknown parameters. A suitable parameter updating law is designed to tackle the unknown parameters. For constructing the modified adaptive sliding-mode control, a simple sliding surface is designed and the stability of the suggested method is proved using Lyapunov stability theory. Finally, the proposed method is applied to gain chaos synchronisation between two different pairs of fractional-order chaotic systems with uncertain parameters. Numerical simulations are performed to demonstrate the robustness and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the dissipativity and stabilization problems via matrix measure strategy. First, we propose a way to construct complex-valued set-valued maps and give the basic framework of complex-valued differential inclusions. In addition, based on matrix measure strategy and the generalized Halanay inequality, we analyze the dissipativity of the addressed discontinuous complex-valued neural networks by using two different ways. Furthermore, we design a set of controllers to guarantee the exponential stability of the studied networks. The main contribution of this paper is an extension of the dissipativity results of traditional neural networks to discontinuous ones. Finally, we give numerical examples to substantiate the merits of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the problem of finite-time stability (FTS) of linear non-autonomous systems with time-varying delays. By constructing an appropriated function, we derive some explicit conditions in terms of matrix inequalities ensuring that the state trajectories of the system do not exceed a certain threshold over a pre-specified finite time interval. Finally, two examples are given to show the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a multi-term variable-order fractional diffusion equation on a finite domain, which involves the Caputo variable-order time fractional derivative of order α(x, t) ∈ (0, 1) and the Riesz variable-order space fractional derivatives of order β(x, t) ∈ (0, 1), γ (x, t) ∈ (1, 2). Approximating the temporal direction derivative by L1-algorithm and the spatial direction derivative by the standard and shifted Grünwald method, respectively, a characteristic finite difference scheme is proposed. The stability and convergence of the difference schemes are analyzed via mathematical induction. Some numerical experiments are provided to show the efficiency of the proposed difference schemes. [ABSTRACT FROM AUTHOR]

We present a fractional-order model of two-species facultative mutualism with harvesting. We investigate the stability of the equilibrium points of the model by using the linearization method for noncoexistence of equilibrium points and the Lyapunov direct method for the positive coexistence of an equilibrium point. In addition, we obtain sufficient conditions to ensure the local asymptotic stability and global uniform asymptotic stability for the model. Finally, we provide illustrated numerical examples to verify the stability results obtained in this study. [ABSTRACT FROM AUTHOR]

We investigate a finite-time synchronization problem of hybrid-coupled delayed dynamical network via pinning control. According to linear feedback principle and finite-time control theory, the finite-time synchronization can be achieved by pinning control with suitable continuous finite-time controller. Some sufficient conditions are given for finite-time synchronization of undirected and directed complex network by applying finite-time stability lemma. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild solutions via fixed point theorems for multivalued functions. Also we characterize the compactness of the solution set. We introduce the concept of generalized Ulam-Hyers stability and present a generalized Ulam-Hyers stability result using multivalued weakly Picard operator theory. Examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper aims to provide the high order numerical schemes for the time-space tempered fractional Fokker-Planck equation in a finite domain. The high order difference operators, called the tempered and weighted and shifted Lubich difference operators, are used to approximate the time tempered fractional derivative. The spatial operators are discretized by the central difference methods. We apply the central difference methods to the spatial operators and obtain that the numerical schemes are convergent with orders $O(\tau^{q} + h^{2})$ ( $q = 1,2,3,4,5$ ). The stability and convergence of the first order numerical scheme are rigorously analyzed. And the effectiveness of the presented schemes is testified with several numerical experiments. Additionally, some physical properties of this diffusion system are simulated. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey model with Holling type-I functional response and multi state impulsive feedback control is established, where the intensity of pesticide spraying and the release amount of natural enemies are linearly dependent on the given threshold in the second impulse. Firstly, the existence of order-1 periodic solution of the system is investigated by successor functions and Bendixson theorem of impulsive differential equations, then the stability of periodic solutions is proved by the analogue of the Poincaré criterion. Furthermore, in order to reduce the actual total cost and obtain the best economic benefit, the optimal economic threshold is obtained, which provides the optimal strategy for the practical application. Finally, numerical simulations for specific examples are carried out to illustrate the feasibility of the above conclusions. [ABSTRACT FROM AUTHOR]

In this paper, a new SEIR (susceptible-exposed-infected-removed) rumor spreading model with demographics on scale-free networks is proposed and investigated. Then the basic reproductive number $R_{0}$ and equilibria are obtained. The theoretical analysis indicates that the basic reproduction number $R_{0}$ has no correlation with the degree-dependent immigration. The globally asymptotical stability of rumor-free equilibrium and the permanence of the rumor are proved in detail. By using a novel monotone iterative technique, we strictly prove the global attractivity of the rumor-prevailing equilibrium. [ABSTRACT FROM AUTHOR]

In this paper, a time-delay rumor propagation model with a saturated control function is established. Regarding time delay as a bifurcation parameter, Hopf bifurcation is studied. By means of the normal form and the center manifold theorem, a formula is put forward to determine both Hopf bifurcation direction and bifurcating periodic solution stability, together with some numerical simulations to illustrate the relevant theoretical results. Simulation results indicate that an appropriate government control could make periodic oscillation behaviors of the system become steady so as to improve the balance of a social system. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical model describing tuberculosis transmission with incomplete treatment and continuous age structure for latently infected and infectious individuals is investigated. It is assumed in the model that the treated individuals may enter either the latent compartment due to the remainder of Mycobacterium tuberculosis or the infectious compartment due to the treatment failure. It is shown that the global transmission dynamics of the disease is fully determined by the basic reproduction number. The asymptotic smoothness of the semi-flow generated by the system is established. By analyzing the corresponding characteristic equations, the local stability of a disease-free steady state and an endemic steady state of the model is established. By using the persistence theory for infinite dimensional system, the uniform persistence of the system is established when the basic reproduction number is greater than unity. By means of suitable Lyapunov functionals and LaSalle's invariance principle, it is proven that if the basic reproduction number is less than unity, the disease-free steady state is globally asymptotically stable; if the basic reproduction number is greater than unity, the endemic steady state is globally asymptotically stable. [ABSTRACT FROM AUTHOR]

In this paper, a predator-prey system with Holling type function response incorporating prey refuge is presented. By applying the analytical approaches, the dynamics behavior of the considered system is investigated, including stability, limit cycle and bifurcation. The results show that the shape of the functional response plays an important role in determining the dynamics of the system. Especially, the interesting conclusion is that the prey refuge has a destabilizing effect under some certain conditions. [ABSTRACT FROM AUTHOR]

Fuzzy fractional differential equations (FFDEs) driven by Liu's process are a type of fractional differential equations. In this paper, we intend to provide and prove a novel existence and uniqueness theorem for the solutions of FFDEs under local Lipschitz and linear growth conditions. We also investigate the stability of solutions to FFDEs by a theorem. Finally, some examples are provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-dimensional predator-prey model with a time delay and square root response function. We analyze the stability of equilibria with the delay τ increasing and the critical value of τ when Hopf bifurcation occurs. Because the model has the term of square root, the zero point is a singularity. In order to clearly study the stability of the zero point, we rescale the variable $x(t)$ , say $x(t)=X^{2}(t)$ . The conclusion is that the zero point is not stable and the instability is not affected by the delay τ. We apply the normal form method and center manifold theorem to obtain the direction and stability of the Hopf bifurcation. Finally, we make several numerical simulations which is consistent with the conclusion of theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper investigates the finite-time anti-synchronization of time-varying delayed neural networks. A simple intermittent adjustment feedback controller is designed to ensure the drive-response systems realize anti-synchronization in a finite time. By employing some differential inequalities and finite-time stability theory, some novel and effective finite-time anti-synchronization criteria are derived based on the Lyapunov functional method. This paper extends some traditional anti-synchronization criteria by using intermittent adjustment feedback control. Finally, two numerical examples are given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

We study a diffusive predator-prey system with a ratio-dependent functional response when a prey population is infected under homogeneous Neumann boundary condition. All non-negative and positive equilibria are investigated, and the conditions that give rise to asymptotic behavior of these equilibria are examined. In particular, we present a biological interpretation of disease-free and total extinction states. A comparison principle and the stability analysis for the parabolic problem are employed. [ABSTRACT FROM AUTHOR]

This paper considers a class of fractional-order complex-valued Hopfield neural networks (CVHNNs) with time delay for analyzing the dynamic behaviors such as local asymptotic stability and Hopf bifurcation. In the case of a neural network with hub and ring structure, the stability of the equilibrium state is investigated by analyzing the eigenvalue of the corresponding characteristic matrix for the hub and ring structured fractional-order time delay models using a Laplace transformation for the Caputo-fractional derivatives. Some sufficient conditions are established to guarantee the uniqueness of the equilibrium point. In addition, conditions for the occurrence of a Hopf bifurcation are also presented. Finally, numerical examples are given to demonstrate the effectiveness of the derived results. [ABSTRACT FROM AUTHOR]

Porcine reproductive and respiratory syndrome (PRRS) gives rise to reproductive disorders in sows and problem with respiratory system in piglets and young pigs. This disease creates serious economic losses to major pork producing countries. The disease, which is characterized by high morbidity and significant mortality, combined with its potential for rapid spread, can devastate the pig industries of the affected countries. However, not much is known about the spatial transmission of PRRSV (porcine reproductive and respiratory syndrome virus) in growing pigs. In previous models, the infection rate has been assumed to be constant with time. Experimental studies on specific cases of this viral infection suggest that this assumption might not hold. A structured model for the spread of PRRSV has therefore been constructed, incorporating time and spatial dimensions as well as the decline of infection rate with time. Using the traveling wave coordinate and the modified extended hyperbolic tangent method, we derive analytical solutions to the model system. Stability and phase plane analyses are also carried out in order to gain insights into the spatial spread of PRRS as time progresses. [ABSTRACT FROM AUTHOR]

In this paper, we devote ourselves to establishing the unconditionally stable and absolutely convergent optimized finite difference Crank-Nicolson iterative (OFDCNI) scheme containing very few degrees of freedom but holding sufficiently high accuracy for the two-dimensional (2D) Sobolev equation by means of the proper orthogonal decomposition (POD) technique, analyzing the stability and convergence of the OFDCNI solutions and using the numerical simulations to verify the feasibility and effectiveness of the OFDCNI scheme. [ABSTRACT FROM AUTHOR]

In this paper, we show that approximate derivations on Banach ∗-algebras are exactly derivations and also show that approximate quadratic ∗-derivations on Banach ∗-algebras are exactly quadratic ∗-derivations by the fixed point theorem. [ABSTRACT FROM AUTHOR]

We prove that the system $\dot{\theta}(t) =\Lambda(t)\theta(t)$ , $t\in\mathbb{R}_{+}$ , is Hyers-Ulam stable if and only if it is uniformly exponentially stable under certain conditions; we take the exact solutions of the Cauchy problem $\dot{\phi}(t)=\Lambda(t)\phi(t)+e^{i\gamma t}\xi(t)$ , $t\in\mathbb{R}_{+}$ , $\phi(0)=\theta_{0}$ as the approximate solutions of $\dot{\theta}(t)=\Lambda(t)\theta(t)$ , where γ is any real number, ξ is a 2-periodic, continuous, and bounded vectorial function with $\xi(0)=0$ , and $\Lambda(t)$ is a 2-periodic square matrix of order l. [ABSTRACT FROM AUTHOR]

A delayed SEIQRS-V model with quarantine describing the dynamics of worm propagation is considered in the present paper. Local stability of the endemic equilibrium is addressed and the existence of a Hopf bifurcation at the endemic equilibrium is established by analyzing the corresponding characteristic equation. By means of the normal form theory and the center manifold theorem, properties of the Hopf bifurcation at the endemic equilibrium are investigated. Finally, numerical simulations are also given to support our theoretical conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we study the finite time stability of delay differential equations via a delayed matrix cosine and sine of polynomial degrees. Firstly, we give two alternative formulas of the solutions for a delay linear differential equation. Secondly, we obtain a norm estimation of the delayed matrix sine and cosine of polynomial degrees, which are used to establish sufficient conditions to guarantee our finite time stability results. Meanwhile, a numerical example is presented demonstrating the validity of our theoretical results. Finally, we extend our study to the same issue of a delay differential equation with nonlinearity by virtue of the Gronwall inequality approach. [ABSTRACT FROM AUTHOR]

In this paper, we propose to study a class of discrete schistosomiasis models with general incidence function. This model is derived from a continuous schistosomiasis model (in Appl. Math. 4:1682-1693, 2003) by using the backward Euler discretization method with step size $h=1$ . We visit some basic properties of this discrete model and we study the stabilities of the equilibria by constructing some appropriate Lyapunov functional for the endemic equilibrium. [ABSTRACT FROM AUTHOR]

This paper proposes the robust iterative learning control (ILC) design for uncertain linear systems with time-varying delays and random packet dropouts. The packet dropout is modeled by an arbitrary stochastic sequence satisfying the Bernoulli binary distribution, which renders the ILC system to be stochastic instead of a deterministic one. The main idea of this paper is to transform the ILC design into robust stability for a two-dimensional (2D) stochastic system described by the Roesser model with a delay varying in a range. A delay-dependent stability condition, which can guarantee mean-square asymptotic stability of such a 2D stochastic system, is derived in terms of linear matrix inequalities (LMIs), and formulas can be given for the ILC law design. An example for the injection molding is given to demonstrate the effectiveness of the proposed ILC method. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fractional order epidemic model for obesity contagion. The population size is assumed to be nonconstant, which is more realistic. The model considers vertical transmission of obesity and also obesity-related death rate. We give local stability analysis of the model. Finally, some numerical examples are presented. [ABSTRACT FROM AUTHOR]

In this paper we show a connection between Levin-Nohel integro-differential equations and ordinary functional differential equations. Based on this connection, we obtain several new conditions for the stability of the solution, including the famous 3/2 stability criterion. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the stability and traveling waves of stage-structured predator-prey reaction-diffusion systems of Beddington-DeAngelis functional response with both nonlocal delays and harvesting. By analyzing the corresponding characteristic equations, the local stability of various equilibria is discussed. We reduce the existence of traveling waves to the existence of a pair of upper-lower solutions by using the cross iteration method and the Schauder's fixed point theorem. The existence of traveling waves connecting the zero equilibrium and the positive equilibrium is then established by constructing a pair of upper-lower solutions. [ABSTRACT FROM AUTHOR]

This paper considers two kinds of novel decoupled algorithms for the non-stationary Stokes-Darcy model. In this way, the considered problem is decoupled into one time-dependent Stokes equations and one linear parabolic equation. For the two algorithms, we establish the stability and the optimal error estimates. Furthermore, the existing result in Mu and Zhu (Math. Comput. 79:707-731, 2010) can be improved to the optimal order $\mathcal{O}(\Delta t)$ following our proof. Finally, some numerical experiments are conducted to validate the established theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a periodic nonlinear competition model governed by impulsive differential equation with infinitely distributed delays and feedback controls. By means of coincidence degree theory and Lyapunov functional, a set of sufficient criteria are obtained to guarantee the existence and globally asymptotic stability of a unique positive periodic solution of the model. Furthermore, applying our main results to some important competition models which have been well studied in the literature, we establish some new criteria to supplement and generalize some well-known results. [ABSTRACT FROM AUTHOR]

This paper concerns the dynamics of a stochastic SIVR epidemic model with imperfect vaccine where, differently from the epidemic model with perfect vaccine, the vaccinated is perturbed by the noise. This difference is the main difficulty to be conquered to give the threshold $R_{0}^{S}$ . Firstly, $R_{0}^{S}>1$ is proved to be sufficient for persistence in mean of the system. Then, three conditions for the disease to die out are given, which improve the previously known results on extinction of the disease. In case that the disease goes extinct, we show that the disease-free equilibrium is almost surely stable by using the nonnegative semimartingale convergence theorem. [ABSTRACT FROM AUTHOR]

In this paper we investigate an impulsive discrete time system with delay. By using Lyapunov stability theory and a Razumikhin type technique, some new criteria for the exponentially practical stability of impulsive discrete time system with delay are established. A numerical example is given to show the effectiveness of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, the Hyers-Ulam stability for a class of first order stochastic differential equations is studied by using the Ito formula. Furthermore, the research results are applied to a class of second order stochastic differential equations with constant coefficients by the substitution method. In the end, the Hyers-Ulam stability of general second order stochastic differential equations is considered by the solutions of two deterministic second order differential equation boundary value problems. [ABSTRACT FROM AUTHOR]

In this paper, the dynamic behaviors of a competition model in the turbidostat with discrete delay are investigated. The stability of the positive equilibrium and the existence of a Hopf bifurcation are discussed by choosing the delay of digestion as a bifurcation parameter. Furthermore, we determine the direction and stability of the bifurcating periodic solutions by the normal form and the center manifold theorem. Moreover, some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper studies the fractional Lotka-Volterra equations for three competitors, since the fractional derivatives possess the properties of good memory and have great biological significance. First of all, the equilibrium points and asymptotic stability for the equations are studied by the stability analysis method. As expected, the fractional-order differential equations are, at least, as stable as their integer-order counterpart. Second, some approximate analytic solutions for this systems are obtained by the stability analysis method and the homotopy perturbation method, which are expressed in the form of the Mittag-Leffler function. The results show that it takes less time for the population to get close to the equilibrium point as the time derivatives increase. Comparing with the classical ones, the fractional Lotka-Volterra equations, no matter whether the fractional derivatives have big or small order, if both take more time, even multiplied time, for the system to reach the equilibrium point, then that may explain the memory properties. Furthermore, some numerical analyses are carried out and verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we investigate synchronization of drive-response neural networks with time-varying delays via impulsive control. Based on impulsive stability theory, we design proper impulsive controllers and derive some sufficient conditions for achieving synchronization. Noticeably, we adopt adaptive strategy to design unified controllers for different neural networks and relax the restrictions on the impulsive interval. All the obtained results are verified by several numerical examples. [ABSTRACT FROM AUTHOR]

We investigate the dynamics of a discrete-time predator-prey system. Firstly, we give necessary and sufficient conditions of the existence and stability of the fixed points. Secondly, we show that the system undergoes a flip bifurcation and a Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, we present numerical simulations not only to show the consistence with our theoretical analysis, but also to exhibit the complex but interesting dynamical behaviors, such as the period-6, -11, -16, -18, -20, -21, -24, -27, and -37 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors, which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Finally, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method. [ABSTRACT FROM AUTHOR]

In this paper, a nonautonomous almost periodic prey-predator system with impulsive effects and multiple delays is considered. By the mean-value theorem of multiple variables, integral inequalities, differential inequalities, and other mathematical analysis skills, sufficient conditions which guarantee the permanence of the system are obtained. Furthermore, by constructing a series of Lyapunov functionals, we derive that there exists a unique almost periodic solution of the system which is uniformly asymptotically stable. Finally, a numerical example and some simulations are presented to support our theoretical results. [ABSTRACT FROM AUTHOR]