Uncertain impulsive functional differential systems of fractional order and almost periodicity

The problem of existence of almost periodic solutions of uncertain impulsive functional differential systems of fractional order is investigated. Using the Lyapunov method combined with the concept of uniformly positive definite matrix functions and Hamilton–Jacobi–Riccati inequalities new criteria are presented. The robust stability of the almost periodic solution is also discussed. We apply our results to an impulsive Lasota–Wazewska type model of fractional order. Our results extend the theory of almost periodic solutions for impulsive delay differential equations to the fractional-order case under uncertainty.     

Stability and Hopf bifurcation analysis of an eco-epidemiological model with delay

In this paper, an eco-epidemiological model with time delay is considered. The asymptotical stability of the three equilibria, the existence of stability switches about both the disease-free planar equilibrium and the positive equilibrium are investigated. It is found that Hopf bifurcation occurs when the delay τ passes through a critical value. Some explicit formulae determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations at the positive equilibrium are obtained by using the normal form theory and center manifold theory. Some numerical simulations for justifying the theoretical analysis are also provided. Finally, biological explanations and main conclusions are given.     

Hopf bifurcation control for delayed complex networks

In this paper, we consider the problem of Hopf bifurcation control for a complex network model with time delays. We know that for the system without control, as the positive gain parameter of the system passes a critical point, Hopf bifurcation occurs. To control the Hopf bifurcation, a time-delayed feedback controller is proposed to delay the onset of an inherent bifurcation when such bifurcation is undesired. Furthermore, we can also change the stability and direction of bifurcating periodic solutions by choosing appropriate control parameters. Numerical simulation results confirm that the new feedback controller using time delay is efficient in controlling Hopf bifurcation.     

Dynamical behavior in a harvested differential-algebraic prey-predator model with discrete time delay and stage structure

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, the dynamic behaviors of the proposed model system with and without discrete time delay are investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; on the other hand, the local stability of the model system with discrete time delay is also studied. The theoretical analysis shows that the discrete time delay has a destabilizing effect in the model of population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Numerical simulations are carried out to show the consistency with theoretical analysis.     

一类状态依赖时滞的脉冲微分方程的周期正解

本文利用锥理论和不动点指数定理,研究了一类具状态依赖时滞的脉冲微分方程的正周期解,获得了关于正周期解存在性的若干新的结果。     

Bifurcation analysis of a competitive system with general toxic production and delayed toxic effects

Population interaction may release poisonous chemicals to inhibit other species’ growth in the ecosystem, especially for the competitive populations. The negative effect of toxic chemical substances may not display immediately and appear with time lag during the species’ growth. In this work, we investigate a competitive system with the delayed toxic effects of the chemicals from species interaction. Theoretical results obtained in this work help us reveal the delayed toxic factors on species’ growth. We first consider the existence and the stability of the equilibria. The influence of delay terms on the positive steady state is validated. The delayed toxic effects here will contribute to the oscillation for the concentration of species when the value of time delay passes through a critical point. Besides, the stability of periodic solutions from the Hopf bifurcation and the direction of the Hopf bifurcation are also determined. Finally, several numerical examples are provided to validate the theoretical conclusions.     

Dynamical behavior in a hybrid stochastic triple delayed prey predator bioeconomic system with Lévy jumps

In this paper, a hybrid triple delayed prey predator bioeconomic system with prey refuge and Lévy jumps is established, where both maturation delay for prey and predator population and gestation delay for predator population are considered. For deterministic system, positivity and uniform permanence of solution are discussed. Local stability of deterministic system around interior equilibrium is investigated due to variations of triple time delays. For stochastic system without time delay, sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed. Existence of stochastic Hopf bifurcation and stochastic stability are investigated. For stochastic system with triple time delays, existence and uniqueness of global positive solution are studied. Finally, combined dynamic effects of triple time delays and Lévy jumps on the hybrid stochastic system are discussed by constructing appropriate Lyapunov functions. Numerical simulations are supported to illustrate theoretical analysis.     

Complex dynamics and stability analysis in a discrete hybrid bioeconomic system with double time delays

In this paper, a discrete hybrid three-species food chain system is proposed, where commercial harvesting on top predator is considered. Two time delays are introduced to represent gestation delay for prey and predator population, respectively. In absence of time delay, sufficient conditions associated with economic interest and step size are derived to show system undergoes flip bifurcation. In presence of double time delays, existence of Neimark–Sacker bifurcation and local stability switch are discussed due to variations of time delays. Furthermore, by utilizing new normal form of delayed discrete hybrid system and center manifold theorem, direction and stability of Neimark–Sacker bifurcation are studied. Numerical simulations are performed not only to validate theoretical analysis, but also exhibit cascades of period-doubling bifurcation, chaotic behavior and stable closed invariant curve.     

Dynamics of impulsive neutral-type BAM neural networks

In this letter, the existence and the global exponential stability of piecewise pseudo almost periodic solutions (PAP_T) for bidirectional associative memory neural networks (BAMNNs) with time-varying delay in leakage (or forgetting) terms and impulsive are investigated by applying contraction mapping fixed point theorem, the exponential dichotomy of linear differential equations and differential inequality techniques. Furthermore, we give an explanatory example to illustrate the efficiency of the theoretical predictions.     

Dynamical behaviors of discrete-time fuzzy cellular neural networks with variable delays and impulses

In this paper, the discrete-time fuzzy cellular neural network with variable delays and impulses is considered. Based on M-matrix theory and analytic methods, several simple sufficient conditions checking the global exponential stability and the existence of periodic solutions are obtained for the neural networks. Moreover, the estimation for exponential convergence rate index is proposed. The obtained results show that the stability and periodic solutions still remain under certain impulsive perturbations for the neural network with stable equilibrium point and periodic solutions. Some examples with simulations are given to show the effectiveness of the obtained results.     

Bifurcation with regard to combined interaction parameter in a life energy system dynamic model of two components with multiple delays

Bifurcation theory is commonly used to study the dynamical behavior of ecosystems. It involves the analysis of points in the parameter space where the stability of the system changes qualitatively. The type of bifurcation that associates equilibria with periodic solution is called Hopf bifurcation. In this paper, a life energy system dynamic model of two components with multiple delays is presented. It is shown that the interaction parameters of the delayed ecosystem play a fundamental role in classifying the rich dynamics and bifurcation phenomena. Regarding the combined interaction parameter as a bifurcation parameter, the bifurcation values in the parameter plane are displayed. The direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by applying the normal form theory and the center manifold theorem. Moreover, the amplitudes of oscillations always increase as the interaction parameters increase, while the robustness of periods occurs as the interaction parameters vary. From an ecological point of view, the existence of Hopf bifurcation expresses periodic oscillatory behavior of the life energy system.     

具有脉冲出生、脉冲接种和双时滞的SEIRS传染病模型

研究具有脉冲出生、脉冲接种和时滞的SEIRS传染病模型,分析无病周期解的存在性和全局稳定性,以及传染病模型的持久性,分析控制传染病传播的主要因素。     

Bifurcation analysis in a predator-prey system with stage-structure and harvesting

In this paper, we consider a predator-prey model with stage-structure and harvesting. This model is the same as the one developed by Kar and Pahari (2007) [9], but we make bifurcation analysis more general than their work. In particular, using the approach of Beretta and Kuang (2002) [4], we show that the positive steady state can be destabilized through a Hopf bifurcation. We also investigate the stability and direction of periodic solutions bifurcating from Hopf bifurcation by using the normal form theory and the center manifold theorem presented in Hassard et al. (1981) [8]. Numerical simulations are then carried out as supporting evidences of our analytical results.     

Spatiotemporal dynamics of a predator-prey system with fear effect

Recent field experiments on vertebrates show that though mere presence of a predator causes a dramatic change in prey demography, the fear of predators increases the survival probability of prey leading to a cost of prey production. Based on the experimental findings, we proposed and analyzed a mathematical model that incorporates the fear-induced birth reduction in the prey population due to presence of predator. A modified and more realistic fear function is proposed in this study. Qualitative behavior of the model is performed including positivity and boundedness of solutions, existence of critical points and their local stability analysis, existence of transcritical and Hopf bifurcation. We analyzed Hopf bifurcation with respect to the prey growth rate and the level of fear. Transcritical bifurcation is analyzed by varying the prey growth rate. Distribution of the population of interacting species in a large scale natural system is heterogeneous and subject to alter for different reasons. Thus, we investigate how behavioral modification in prey population due to fear for predators and mutual interference among predator species can create various spatiotemporal pattern formation in population distribution. In the spatially extended system, we provide a detailed stability analysis and obtain the conditions for Turing instability. Numerical simulations are performed to validate analytical results for both non-spatial and spatial models. Warm spot patterns are obtained by considering three different types of initial data and discussed the biological significance of these patterns for the two-dimensional spatial model. Our numerical simulation demonstrates that the fear effect in a diffusive predator-prey system with mutual interference between predators may exhibit more complicated dynamics.     

Existence, continuation, and uniqueness problems of stochastic impulsive systems with time delay

This paper studies stochastic impulsive systems with time delay, where the impulse times are state-dependent. Using Itô calculus, we develop the essential foundation of the theory of the mentioned system. In particular, we establish results on local and global existence, forward continuation, and uniqueness of adapted solutions.     

Robust exponential stability of uncertain impulsive delay difference equations with continuous time

In this paper, the robust exponential stability of uncertain impulsive delay difference equations is investigated. First, some robust exponential stability criteria for uncertain impulsive delay difference equations with continuous time in which the state variables on the impulses may relate to the time-varying delays are provided. Then a robust exponential stability result for uncertain linear impulsive delay difference equations with discrete time is given. Some examples, including an example which cannot be studied by the existing results, are also presented to illustrate the effectiveness of the obtained results.     

Effect of seasonality on the dynamical behavior of an ecological system with impulsive control strategy

In this paper, on the basis of the theories and methods of ecology and ordinary differential equation, an ecological model consisting of two preys and one predator with impulsive control strategy and seasonal effects is established. Conditions which guarantee the global asymptotical stability of the prey-eradication periodic solution are obtained using the theory of impulsive equations, small amplitude perturbation skills, and comparison techniques. Further, the influences of the impulsive perturbation and seasonal effects on the inherent oscillation are studied numerically. These show to be consistent with the theoretical analysis and rich complex population dynamics, such as species extinction and permanence. Moreover, the population dynamical behavior of the model is demonstrated by the computed largest Lyapunov exponent. By investigating the strange attractors through their computed Fourier spectra, we know that seasonality has a profound effect on the population dynamical behavior. All these results are expected to be of use in the study of dynamic complexity of ecosystems.     

状态依赖时滞型微分方程正周期解的存在性

本文利用Krasnoselskii不动点定理,考虑了状态依赖时滞性微分方程x′(t) = &;#8722;A(t, x(t))x(t) + B(x(t))F(x(t &;#8722;  (t, x(t))))正周期解的存在性, 得到了该方程存在与不存在正周期解的充分条件.     

Exponential stability of non-linear neutral stochastic delay differential system with generalized delay-dependent impulsive points

This work aims to analyze the exponential stability of a non-linear impulsive neutral stochastic delay differential system. In this study, impulse perturbation is considered a delay-dependent state variable. The solution of the delay-dependent impulsive neutral stochastic delay differential system is associated with the solution of the system without impulses. First, we developed a relation connecting the solution of the neutral stochastic delay differential system without impulses and the solution of the corresponding system with impulses. Then, the conditions of the exponential stability of the proposed impulsive system are derived by determining the stability analysis of the respective system without impulse. The numerical approach for the neutral stochastic delay system without impulses is generated using the Euler-Maruyama method and adopted for the corresponding impulsive system. Finally, the achieved theoretical results are illustrated for applying the Malthusian single species neutral stochastic delay population model with immigration impulses.     

Global dynamics analysis of a water hyacinth fish ecological system under impulsive control

Control of a water hyacinth-fish ecological system is required for a healthy and sustainable environment. This paper aims to investigate the global dynamics of a water hyacinth fish ecological system under ratio-dependent state impulsive control. First, we study the positivity and boundedness of the solution of the controlled system. By studying the local stability of the equilibrium, we find that the system has two situations. One is that there are two equilibria, namely a saddle point and a boundary equilibrium. In the second case, there are four equilibria, namely, two saddle points, a boundary equilibrium, and a focus point. For the first case, when we select an appropriate ratio-dependent control threshold, the trajectory will globally converge to the boundary equilibrium. For the second case, when the control line is located below the focus point, by using Poincare mapping method, flip bifurcation theory, and vector field analysis techniques, we find that the solution of the controlled system either globally converges to the boundary equilibrium, order-1 periodic solution, or order-2 periodic solution under certain conditions. When the control line is located above the focus point, the solution of the controlled system either globally converges to the focus point, order-1 or order-2 periodic solution. Finally, we use examples to verify the correctness and validity of the theoretical results.