82 results
Search Results
2. Global dynamics of two-species reaction–diffusion competition model with Gompertz growth.
- Author
-
Yang, Yefen, Ma, Li, Duan, Banxiang, and Zou, Rong
- Subjects
GOMPERTZ functions (Mathematics) ,DYNAMICAL systems ,SPATIAL variation ,EIGENVALUES ,MATHEMATICS - Abstract
In this paper, we investigate a two-species reaction–diffusion competition model with Gompertz growth, where the intrinsic growth rates and carrying capacities of environments are heterogeneous. At firstly, assuming two competing species only admit different diffusive rates, we show that 'slower diffuser prevails', which is consistent with the well-known result in Dockery J, Hutson V, Mischaikow K, Pernarowski M. [The evolution of slow dispersal rates: a reaction–diffusion model. J Math Biol. 1998;37(1):61–83; Hastings A. Can spatial variation alone lead to selection for dispersal? Theor Popul Biol. 1983;24:244–251]. Then, for the "weak competition" case, we establish a prior estimate, which combined with the theory of monotone dynamical system and spectral analysis implies that the model admits a unique coexistence steady state, which is globally asymptotically stable. Finally, for the "strong–weak competition" case, we give the expression of critical competition intensity and the weak competitor will be wiped out. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Dynamics of a Predator–Prey System with Impulsive Stocking Prey and Nonlinear Harvesting Predator at Different Moments.
- Author
-
Zhou, Zeli, Jiao, Jianjun, Dai, Xiangjun, and Wu, Lin
- Subjects
FLOQUET theory ,NATURAL resources ,SYSTEM dynamics ,RESOURCE management ,COMPUTER simulation - Abstract
In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
4. Global attractivity of a rational difference equation with higher order and its applications.
- Author
-
Li, Xianyi and Lv, Luyao
- Subjects
DIFFERENCE equations ,MATHEMATICAL formulas ,MATHEMATICAL models ,LOGICAL prediction ,GENERALIZATION - Abstract
We study in this paper the global attractivity for a higher order rational difference equation. As application, our results not only include and generalize many known ones, but also formulate some new results for several conjectures presented by Camouzis and Ladas, et al. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
5. Markus–Yamabe conjecture for planar piecewise linear refracting system
- Author
-
Zhechen, Jiang and Shimin, Li
- Published
- 2024
- Full Text
- View/download PDF
6. Delayed nonmonotonic immune response in HIV infection system.
- Author
-
Shanshan Wang and Shaoli Wang
- Subjects
- *
HIV infections , *IMMUNE response , *HOPF bifurcations , *IMMUNE system - Abstract
In this paper, we study a delayed HIV infection model with nonmonotonic immune response and perform stability and bifurcation analysis. Our results show that the delayed HIV infection system with nonmonotonic immune response has bistability and stable periodic solution appear. We find that both the uninfected and immune-free equilibria are globally asymptotically stable under certain conditions which are not affected by time delay. However, the time delay makes one immune equilibrium always unstable for τ ≥ 0 and also makes another immune equilibrium appear stability switches; meanwhile, the system will exhibit local Hopf bifurcation, global Hopf bifurcation, and saddle-node-Hopf bifurcation. Numerical simulations are carried out to verify our results. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
7. Qualitative Behavior of the Difference Equation xn+1 = αxn-m+ηxn-k+σxn-l+δxn/β+γxn-kxn-l(xn-k+xn-l).
- Author
-
El-Moneam, Mohamed Abd
- Subjects
QUALITATIVE research ,NONLINEAR differential equations ,SUBMANIFOLDS ,REAL numbers ,EQUILIBRIUM - Abstract
In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation x
n+1 = αxn-m +ηxn-k +σxn-l +δxn /β+γxn-k xn-l (xn-k +xn-l ), n = 0,1,2... where the parameters α, β, γ, η, σ ∊(0,∞) while m;k; l are positive integers, such that m, k, l. The initial conditions x-m ....,x-k ...,x-l ...,x-1 ...,x0 are arbitrary positive real numbers. We will give some numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]- Published
- 2023
- Full Text
- View/download PDF
8. Robust H∞ Control of Switched Nonlinear Systems Under Sampled Data
- Author
-
Zhao, Hongpeng and Wang, Xingtao
- Published
- 2022
- Full Text
- View/download PDF
9. Control of periodic sampling systems subject to actuator saturation.
- Author
-
Yang, Hongjiu, Li, Zhiwei, Shi, Peng, and Hua, Changchun
- Subjects
ACTUATORS ,ROBUST control ,CONTROL theory (Engineering) ,TIME delay systems ,FEEDBACK control systems - Abstract
In this paper, the control problem of linear systems with periodic sampling period subject to actuator saturation is considered via delta operator approach. Using periodic Lyapunov function, sufficient conditions of local stabilization for periodic sampling systems are given. By solving an optimization problem, we derive the periodic feedback control laws and the estimate of the domain of attraction. As the saturation function sat. belongs to the sector OE0; 1, sufficient conditions are derived by constructing saturation-dependent Lyapunov functions to ensure that the periodic sampling system is globally asymptotically stable. A numerical example is given to illustrate the theoretical results proposed in this paper. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
10. Robust H∞ Control of Switched Nonlinear Systems Under Sampled Data.
- Author
-
Zhao, Hongpeng and Wang, Xingtao
- Abstract
This paper investigates the globally asymptotically stable and L
2 -gain of robust H∞ control for switched nonlinear systems under sampled data. By considering the relationship between the sampling period and the dwell time, the non-switching and one switching are discussed in the sampling interval, respectively. Firstly, a state feedback sampled-data controller is constructed by the back-stepping method, and the switching converts to asynchronous switching if it happens within the sampling interval. Then, under the limiting conditions of the sampling period, which are obtained by the average dwell time method, the closed-loop system is globally asymptotically stable and has L2 -gain. Finally, two numerical examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]- Published
- 2022
- Full Text
- View/download PDF
11. Global stability of a diffusive HCV infections epidemic model with nonlinear incidence.
- Author
-
Su, Ruyan and Yang, Wensheng
- Abstract
In this paper, we study a diffusive HCV infections epidemic model with nonlinear incidence rate and analyze the stability of the two kinds of equilibria. By constructing various Lyapunov functions, we prove that the disease-free equilibrium is globally asymptotically stable when the basic reproduction number R 0 < 1 and the endemic equilibrium is globally asymptotically stable when the basic reproduction number R 0 > 1 . Finally, some numerical simulations are given to confirm the theoretical analysis. The results show that when other parameters are the same, the linear infection rate and the non-linear infection rate have different effects on disease spread. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
12. Evolution of dispersal in advective homogeneous environments.
- Author
-
Ma, Li and Tang, De
- Subjects
ADVECTION-diffusion equations ,NEUMANN boundary conditions ,PARTIAL differential equations ,ADVECTION - Abstract
The effects of weak and strong advection on the dynamics of reaction-diffusion models have long been investigated. In contrast, the role of intermediate advection still remains poorly understood. This paper is devoted to studying a two-species competition model in a one-dimensional advective homogeneous environment, where the two species are identical except their diffusion rates and advection rates. Zhou (P. Zhou, On a Lotka-Volterra competition system: diffusion vs advection, Calc. Var. Partial Differential Equations, 55 (2016), Art. 137, 29 pp) considered the system under the no-flux boundary conditions. It is pointed that, in this paper, we focus on the case where the upstream end has the Neumann boundary condition and the downstream end has the hostile condition. By employing a new approach, we firstly determine necessary and sufficient conditions for the persistence of the corresponding single species model, in forms of the critical diffusion rate and critical advection rate. Furthermore, for the two-species model, we find that (i) the strategy of slower diffusion together with faster advection is always favorable; (ii) two species will also coexist when the faster advection with appropriate faster diffusion. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
13. Nonlinear Dynamical Analysis and Optimal Control Strategies for a New Rumor Spreading Model with Comprehensive Interventions.
- Author
-
Li, Tingting and Guo, Youming
- Abstract
In the current era, information dissemination is more convenient, the harm of rumors is more serious than ever. At the beginning of 2020, COVID-19 is a biochemical weapon made by a laboratory, which has caused a very bad impact on the world. It is very important to control the spread of these untrue statements to reduce their impact on people’s lives. In this paper, a new rumor spreading model with comprehensive interventions (background detection, public education, official debunking, legal punishment) is proposed for qualitative and quantitative analysis. The basic reproduction number with important biological significance is calculated, and the stability of equilibria is proved. Through the optimal control theory, the expression of optimal control pairs is obtained. In the following numerical simulation, the optimal control under 11 control strategies are simulated. Through the data analysis of incremental cost-effectiveness ratio and infection averted ratio of all control strategies, if we consider the control problem from different perspectives, we will get different optimal control strategies. Our results provide a flexible control strategy for the security management department. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
14. Improved results on $\mathcal{H}_{\infty}$ state estimation of static neural networks with interval time-varying delay.
- Author
-
Shu, Yanjun and Liu, Xinge
- Subjects
ARTIFICIAL neural networks ,TIME-varying systems ,ESTIMATION theory ,ERROR analysis in mathematics ,PERFORMANCE evaluation - Abstract
This paper is concerned with the problem of the guaranteed $\mathcal{H_{\infty}}$ performance state estimation for static neural networks with interval time-varying delay. Based on a modified Lyapunov-Krasovskii functional and the linear matrix inequality technique, a novel delay-dependent criterion is presented such that the error system is globally asymptotically stable with guaranteed $\mathcal{H_{\infty}}$ performance. In order to obtain less conservative results, Wirtinger's integral inequality and reciprocally convex approach are employed. The estimator gain matrix can be achieved by solving the LMIs. Numerical examples are provided to demonstrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
15. Optimal control strategies for an online game addiction model with low and high risk exposure.
- Author
-
Guo, Youming and Li, Tingting
- Subjects
VIDEO games ,STRATEGY games ,OPTIMAL control theory ,RISK exposure ,BASIC reproduction number - Abstract
In this paper, we establish a new online game addiction model with low and high risk exposure. With the help of the next generation matrix, the basic reproduction number R0 is obtained. By constructing a suitable Lyapunov function, the equilibria of the model are Globally Asymptotically Stable. We use the optimal control theory to study the optimal solution problem with three kinds of control measures (isolation, education and treatment) and get the expression of optimal control. In the simulation, we first verify the Globally Asymptotical Stability of Disease-Free Equilibrium and Endemic Equilibrium, and obtain that the different trajectories with different initial values converges to the equilibria. Then the simulations of nine control strategies are obtained by forward-backward sweep method, and they are compared with the situation of without control respectively. The results show that we should implement the three kinds of control measures according to the optimal control strategy at the same time, which can effectively reduce the situation of game addiction. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
16. A nonstandard finite difference scheme for a multi-group epidemic model with time delay.
- Author
-
Xu, Jinhu and Geng, Yan
- Subjects
FINITE differences ,TIME delay systems ,DYNAMICS ,DISCRETIZATION methods ,SYNCHRONIZATION - Abstract
In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if $\mathfrak{R}_{0}\leq1$ , then the disease-free equilibrium $P_{0}$ is globally asymptotically stable; if $\mathfrak{R}_{0}>1$ , then the infection equilibrium $P_{*}$ is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
17. Extinction and permanence of the predator-prey system with general functional response and impulsive control.
- Author
-
Liu, Juan, Hu, Jie, and Yuen, Peter
- Subjects
- *
HARVESTING , *PREDATION , *BIOLOGICAL pest control , *INTEGRATED pest control , *PREY availability , *PEST control - Abstract
• Modelling the predator-prey ecosystem using a 'generalized' impulsive response function for the first time. • Dynamical behaviors such as sufficient conditions for the stability of prey-eradication and the permanence have been given. • The generalized solutions under asymptotically stable conditions are validated through various specific response functions. • The correctness of the derived theoretical results are further validated using numerical simulations. • Unlike previous research this paper also models the non-simultaneous occurrence of predator stocking and prey harvesting. Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environmental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a 'generalized' impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the harvesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradication periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a 'generalized' impulsive response function. Subsequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the 'general impulsive response function' under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response function (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elimination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator parameter and the incremental stocking of the predator parameter. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
18. Optimal diffusion rate of species in flowing habitat.
- Author
-
Xu, Benlong and Liu, Nannan
- Subjects
BOUNDARY value problems ,ASYMPTOTIC expansions ,DEPENDENCE (Statistics) ,DIFFUSION processes ,GLOBAL asymptotic stability - Abstract
It is widely accepted that diffusive dispersal can permit persistence in an advective environment. This paper studies in some sense the optimal diffusion rate of species in a flowing habitat with hostile downstream boundary conditions. Firstly, we study the dependence of the critical length of the habitat on the dispersal rate d. It is shown that the critical length first decreases and then increases and asymptotically tends to infinity. Then there is a unique optimal diffusion rate $d_{0}$ for a single species to evolve. Then, by using this observation, we study the competition system of two species which are the same but only with different dispersal rates. We get an open finite interval, which is a neighborhood of $d_{0}$ , such that, if one of the dispersal rates lies within the interval but the other rate falls outside, then competition exclusion occurs. If the two dispersal rates both lie within the interval, the one with an intermediate dispersal rate can always invade the other with its dispersal rate near the ends of the interval. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
19. Dynamical analysis for delayed virus infection models with cell-to-cell transmission and density-dependent diffusion.
- Author
-
Wang, Shaoli, Zhang, Achun, and Xu, Fei
- Subjects
BASIC reproduction number ,VIRUS diseases ,NEUMANN boundary conditions ,GLOBAL asymptotic stability ,DIFFUSION - Abstract
In this paper, certain delayed virus dynamical models with cell-to-cell infection and density-dependent diffusion are investigated. For the viral model with a single strain, we have proved the well-posedness and studied the global stabilities of equilibria by defining the basic reproductive number R 0 and structuring proper Lyapunov functional. Moreover, we found that the infection-free equilibrium is globally asymptotically stable if R 0 < 1 , and the infection equilibrium is globally asymptotically stable if R 0 > 1. For the multi-strain model, we found that all viral strains coexist if the corresponding basic reproductive number R j 𝜖 > 1 , while virus will extinct if R j 𝜖 < 1. As a result, we found that delay and the density-dependent diffusion does not influence the global stability of the model with cell-to-cell infection and homogeneous Neumann boundary conditions. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
20. Transmission Dynamics of an Epidemic Model with Vaccination, Treatment and Isolation.
- Author
-
Cui, Qianqian, Zhang, Qiang, Qiu, Zhipeng, and Yang, Xiaomin
- Subjects
GLOBAL asymptotic stability ,VACCINATION ,COMMUNICABLE diseases ,BASIC reproduction number ,LYAPUNOV functions - Abstract
This paper focuses on the global stability of an epidemic model with vaccination, treatment and isolation. The basic reproduction number R 0 is derived. By constructing suitable Lyapunov functions, sufficient conditions for the global asymptotic stability of equilibria are obtained. Numerical simulations are performed to verify and complement the theoretical results. Furthermore, we consider the uncertainty and sensitivity analysis of the basic reproduction number R 0 . The results show that the transmission rate, the fraction of infected receives treatment, vaccination rate, the isolation rate are crucial to prevent the spread of infectious diseases. These suggest that public health workers design the control strategies of disease should consider the influence of vaccination, treatment and isolation. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. Global analysis of a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate.
- Author
-
Su, Yongmei, Sun, Deshun, and Zhao, Lei
- Subjects
GLOBAL analysis (Mathematics) ,CELLULAR immunity ,IMMUNE response ,T cells ,LYAPUNOV functions - Abstract
In this paper, a humoral and cellular immunity virus dynamics model with the Beddington-DeAngelis incidence rate is set up. We derive the basic reproductive number R
0 , the cytotoxic T lymphocytes immune response reproductive number R1 , the humoral immune response reproductive number R2 , humoral immune response competitive reproductive number R3 , and cytotoxic T lymphocytes immune response competitive reproductive number R4 , and a full description of the relation between the existence of the equilibria and reproductive numbers is given. The global properties of the five equilibria are obtained by constructing Lyapunov functions. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]- Published
- 2015
- Full Text
- View/download PDF
22. Global stability of a discrete SIR epidemic model with vaccination and treatment.
- Author
-
Cui, Qianqian and Zhang, Qiang
- Subjects
EPIDEMIOLOGICAL models ,DISEASE susceptibility ,VACCINATION ,COMPUTATIONAL mathematics ,APPLIED mathematics - Abstract
In this paper, we study a discrete susceptible-infected-removed epidemic model with vaccination and treatment and it is shown that the global dynamics is determined by the basic reproduction number. If, then the disease-free equilibrium is globally asymptotically stable and if, then the endemic equilibrium is globally asymptotically stable. [ABSTRACT FROM PUBLISHER]
- Published
- 2015
- Full Text
- View/download PDF
23. A spatial echinococcosis transmission model with time delays: Stability and traveling waves.
- Author
-
Xu, Zhiting and Ai, Cuihua
- Subjects
ECHINOCOCCOSIS ,TIME delay systems ,TRAVELING waves (Physics) ,LYAPUNOV functions ,MATHEMATICS theorems ,INFECTIOUS disease transmission - Abstract
In this paper, we derive a time-delayed and diffusive echinococcosis transmission model. We first address the well-posedness to the initial-value problem for the model and give the basic reproduction number . In the case of a bounded spatial domain, we establish the local stability as well as the global stability of the disease-free and disease equilibria of the model. The methods to prove the local and the global stability are to analyze the corresponding characteristic equations and construct Lyapunov functionals, respectively. In the case of an unbounded spatial domain, by applying Schauder's fixed point theorem and the limiting arguments, we show that when , there exists a constant such that the model admits positive traveling wave solutions connecting the disease-free and endemic equilibrium for , and when and , the model has no positive traveling wave solutions connecting them. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
24. Stability preserving NSFD scheme for a delayed viral infection model with cell-to-cell transmission and general nonlinear incidence.
- Author
-
Xu, Jinhu and Geng, Yan
- Subjects
FINITE difference method ,VIRUS diseases ,IMMUNE response - Abstract
In this paper, we designed and analysed a discrete model to solve a delayed within-host viral infection model by using non-standard finite difference scheme. The original model that we considered was a delayed viral infection model with cell-to-cell transmission, cell-mediated immune response and general nonlinear incidence. We show that the discrete model has equilibria which are exactly the same as those of the original continuous model and the conditions for those equilibria to be globally asymptotically stable are consistent with the original continuous model with no restriction on the time step size. The results imply that the discretization scheme can efficiently preserves the qualitative properties of solutions for corresponding continuous model. [ABSTRACT FROM PUBLISHER]
- Published
- 2017
- Full Text
- View/download PDF
25. Global stability of a multi-group model with distributed delay and vaccination.
- Author
-
Xu, Jinhu, Geng, Yan, and Zhou, Yicang
- Subjects
EPIDEMIOLOGICAL models ,VACCINATION ,INCIDENCE functions ,LYAPUNOV functions ,EPIDEMICS - Abstract
A delayed multi-group SVEIR epidemic model with vaccination and a general incidence function has been formulated and studied in this paper. Mathematical analysis shows that the basic reproduction number [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
26. An inverse-free dynamical system for solving the absolute value equations.
- Author
-
Chen, Cairong, Yang, Yinong, Yu, Dongmei, and Han, Deren
- Subjects
- *
DYNAMICAL systems , *ABSOLUTE value , *EQUATIONS , *GLOBAL asymptotic stability , *COMPUTER simulation , *EIGENVALUES - Abstract
In this paper, an inverse-free dynamical system is built to solve the absolute value equations (AVEs), whose equilibrium points coincide with the solutions of the AVEs. Under proper assumptions, the equilibrium points of the dynamical system exist and could be (globally) asymptotically stable. In addition, with strongly monotone property, a global projection-type error bound is provided to estimate the distance between any trajectories and the unique equilibrium point. Compared with four existing dynamical systems for solving the AVEs, our method is inverse-free and is still valid even if 1 is an eigenvalue of the coefficient matrix. Some numerical simulations are given to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
27. Globally attracting positive periodic solution of the [formula omitted]-dimensional periodic Ricker system.
- Author
-
Zhang, Yuhong, Song, Yuheng, and Niu, Lei
- Subjects
- *
PERIODIC functions , *LIMIT cycles , *GLOBAL analysis (Mathematics) , *LOTKA-Volterra equations - Abstract
In this paper, we provide a criterion to guarantee the existence of a globally attracting positive periodic solution for the nonautonomous Ricker system whose coefficients are all periodic functions of a common period, that is, under certain circumstances, all species will coexist in a limit cycle fashion. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Dynamic behaviors and non-instantaneous impulsive vaccination of an SAIQR model on complex networks.
- Author
-
Fu, Xinjie and Wang, JinRong
- Subjects
- *
BASIC reproduction number , *VACCINATION - Abstract
We establish an SAIQR epidemic network model, in which asymptomatic infected people (A) are as contagious as infected people (I). The basic reproductive number R 0 is calculated, and the globally asymptotically stable of the disease-free equilibrium, the globally attractive and globally asymptotically stable of the endemic equilibrium are obtained. For the control of epidemic transmission, we take into account the non-instantaneous impulsive vaccination in the model, calculate the basic reproduction number R 0 ⁎ of the model, and demonstrate that the disease-free T -periodic solution is globally attractive and the model is permanent. Finally, we choose scale-free network to simulate numerically and validate the results of this paper. • An SAIQR epidemic network model is established, in which asymptomatic infected people are as contagious as infected people. • The globally asymptotically stable of the disease-free equilibrium and the endemic equilibrium are obtained. • Non-instantaneous impulsive vaccination is considered in the system, and the permanence of the system are investigated. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Asymptotic behavior of the principal eigenvalue for cooperative periodic-parabolic systems and applications.
- Author
-
Bai, Xueli and He, Xiaoqing
- Subjects
- *
EIGENVALUES , *DIFFUSION coefficients , *ORDINARY differential equations , *POPULATION dynamics , *BEHAVIOR - Abstract
The effects of spatial heterogeneity on population dynamics have been studied extensively. However, the effects of temporal periodicity on the dynamics of general periodic-parabolic reaction-diffusion systems remain largely unexplored. As a first attempt to understand such effects, we analyze the asymptotic behavior of the principal eigenvalue for linear cooperative periodic-parabolic systems with small diffusion rates. As an application, we show that if a cooperative system of periodic ordinary differential equations has a unique positive periodic solution which is globally asymptotically stable, then the corresponding reaction-diffusion system with either the Neumann or regular oblique derivative boundary condition also has a unique positive periodic solution which is globally asymptotically stable, provided that the diffusion coefficients are sufficiently small. The role of temporal periodicity, spatial heterogeneity and their combined effects with diffusion will be studied in subsequent papers for further understanding and illustration. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
30. Study of two-nutrient and two-micro-organism chemostat model with pulsed input in a polluted environment.
- Author
-
Jia, Jianwen and Lv, Tingting
- Subjects
CHEMOSTAT ,FLOQUET theory ,PERTURBATION theory ,COMPUTER simulation ,MICROORGANISMS - Abstract
In this paper, a model of Beddington-DeAngelies chemostat involving two species of micro-organism competing for two perfectly complementary growth-limiting nutrients and pulsed input of toxicant in the polluted environment was studied. Using Floquet theory and small amplitude perturbation method, a conclusion was that there exists two-micro-organism eradication periodic solution and which is global asymptotical stability. At the same time, the condition of the permanence for system was obtained. From the biological point of view, the method for protecting species is to improve the amount of impulsive period, and control the amount of toxicant input to the chemostat. Finally, our results are illustrated by numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
31. Structure and dynamics of acyclic networks.
- Author
-
Veliz-Cuba, Alan, Murrugarra, David, and Laubenbacher, Reinhard
- Abstract
Acyclic networks are dynamical systems whose dependency graph (or wiring diagram) is an acyclic graph. It is known that such systems will have a unique steady state and that it will be globally asymptotically stable. Such result is independent of the mathematical framework used. More precisely, this result is true for discrete-time finite-space, discrete-time discrete-space, discrete-time continuous-space and continuous-time continuous-space dynamical systems; however, the proof of this result is dependent on the framework used. In this paper we present a novel and simple argument that works for all of these frameworks. Our arguments support the importance of the connection between structure and dynamics. [ABSTRACT FROM AUTHOR]
- Published
- 2014
- Full Text
- View/download PDF
32. Rich and complex dynamics of a time-switched differential equation model for wild mosquito population suppression with Ricker-type density-dependent survival probability.
- Author
-
Zhongcai Zhu and Xue He
- Subjects
DIFFERENTIAL equations ,ORDINARY differential equations ,MOSQUITOES - Abstract
Dengue presents over 390 million cases worldwide yearly. Releasing Wolbachiainfected male mosquitoes to suppress wild mosquitoes via cytoplasmic incompatibility has proven to be a promising method for combating the disease. As cytoplasmic incompatibility causes early developmental arrest of the embryo during the larval stage, we introduce the Ricker-type survival probability to assess the resulting effects. For periodic and impulsive release strategies, our model switches between two ordinary differential equations. Owing to a Poincar'e map and rigorous dynamical analyses, we give thresholds T*; c* and c**(> c*) for the release period T and the release amount c. Then, we assume c > c* and prove that our model admits a globally asymptotically stable periodic solution, provided T > T*, and it admits at most two periodic solutions when T < T*. Moreover, for the latter case, we assert that the origin is globally asymptotically stable if c ≥ c**, and there exist two positive numbers such that whenever there is a periodic solution, it must initiate in an interval composed of the aforementioned two numbers, once c* < c < c**. We also offer numerical examples to support the results. Finally, a brief discussion is given to evoke deeper insights into the Ricker-type model and to present our next research directions. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. Global dynamic behavior of a plant disease model with ratio dependent impulsive control strategy.
- Author
-
Li, Wenjie, Huang, Lihong, Guo, Zhenyuan, and Ji, Jinchen
- Subjects
- *
BASIC reproduction number , *PLANT diseases , *MEDICAL model - Abstract
In this paper, we consider the dynamics of a plant disease model with a ratio-dependent state impulsive control strategy. It is shown that the boundary equilibrium point of the controlled system is globally asymptotically stable. By combining LaSalle's invariant theorem, Brouwer's fixed point theorem and some analysis techniques, we are able to determine the basic reproduction number, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability of the equilibria. Most interestingly, we find that in the case that the basic reproduction number is more than unity and the endemic equilibrium locates above the impulsive control strategy, we can obtain a unique k-order periodic solution and the critical values between 1-order and 2-order periodic solutions. Furthermore, it is found that the endemic equilibrium point is also globally asymptotically stable under the control strategy. Finally, we present a numerical example to substantiate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]
- Published
- 2020
- Full Text
- View/download PDF
34. Dynamics of a Predator–Prey System with Impulsive Stocking Prey and Nonlinear Harvesting Predator at Different Moments
- Author
-
Zeli Zhou, Jianjun Jiao, Xiangjun Dai, and Lin Wu
- Subjects
predator–prey system ,impulsive stocking prey ,impulsive nonlinear harvesting ,globally asymptotically stable ,permanence ,Mathematics ,QA1-939 - Abstract
In this article, we study a predator–prey system, which includes impulsive stocking prey and a nonlinear harvesting predator at different moments. Firstly, we derive a sufficient condition of the global asymptotical stability of the predator–extinction periodic solution utilizing the comparison theorem of the impulsive differential equations and the Floquet theory. Secondly, the condition, which is to maintain the permanence of the system, is derived. Finally, some numerical simulations are displayed to examine our theoretical results and research the effect of several important parameters for the investigated system, which shows that the period of the impulse control and impulsive perturbations of the stocking prey and nonlinear harvesting predator have a significant impact on the behavioral dynamics of the system. The results of this paper give a reliable tactical basis for actual biological resource management.
- Published
- 2024
- Full Text
- View/download PDF
35. An efficient approach to stabilization for linear systems subject to output saturation.
- Author
-
Xie, Pengfei and Liu, Wanquan
- Subjects
- *
TRAFFIC safety - Abstract
This paper studies the stabilization problem for controllable and observable linear systems subject to output saturation. An efficient approach to global asymptotical stabilization is proposed, where multiple output components can be collectively steered to escape from saturation. This differs from previous approaches in which each output component is driven one by one. Sufficient conditions for collectively driving all output elements are presented in single-batch driven strategy. In the case of the above sufficient conditions not being satisfied, a multi-batch driven strategy is constructed to guarantee that each component is steered away from saturation at least once. Moreover, an algorithm is put forward for selecting controller parameters in the multi-batch driven case. Finally, numerical examples are given to demonstrate the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
36. Global dynamics and control of malicious signal transmission in wireless sensor networks.
- Author
-
Li, Wenjie, Ji, Jinchen, Huang, Lihong, and Zhang, Lingling
- Abstract
This paper studies the global dynamics of a discontinuous delayed model of malicious signal transmission in wireless sensor networks under the framework of differential inclusion. The local stability of two types of steady states are investigated for the discontinuous system by studying the corresponding characteristic equation. The sufficient conditions for the existence of two types of globally asymptotically stable steady states are obtained for the discontinuous system by using the comparison arguments method. Furthermore, the optimal control of the discontinuous system is investigated by using Pontryagin's maximum principle. Numerical simulations of two examples are carried out to illustrate the main theoretical results. The obtained results can help us to better control and predict the spread of malicious signal transmission in wireless sensor networks. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. A mathematical model study on plant root pest management.
- Author
-
Lizhuang Huang, Yuan Zhuang, and Qiong Liu
- Subjects
PLANT parasites ,PLANT roots ,MATHEMATICAL models ,EVIDENCE gaps - Abstract
Unlike conventional methods of pests control, introducing in an appropriate mathematical model can contribute a batter performance on pests control with higher efficiency while lest damage to ecosystem. To fill the research gap on plant root pest control, we propose a plant root pest management model with state pulse feedback control. Firstly, the stability of the equilibrium point of the model (1.3) is analyzed by using the linear approximate equation, given that the only positive equilibrium point of model (1.3) is globally asymptotically stable. Moreover, the existence and uniqueness of order 1 periodic solutions of model (1.3) are discussed in detail according to the geometric theory of semi-continuous dynamical systems, successor functions method and the qualitative theory of differential equations. Finally, with further analysis in different methods, the asymptotic stability of the order 1 periodic solution of model (1.3) is obtained by using Similar Poincare Criterion or interval set theorem. The results show that this model can effectively control the number of pests below the economic level of damage. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
38. Global Asymptotic Stability of a System of Difference Equations with Quadratic Terms.
- Author
-
El-Moneam, Mohamed Abd
- Subjects
GLOBAL asymptotic stability ,DIFFERENCE equations ,QUADRATIC equations - Abstract
In this article, we discuss the global asymptotic stability of following system of difference equations with quadratic terms:... where α, β are positive numbers and the initial values are positive numbers. We also study the rate of convergence and oscillation behaviour of the solutions of related system. We will give also, some numerical examples to illustrate our results. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. Improved results on H ∞ state estimation of static neural networks with interval time-varying delay
- Author
-
Shu, Yanjun and Liu, Xinge
- Published
- 2016
- Full Text
- View/download PDF
40. Mathematical model of Alzheimer's disease with prion proteins interactions and treatment.
- Author
-
Li, Huixia and Zhao, Hongyong
- Subjects
- *
ALZHEIMER'S disease , *PRION diseases , *PROTEIN-protein interactions , *GLOBAL asymptotic stability , *PRIONS , *ELASTIC analysis (Engineering) - Abstract
• Studied an Alzheimer's disease model with discrete sizes and two types of A β , the interaction with P r p c as well as the treatment. • Proved the global asymptotic stability. • Presented numerical simulations and investigate the contributions from two types of A β. • Explained A β 42 paranuclei can be used as a therapeutic target. The accumulation of β -amyloid (A β) is one of the most important pathogenic factors in the occurrence of Alzheimer's disease (AD). Studies have shown that oligomers are more toxic in the process of A β aggregation because oligomers can interact with receptors such as prion proteins (P r p c) and the interaction causes P r p c to be misfolded into pathogenic oligomeric prion proteins (P r p o l ). In this paper, we propose an AD model including two types of A β , the interaction of oligomers with P r p c and anti- A β drugs treatment. The existence, uniqueness, and non-negativity of the solutions are analyzed. Furthermore, we prove that the model admits a unique globally asymptotically stable equilibrium, which means the drug cannot cure AD completely. Finally, we present some numerical simulations and investigate the relative contributions to AD with two types of A β , A β 40 and A β 42. In addition, elastic analysis description A β 42 paranuclei can be used as a therapeutic target. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
41. Asymptotic behavior of a three species eco-epidemiological model perturbed by white noise.
- Author
-
Zhang, Qiumei, Jiang, Daqing, Liu, Zhenwen, and O'Regan, Donal
- Subjects
- *
PERTURBATION theory , *WHITE noise theory , *STOCHASTIC differential equations , *LYAPUNOV functions , *PARAMETERS (Statistics) - Abstract
This paper considers a three species eco-epidemiological model perturbed by white noise. Stochastic stability and long time behavior around the equilibrium of deterministic eco-epidemiological model are analyzed by Lyapunov methods. Numerical simulations for a set of parameter values are presented to illustrate the analytical findings. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
42. An efficient projection neural network for solving bilinear programming problems.
- Author
-
Effati, Sohrab, Mansoori, Amin, and Eshaghnezhad, Mohammad
- Subjects
- *
ARTIFICIAL neural networks , *MIXED integer linear programming , *BILINEAR forms , *LINEAR complementarity problem , *MATHEMATICAL programming - Abstract
In this paper the application of projection neural network for solving bilinear programming problems (BLPs) is obtained. So far as we know, no study has yet been attempted for these problems via neural network. In fact, some interesting reformulations of BLP and mixed-integer bilinear programming problem (MIBLP) with a binary vector to linear complementarity problem (LCP) are given. Additionally, we show that the special type of MIBLP with a binary vector is equal to a quadratic program and on the other hand, it is equal to a mixed-integer linear program (MILP). Finally, we use a neural network to solve projection equation which has the same solution with LCP. Then, by presenting a Lyapunov function, we show that the proposed neural network is globally asymptotically stable. Illustrative examples are given to show the effectiveness and efficiency of our method. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
43. On the dynamics of the nonlinear rational difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $
- Author
-
A. M. Alotaibi and M. A. El-Moneam
- Subjects
difference equations ,rational difference equations ,qualitative properties of solutions of difference equations ,equilibrium ,oscillates ,prime period two solution ,globally asymptotically stable ,points ,Mathematics ,QA1-939 - Abstract
In this paper, we discuss some qualitative properties of the positive solutions to the following rational nonlinear difference equation $ { x_{n+1}} = \frac{{\alpha {x_{n-m}}} \ \ + \ \ \delta {{x_{n}}}}{{\beta +\gamma {x_{n-k}} \ \ { x_{n-l}} \ \ \left({{x_{n-k}} \ \ + \ \ {x_{n-l}}} \ \ \right) }} $, $ n = 0, 1, 2, ... $ where the parameters $ \alpha, \beta, \gamma, \delta \in (0, \infty) $, while $ m, k, l $ are positive integers, such that $ m < k < l. $ The initial conditions $ {x_{-m}}, ..., {x_{-k}}, ..., {x_{-l}}, ..., {x_{-1}}, ..., {x_{0}} $ are arbitrary positive real numbers. We will give some numerical examples to illustrate our results.
- Published
- 2022
- Full Text
- View/download PDF
44. Feedback control for a class of second order hyperbolic distributed parameter systems
- Author
-
Fu, Qin, Gu, Weiguo, Gu, Panpan, and Wu, Jianrong
- Published
- 2016
- Full Text
- View/download PDF
45. Global dynamics of a two-strain flu model with delay.
- Author
-
Xu, Zhiting, Qu, Liangcheng, and Huang, Yehui
- Subjects
- *
DYNAMICAL systems , *LYAPUNOV functions , *COMPUTER simulation , *VACCINATION , *NUMBER theory - Abstract
In this paper, we deal with the global dynamics of a two-strain flu model with delay. Using the method of Lyapunov functional, we show that if the basic reproduction number is less than one, then both strains die out; but when the number is larger than one, one or both of the strains become endemic. The main results are confirmed by some numerical simulations. The theoretical results obtained here provide some useful information on the impact of the vaccination rate of a single vaccine for one strain on the dynamics of the two strains. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
46. Global stability and persistence of HIV models with switching parameters and pulse control.
- Author
-
Wang, Xiying, Liu, Xinzhi, Xie, Wei-chau, Xu, Wei, and Xu, Yong
- Subjects
- *
STABILITY theory , *HIV infections , *PARAMETER estimation , *REVERSE transcriptase inhibitors , *PROTEASE inhibitors - Abstract
This paper studies some HIV (Human Immunodeficiency Virus) models with switching parameters and pulse control. By taking into account of the effects of reverse transcriptase inhibitor (RTI) drugs, protease inhibitor (PI) drugs and the variable transmission rate, we propose a new HIV model with switching parameters and derive a general threshold value that measures the persistence of the disease. Furthermore, pulse vaccination is applied to the HIV model and some novel threshold conditions are established to ensure the existence and stability of the periodic infection-free solution. In contrast to the standard HIV models, it is shown that our proposed models are more practical and useful. Moreover, pulse vaccination strategies are proven to be more effective in determining whether or not the disease is eradicated. Numerical simulations are carried out to illustrate the effectiveness of the obtained results, and future research directions are suggested. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
47. Extinction and permanence of the predator-prey system with general functional response and impulsive control
- Author
-
Juan Liu, Jie Hu, and Peter W. T. Yuen
- Subjects
Integrated pest management ,Floquet theory ,Impulsive ,education.field_of_study ,Computer simulation ,Applied Mathematics ,Population ,Functional response ,Perturbation (astronomy) ,02 engineering and technology ,Globally asymptotically stable ,01 natural sciences ,Permanence ,Predation ,020303 mechanical engineering & transports ,0203 mechanical engineering ,Modeling and Simulation ,Stability theory ,0103 physical sciences ,Applied mathematics ,education ,010301 acoustics ,General functional response ,Mathematics - Abstract
Traditional approach for modelling the evolution of populations in the predator-prey ecosystem has commonly been undertaken using specific impulsive response function, and this kind of modelling is applicable only for a specific ecosystem under certain environ- mental situations only. This paper attempts to fill the gap by modelling the predator-prey ecosystem using a ‘generalized’ impulsive response function for the first time. Different from previous research, the present work develops the modelling for an integrated pest management (IPM) especially when the stocking of predator (natural enemy) and the har- vesting of prey (pest) occur impulsively and at different instances of time. The paper firstly establishes the sufficient conditions for the local and the global stabilities of prey eradica- tion periodic solution by applying the Floquet theorem of the Impulsive different equation and small amplitude perturbation under a ‘generalized’ impulsive response function. Sub- sequently the sufficient condition for the permanence of the system is given through the comparison techniques. The corollaries of the theorems that are established by using the ‘general impulsive response function’ under the locally asymptotically stable condition are found to be in excellent agreement with those reported previously. Theoretical results that are obtained in this work is then validated by using a typical impulsive response func- tion (Holling type-II) as an example, and the outcome is shown to be consistent with the previously reported results. Finally, the implication of the developed theories for practical pest management is illustrated through numerical simulation. It is shown that the elim- ination of either the preys or the pest can be effectively deployed by making use of the theoretical model established in this work. The developed model is capable to predict the population evolutions of the predator-prey ecosystem to accommodate requirements such as: the combinations of the biological control, chemical control, any functional response function, the moderate impulsive period, the harvest rate for the prey and predator pa- rameter and the incremental stocking of the predator parameter
- Published
- 2020
48. Global dynamics of a two-strain flu model with a single vaccination and general incidence rate
- Author
-
Arturo J. Nic-May and Eric J. Avila-Vales
- Subjects
general nonlinear incidence rate ,mathematical model ,basic reproduction number ,lyapunov functional ,globally asymptotically stable ,vaccination ,influenza ,Biotechnology ,TP248.13-248.65 ,Mathematics ,QA1-939 - Abstract
Influenza remains one of the major infectious diseases that target humankind, therefore, understand transmission mechanisms and control strategies can help us obtain more accurate predictions. There are many control strategies, one of them is vaccination. In this paper, our purpose is to extend the incidence rate of a two-strain flu model with a single vaccination, which includes a wide range of incidence rates among them, some cases are not monotonic nor concave, which may be used to reflect media education or psychological effect. Our main aim is to mathematically analyze the effect of the vaccine for strain 1, the general incidence rate of strain 1 and the general incidence rate of strain 2 on the dynamics of the model. Four equilibrium points were obtained and the global dynamics of the model are completely determined via suitable Lyapunov functions. We illustrate our results by some numerical simulations. Our results showed that the vaccination is always beneficial for controlling strain 1, its impact on strain 2 depends on the force of infection of strain 2. Also, the psychological effect is always beneficial for controlling the disease.
- Published
- 2020
- Full Text
- View/download PDF
49. Dynamic analysis of a soil organic matter and plant system with reaction-diffusion.
- Author
-
Pan, Shiliang, Zhang, Qimin, and Meyer-Baese, Anke
- Subjects
- *
SOIL testing , *BASIC reproduction number , *ORGANIC compounds , *PLANT growth , *DYNAMIC models - Abstract
Soil organic matter (SOM) is considered to be a driving factor in determining plant patterns, but few dynamic models include both plant and SOM. In this paper, we presented a plant-SOM model with reaction diffusion to understand how SOM affects dynamic changes of plant growth. We proposed the existence and uniqueness of the global positive solution of the model and then defined the basic reproduction number ℜ 0 : when ℜ 0 ≤ 1 , the plant-free equilibrium was globally asymptotically stable; when ℜ 0 > 1 , the solution to the system was uniformly persistent. We also confirmed the global attractivity of a unique positive equilibrium when all the parameters of this model are constant. Finally, we conducted numerical simulations to verify the effectiveness of the theoretical analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
50. A nonstandard finite difference scheme for a multi-group epidemic model with time delay
- Author
-
Jinhu Xu and Yan Geng
- Subjects
multi-group ,time delay ,NSFD scheme ,globally asymptotically stable ,Mathematics ,QA1-939 - Abstract
Abstract In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stability are identical in case of both the discrete and the corresponding continuous models. Specifically, if R 0 ≤ 1 $\mathfrak{R}_{0}\leq1$ , then the disease-free equilibrium P 0 $P_{0}$ is globally asymptotically stable; if R 0 > 1 $\mathfrak{R}_{0}>1$ , then the infection equilibrium P ∗ $P_{*}$ is globally asymptotically stable. The results imply that the discretization scheme can efficiently preserve the global dynamics of the original continuous model.
- Published
- 2017
- Full Text
- View/download PDF
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.