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For a class of scalar delay differential equations with impulses and satisfying a Yorke-type condition, criteria for the global asymptotic stability of the zero solution are established. These equations possess a non-delayed feedback term, which will be used to refine the general results on stability presented in recent literature. The usual requirements on the impulses are also relaxed. As an application, sufficient conditions for the global attractivity of a periodic solution for an impulsive periodic model are given.

In this paper, we study a time-periodic and delayed reaction-diffusion system with quiescent stage in both unbounded and bounded habitat domains. In unbounded habitat domain $\mathbb{R}$, we first prove the existence of the asymptotic spreading speed and then show that it coincides with the minimal wave speed for monotone periodic traveling waves. In a bounded habitat domain $\Omega\subset\mathbb{R}^N\ (N\geq1)$, we obtain the threshold result on the global attractivity of either the zero solution or the unique positive time-periodic solution of the system.

Amensalism, a rare yet impactful symbiotic relationship in ecological systems, is the focus of this study. We examine a discrete-time amensalism system by incorporating the fear effect on the first species. We identify the plausible equilibrium points and analyze their local stability conditions. The global attractivity of the positive equilibrium, $ E^* $, and the boundary equilibrium, $ E_1 $, are analyzed by exploring threshold conditions linked to the level of fear. Additionally, we analyze transcritical bifurcations and flip bifurcations exhibited by the boundary equilibrium points analytically. Considering some biologically feasible parameter values, we conduct extensive numerical simulations. From numerical simulations, it is observed that the level of fear has a stabilizing effect on the system dynamics when it increases. It eventually accelerates the extinction process for the first species as the level of fear continues to increase. These findings highlight the complex interplay between external factors and intrinsic system dynamics, enriching potential mechanisms for driving species changes and extinction events.

We study the long-term behavior of positive solutions of the cyclic system of difference equations $$ x^{(i)}_{n+1}=\max\Big\{\alpha,\frac{(x^{(i+1)}_n)^p}{(x^{(i+2)}_{n-1})^q}\Big\}, \quad i=1,\ldots,k,\; n\in\mathbb{N}_0, $$ where $k\in\mathbb{N}$, $\min\{\alpha, p, q\}>0$ and where we regard that $x^{(i_1)}_n=x^{(i_2)}_n$ when $i_1\equiv i_2$ (mod $k$). We determine the set of parameters $\alpha$, p and q in $(0,\infty)^3$ for which all such solutions are bounded. In the other cases we show that the system has unbounded solutions. For the case p=q we give some sufficient conditions which guaranty the convergence of all positive solutions. The main results in this paper generalize and complement some recent ones.

Abstract In this paper, we propose a viral infection model with multi-target cells on a heterogeneous environment. Then we use the semigroup theory and a variational characterization to compute the local reproduction number R (x) and the basic reproduction number R 0. Furthermore, the model exhibits a threshold dynamics: the virus-free steady state E 0 is globally asymptotically stable if R 0 < 1 ; otherwise, the endemic steady state E ∗ is globally asymptotically stable. Finally, we perform some numerical examples to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the global behavior of a multi-group SIR epidemic model with age structure. A similar model was studied in [T. Kuniya, J. Wang, and H. Inaba, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), pp. 3515--3550] under some restrictive assumptions. In this paper, we weaken some of these assumptions for the purpose of application, and show that the global behavior of our model is completely determined by the basic reproduction number \scrR 0: if \scrR 0 < 1, then the disease-free equilibrium is globally attractive, whereas if \scrR 0 > 1, then the endemic equilibrium is globally attractive. The proofs are done by constructing suitable Lyapunov functions for total trajectories in compact attractors. In the application, we consider the chlamydia epidemic in Japan in 2015, and compare the estimation results of \scrR 0 in four special cases of our model: a homogeneous model, an age-independent two-sex model, an age-dependent onesex model, and an age-dependent two-sex model. In conclusion, we see that \scrR 0 for the chlamydia epidemic in Japan in 2015 is in the range 1.0148--1.0535, and the introduction of the age structure has more influence on the value of \scrR 0 than that of the two-group structure. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear fractional differential equation with Hadamard fractional derivative. By using comparision results sufficient conditions are obtained for the global attractivity of the solutions for nonlinear fractional differential equations in weighted spaces. [ABSTRACT FROM AUTHOR]

A general diffusive predator-prey model is investigated in this paper. We prove the global attractivity of constant equilibria when the conversion rate is small, and the non-existence of non-constant positive steady states when the conversion rate is large. The results are applied to several predator-prey models and give some ranges of parameters where complex pattern formation cannot occur. [ABSTRACT FROM AUTHOR]

The primary focus of this paper is to investigate the boundedness and asymptotic behavior of the following symmetric system of max-type difference equations Xn+1 = max{c, ypn/zqn-1}, yn+1 = max {c, zpn/xqn-1}, zn+1 = max {c, xpn/yqn-1}, n ∊ N0, where the parameters c, p,q ∊ (0, ∞) and the initial conditions x-1, x0, y-1, y0, z-1, z0 are arbitrary positive real numbers. Our main results considerably improve results appearing in the literature (see, Stević, (2014) [29]). [ABSTRACT FROM AUTHOR]

This paper aims to investigate the global behaviours of a diffusive viral model with spatial heterogeneity on the general bounded domain and general incidence function. On the basis of the theory of uniform persistence, we obtain the system is uniformly persistent under appropriate conditions. By constructing Lyapunov functions and using LaSalle invariance principle for PDEs, we obtain that the global behaviours is completely determined by the value of the basic reproduction number for viral infection in the special case. Some numerical simulations are carried out to illustrate the analytical predictions. [ABSTRACT FROM AUTHOR]

In this study, a class of delayed prey–predator model with Hassell–Varley type functional response is investigated, in which the harvesting policies are modeled by discontinuous functions. Based on functional differential inclusions and set-valued analysis theories, the local and global existence of positive solution in sense of Filippov is given. By employing theory of functional differential inclusions and generalized differential inequalities, some sufficient conditions which guarantee the permanence of the system are obtained. According to non-smooth analysis theory with generalized Lyapunov functional approach, a series of useful criteria on existence, uniqueness and global asymptotic stability of the almost positive periodic solution to the system are derived. Some suitable examples together with their numerical simulations are given to illustrate the effectiveness of our results by using MATLAB. [ABSTRACT FROM AUTHOR]

This paper deals with a class of fuzzy cellular neural networks with multi-proportional delays. By applying the contraction mapping fixed point theorem and differential inequality techniques, a set of easily verifiable sufficient conditions are established for the existence and global attractivity of a unique pseudo almost periodic solution for the model, which improve and supplement previously known researches. Moreover, a numerical example is given to illustrate the feasibility and application of the obtained results. [ABSTRACT FROM AUTHOR]

Rumor spreading is an important form of social communication, and it will cause unnecessary panic and conflict to society. This paper aims to investigate the combined impact of the internal and external factors on the efficiency of rumor spreading over the social networks. In this paper, firstly, a modified I S R rumor spreading model on heterogeneous networks by taking into account the activity and infectivity of nodes and propagation environment was introduced. Then, through theoretical analysis, we derive the mean field equations with considering nonlinear spreading rate and active network nodes. Next, we calculate the basic reproduction number ℜ 0 based on the next generation matrix method whose results reveal that if ℜ 0 < 1 , the rumor-free equilibrium is global asymptotically stable, the rumors will disappear. If ℜ 0 > 1 , the rumor-endemic equilibrium is global attractivity, then the number of spreaders will remain stable and the rumors will become endemic. Further, some numerical simulations are performed, which is consistent with the deterministic mean-field approach. Our results show that it is very important to consider the internal and external factors to control the spread of the rumors. [ABSTRACT FROM AUTHOR]

The Lasota-Wazewska model is often used to describe the regeneration of red blood cells in animals. Based on the Banach contraction mapping principle and through construction of the Lyapunov function，the existence，uniqueness and global attractivity of pseudo almost periodic solutions to a class of Lasota-Wazewska models were studied. The results have some advantages，and can enrich the characterization of the dynamic behavior of the Lasota-Wazews- ka model. [ABSTRACT FROM AUTHOR]

In this paper, by applying the fluctuation lemma and some differential inequality technique, delay‐dependent criteria are obtained for the global attractivity of a differential neoclassical growth model with patch structure and multiple time‐varying delays. Moreover, some numerical examples are provided to illustrate the validity of the obtained results. [ABSTRACT FROM AUTHOR]

Insecticide-treated bed nets (ITNs) are among the most important and effective intervention measures against malaria. In order to investigate the impact of bed net use on disease control, we formulate a periodic vector-bias malaria model incorporating the juvenile stage of mosquitoes and the use of ITNs. We derive the vector reproduction ratio Rv and the basic reproduction ratio R0. We show that the global dynamics of the model is completely determined by these two reproduction ratios. More precisely, the mosquito-free periodic solution is globally attractive if Rv<1; the unique disease-free periodic solution is globally attractive if Rv>1 and R0<1; and the model admits a unique positive periodic solution and it is globally attractive if Rv>1 and R0>1. Numerically, we study the malaria transmission case in Port Harcourt, Nigeria. Our findings show that the use of ITNs has a positive effect on reducing R0, and that malaria may be eliminated from this area if over 75% of the human population were to use ITNs. The simulation about the long term behavior of solutions has good agreement with the obtained analytic result. Moreover, we find that the ignorance of the vector-bias effect may result in underestimation of the basic reproduction ratio R0. Another notable result is that the infection risk would be underestimated if the basic reproduction ratio [R0] of the time-averaged autonomous system were used. [ABSTRACT FROM AUTHOR]

The main objective of this paper is to study the global attractivity, and the boundedness for the solutions of the rational difference equation ... where the parameters &#945, &#945, &#945, A,B,p and q ∊ (0,∞) and the initial conditions x-l; x-l+1; :::; x-1; x0 where l = max{km} are positive real numbers. [ABSTRACT FROM AUTHOR]

In this study, we investigate two kinds of SIS epidemic models with an age structure and spatial heterogeneity. The first is the SIS epidemic model with age and patch structures, and the second is the SIS epidemic model with age structure and spatial diffusion. For both models, we prove that the global attractivity of each equilibrium is completely determined by the basic reproduction number R0; that is, the disease-free equilibrium is globally attractive if R0<1, whereas the endemic equilibrium uniquely exists and is globally attractive if R0>1. In particular, we provide a theoretical justification for the basic reproduction number being the spectral radius of the next generation operator. This enables us to perform numerical simulations that verify our theoretical results, which is important from the viewpoint of applications. [ABSTRACT FROM AUTHOR]

Consider the following higher order difference equation... where/(n, x) and g(n, x): {0,1, . . .} x [0, ∞) → [0, ∞) are continuous functions in x and periodic functions in n with period p, and k is a nonnegative integer. We show the existence of a periodic solution (...(n)} under certain conditions, and then establish a sufficient condition for {cc(n)} to be a global attractor of all nonnegative solutions of the equation. Applications to Riccati difference equation and some other difference equations derived from mathematical biology are also given. [ABSTRACT FROM AUTHOR]

This paper considers a stochastic predator-prey model with infinite delay and impulsive perturbations. Sufficient conditions for permanence in time average are established as well as extinction, stability in time average and global attractivity of the stochasic model. Some simulation figures, which are obtained by the split-step θ-method to discretize the stochasic model, are introduced to support the analytical findings. Our results demonstrate that, firstly, impulsive perturbations which may represent human factor play a key role in maintaining ecological balance; secondly, environmental noise, which can be modelled by Brownian motion, is disadvantageous to population survival; finally, infinite delay has not affect permanence in time average, extinction, stability in time average and global attractivity of the stochasic model. [ABSTRACT FROM AUTHOR]

This paper investigates the asymptotic behavior of a delayed Nicholson‐type system involving patch structure and nonlinear density‐dependent mortality terms. Based on the fluctuation lemma and some differential inequality techniques, delay‐dependent criteria are deduced for the global attractivity of the addressed system, which indicate that small delays have no effect on the attractivity of the system. Meanwhile, an illustrative numerical example is given to illustrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we establish two new stochastic switched epidemic models with continuous and impulsive control. The stochastic perturbations are considered for the natural death rate in each equation of the models. Firstly, a stochastic switched S I L I model with continuous control schemes is investigated. By using Lyapunov–Razumikhin method, the sufficient conditions for extinction in mean are established. Our result shows that the disease could be die out theoretically if threshold value R is less than one, regardless of whether the disease-free solutions of the corresponding subsystems are stable or unstable. Then, a stochastic switched S I L I model with continuous control schemes and pulse vaccination is studied. The threshold value R is derived. The global attractivity of the model is also obtained. At last, numerical simulations are carried out to support our results. [ABSTRACT FROM AUTHOR]

As compared to the traditional advertising, word-of-mouth (WOM) communications have striking advantages such as significantly lower cost and much faster propagation, and this is especially the case with the popularity of online social networks. This paper focuses on the modeling and analysis of the WOM marketing. A dynamic model, known as the SIPNS model, capturing the WOM marketing processes with both positive and negative comments is established. On this basis, a measure of the overall profit of a WOM marketing campaign is proposed. The SIPNS model is shown to admit a unique equilibrium, and the equilibrium is determined. The impact of different factors on the equilibrium of the SIPNS model is illuminated through theoretical analysis. Extensive experimental results suggest that the equilibrium is much likely to be globally attracting. Finally, the influence of different factors on the expected overall profit of a WOM marketing campaign is ascertained both theoretically and experimentally. Thereby, some promotion strategies are recommended. To our knowledge, this is the first time the WOM marketing is treated in this way. [ABSTRACT FROM AUTHOR]

Natural enemies of insects are extremely important in preventing pest outbreaks in crop fields. Therefore, we investigate the dynamical behavior of a pest-dependent consumption pest–natural enemy (i.e., prey–predator) SEIN model concerning diseases in pest population with three classes (susceptible–exposed–infectious) and impulsive releasing of infectious pests and natural enemies at fixed moments of time. We prove that all solutions of the system are uniformly ultimately bounded. In first part of the main results, the sufficient conditions for local as well as global asymptotic stability of the susceptible and exposed pest extinction periodic solution are determined using a Floquet’s theorem of impulsive differential equations, small-amplitude perturbation skills and comparison theorem. In second part, the sufficient condition for the permanence of a system is determined. These dynamics imply that susceptible and exposed pest populations become extinct when impulse period is less than some critical value and pests coexist with natural enemies at low level when impulse period crosses the critical value. Thus, our results provide some reliable theoretical tactics for pest management and finally these are verified by performing some numerical simulations. [ABSTRACT FROM AUTHOR]

The present paper aims to provide a supplementary result on a foot-and-mouth disease model proposed in Sun et al. (2022). We show that the endemic equilibrium for the model is globally attractive when the basic reproduction number is greater than one, which is an unsolved problem in the previously mentioned paper. [ABSTRACT FROM AUTHOR]

We present fixed points results of multivalued Prešić type k-step iterative mappings satisfying generalized weakly contraction conditions in metric spaces. An example is presented to support the main result proved herein. The stability of fixed point sets of multivalued Prešić type weakly contractive mappings are also established. Global attractivity result for the class of matrix difference equations is derived as application of the result presented herein. These results generalize and extend various comparable results in the existing literature.

In this paper, we are concerned with modeling and analyzing the dynamics for a two-stage species that lives on a half plane. We first derive a spatially nonlocal and temporally delayed differential equation that describes the mature population on a semi-infinite environment with a homogeneous Dirichlet condition. For the derived model, we are able to show that the solutions induce a k -set contraction semiflow with respect to the compact open topology on a bounded positive invariant set attracting every solution of the equation. To describe the global dynamics, we first establish a priori estimate for nontrivial solutions after exploring the delicate asymptotic properties of the nonlocal delayed effect, which enables us to show the repellency of the trivial equilibrium. Using the estimate, k -set contracting property as well as Schauder fixed point theorem, we then establish the existence of a positive spatially heterogeneous steady state. At last, we show global attractivity of the nontrivial steady state by employing dynamical system approaches. [ABSTRACT FROM AUTHOR]

This paper aims to study the combined impact of countermeasure and network topology on virus diffusion and optimal dynamic countermeasure. A novel heterogenous propagation model and its optimal control problem are proposed and analyzed. Qualitative analysis shows that the unique equilibrium of the proposed model is globally attractive and the optimal control problem has an optimal control. Some simulation experiments are also performed. Specifically, it is found that our obtained results are contrary to some previous results and countermeasure dissemination to higher-degree nodes is more effective than that to lower-degree nodes. The related explanations are also made. This indicates that countermeasures and network topology play an important role in suppressing viral spread. [ABSTRACT FROM AUTHOR]

In this paper, the effect of delay on the asymptotic behavior of a Lasota-Wazewska model with multiple time-varying delays is investigated. By employing the fluctuation lemma and some differential inequality techniques, delay-dependent criteria are obtained for the global attractivity of the considered model. An example is also given in the end of this paper to show the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonautonomous impulsive plankton model with mutual help of preys. Sufficient conditions ensuring permanence and global attractivity of the model are established by the relation between solutions of impulsive system and corresponding nonimpulsive system. Also, we propose the conditions for which the species of system are driven to extinction. Numerical simulations are given to verify the main results. [ABSTRACT FROM AUTHOR]

In this paper, a nonautonomous delayed Gilpin-Ayala competition system without instantaneous negative feedbacks (i.e., pure-delay-type system) is investigated. By the techniques of comparison arguments and constructing Lyapunov functionals something different to usual case, several results to guarantee the permanence of the system are derived by means of Ahmad and Lazer's definitions of lower and upper averages of a function. Moreover, the sufficient conditions for the global attractivity of the positive solution are also obtained, in which it is not necessarily to require the exponent of nonlinear intraspecific interference to exceed that of nonlinear interspecific interactions. These results are more general and practical, and possess a wide range of applications. Obviously, they are basically an extension of many existing conclusions for nonlinear competitive systems. [ABSTRACT FROM AUTHOR]

We study the existence, uniqueness, global attractivity and bifurcation of time-periodic patterns for a seasonal phytoplankton model with self-shading effect. By the comparison principle, we obtain the globally asymptotical stability of the zero solution when the principal eigenvalue λ 1 is less than zero. When λ 1 > 0 , by transforming the model into a new system, we successfully prove the conjecture in previous studies on the uniqueness and attractivity of the positive periodic solution. The positive periodic pattern bifurcating from the zero solution is a very interesting phenomenon. Here we apply the Crandall and Rabinowiz's theory to prove rigorously the existence of a bifurcation point. By way of asymptotic analysis, we derive an asymptotic formula for the positive periodic pattern. Based on the solution formula, we find the linear stability of this positive pattern. Finally, we provide a numerical scheme for the calculations of the principal eigenvalue and the simulations of the solution. The simulations corroborate our theoretical analysis. [ABSTRACT FROM AUTHOR]

Abstract. We analyse the stability and attractivity of a class of n-dimensional Nicholson systems with constant coefficients and multiple time-varying delays. Delayindependent sufficient conditions on the coefficients are given, for the existence and absolute global exponential stability of a unique positive equilibrium N*, generalizing and improving known results for autonomous systems. We further establish delaydependent criteria for N* to be a global attractor of all positive solutions. In the latter case, upper bounds on the size of the delays which do not require an a priori explicit knowledge of the equilibrium N* are also derived. [ABSTRACT FROM AUTHOR]

In this paper, a class of fuzzy cellular neural networks with multi-proportional delays is investigated. By applying contraction mapping fixed point theorem and differential inequality techniques, some sufficient conditions are established for the existence and global attractivity of a unique almost periodic solution for the model, which improve and supplement existing ones. Moreover, a numerical example is given to illustrate the feasibility and application of the obtained results. [ABSTRACT FROM AUTHOR]

A nonautonomous system of Volterra integrodifferential equations is studied in this paper. It is shown that if the coefficients are continuous, bounded above and below by positive constants and satisfy certain inequalities, then one of the components will be driven to extinction while the other one will stabilize at the certain positive solution of a nonlinear single species model. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the effect of delay on the asymptotic behavior of Nicholson's blowflies model with patch structure and multiple time-varying delays. By using the fluctuation lemma and some differential inequality technique, delay-dependent criteria are obtained for the global attractivity of the addressed system. Meanwhile, some numerical examples are given to illustrate the feasibility of the theoretical results. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper is concerned with the effect of delay on the asymptotic behavior of a differential neoclassical growth model with multiple time-varying delays. By using the fluctuation lemma and some differential inequality technique, delay-dependent criteria are obtained for the global attractivity of the addressed system. Meanwhile, some numerical examples are given to illustrate the feasibility of the theoretical results. [ABSTRACT FROM AUTHOR]

By using some analytical techniques, modified inequalities and Mawhin's continuation theorem of coincidence degree theory, some sufficient conditions for the existence of at least one positive almost periodic solution of a kind of fishing model with delay are obtained. Further, the global attractivity of the positive almost periodic solution of this model is also considered. Finally, three examples are given to illustrate the main results of this paper. [ABSTRACT FROM AUTHOR]

Malaria is an infectious disease caused by Plasmodium parasites and is transmitted among humans by female Anopheles mosquitoes. Climate factors have significant impact on both mosquito life cycle and parasite development. To consider the temperature sensitivity of the extrinsic incubation period (EIP) of malaria parasites, we formulate a delay differential equations model with a periodic time delay. We derive the basic reproduction ratio $$R_0$$ and establish a threshold type result on the global dynamics in terms of $$R_0$$ , that is, the unique disease-free periodic solution is globally asymptotically stable if $$R_0<1$$ ; and the model system admits a unique positive periodic solution which is globally asymptotically stable if $$R_0>1$$ . Numerically, we parameterize the model with data from Maputo Province, Mozambique, and simulate the long-term behavior of solutions. The simulation result is consistent with the obtained analytic result. In addition, we find that using the time-averaged EIP may underestimate the basic reproduction ratio. [ABSTRACT FROM AUTHOR]

A new SAIRS alcoholism model with birth and death on complex heterogeneous networks is proposed. The total population of our model is partitioned into four compartments: the susceptible individual, the light problem alcoholic, the heavy problem alcoholic and the recovered individual. The spread of alcoholism threshold R 0 is calculated by the next generation matrix method. When R 0 < 1 , the alcohol free equilibrium is globally asymptotically stable, then the alcoholics will disappear. When R 0 > 1 , the alcoholism equilibrium is global attractivity, then the number of alcoholics will remain stable and alcoholism will become endemic. Furthermore, the modified SAIRS alcoholism model on weighted contact network is introduced. Dynamical behavior of the modified model is also studied. Numerical simulations are also presented to verify and extend theoretical results. Our results show that it is very important to treat alcoholics to control the spread of the alcoholism. [ABSTRACT FROM AUTHOR]

In the present paper, a stochastic mutualism model subject to white noises is established. We first investigate the existence and uniqueness of globally positive solution of the stochastic model. Then we study its asymptotic behavior, such as stochastic permanence and extinction, and estimate the limit of the average in time of the sample paths of every component. We also show that the stochastic system is globally attractive under some appropriate conditions. Finally, numerical simulations are presented to justify the analytical results. [ABSTRACT FROM AUTHOR]

A mathematical model describing continuous microbial culture and harvest in a chemostat, incorporating a control strategy and defined by impulsive differential equations, is presented and investigated. Theoretical results indicate that the model has a microbe-extinction periodic solution, which is globally attractive if the threshold $R_{1}$ is less than unity, and the model is permanent if the threshold $R_{2}$ is greater than unity. Further, we consider the control strategy under time delay and periodical impulsive effect. Analysis shows that continuous microbial culture and harvest process can be implemented by adjusting time delay, impulsive period or input amount of flocculant. Finally, we give an example with numerical simulations to illustrate the control strategy. [ABSTRACT FROM AUTHOR]