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In this paper, we formulate a nonlocal and time-delayed reaction–diffusion vector-borne model with the saturated incidence with respect to infectious terms and derive the basic reproduction number R 0 , which serves as a threshold parameter that predicts whether the vector-borne disease will spread. Furthermore, the global attractivity of the endemic equilibrium is obtained by Lyapunov functionals if all the parameters are positive constants. Also, we can get the globally asymptotical stability for the corresponding time-delayed ODE system. In the case of an unbounded spatial habitat, we show the existence and non-existence of travelling wave solutions for the spatially homogeneous model. [ABSTRACT FROM AUTHOR]

In this paper, a new numerical scheme, which we call the impulsive linearly implicit Euler method, for the SIR epidemic model with pulse vaccination strategy is constructed based on the linearly implicit Euler method. The sufficient conditions for global attractivity of an infection-free periodic solution of the impulsive linearly implicit Euler method are obtained. We further show that the limit of the disease-free periodic solution of the impulsive linearly implicit Euler method is the disease-free periodic solution of the exact solution when the step size tends to 0. Finally, two numerical experiments are given to confirm the conclusions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a nonlocal reaction–diffusion model of Chikungunya disease with periodic time delays. We establish two threshold type results on the global dynamics for the growth of mosquitoes and the disease transmission, respectively. Further, we obtain the global attractivity of a positive steady state for a simplified nonlocal and time-delayed system with constant coefficients. We also conduct numerical simulations for the Chikungunya transmission in Ceará, Brazil to investigate the effects of spatial heterogeneity on the disease transmission. [ABSTRACT FROM AUTHOR]

We present a new mathematical model of the Coronavirus Disease of 2019 using a system of reaction-diffusion partial differential equations which defines multiple stages of individual infectivity and accounts for environmental modes of transmission. Inspired by biological results that demonstrate three distinct categories of infected individuals, this model accounts for exposed presymptomatic individuals becoming either asymptomatically infectious or symptomatically infectious. Additionally, we include a host immunity-loss term in accordance with the known weakening of resistance to the virus in recovered individuals over time, so that the model may be used for long-term disease analysis. We prove that if all human diffusion coefficients are the same, then the system is globally well-posed, and moreover show that in a one-dimensional setting the system remains well-posed even without the assumption on the coefficients. Furthermore, we derive the basic reproduction number of the system, and under the same condition on the diffusion coefficients we demonstrate global stability of the disease-free-equilibrium when this number is below one, as well as uniform persistence of the disease with a stable endemic equilibrium when it is above one. By requiring that only one, rather than all, of the infection-related compartments needs to initially be nonzero to establish uniform persistence, this result is mathematically stronger than those in the literature of partially diffusive models of infectious diseases; we accomplish this improvement by discovering new minimum principle type results for solutions to nondiffusive equations. Finally, we generalize this new minimum principle for future applications to other dynamical systems, especially those in epidemiology. [ABSTRACT FROM AUTHOR]

The impact of multiple time delays and the variation of populations between patches make population dynamics research more realistic and challenging. The main goal of this work is to study the effect of delay on the asymptotic behavior of tick population dynamics model with patch structure and multiple time-varying delays. We make use of the fluctuation lemma and inequality techniques to tackle the positivity, persistence and the global attractivity of the considered high-dimensional system. Finally, we give an example with three numerical simulations to verify that our theoretical results are completely new and complement some earlier existing ones. Our work is conducive to further understanding the more complex laws of tick population dynamics. [ABSTRACT FROM AUTHOR]

This study investigated an n-species cooperative Lotka-Volterra system with impulsive controls and time delays. In the case of constructing appropriate Lyapunov functionals and using inequality techniques, some conditions for the permanence and global attractiveness of the system are obtained. In addition, sufficient conditions for the existence of positive periodic solutions were established using the coincidence degree theory. Finally, numerical simulation results illustrate the effectiveness of our findings. [ABSTRACT FROM AUTHOR]

In this paper, we study an SIS epidemic reaction-diffusion model with mass action infection mechanism and subject to homogeneous Dirichlet boundary conditions. We define a basic reproduction number $ \mathcal{R}_0 $ for the model and prove that there exists a unique endemic equilibrium if $ \mathcal{R}_0> 1 $. We then establish the global attractivity of the disease-free equilibrium and the endemic equilibrium for a special case. Furthermore, we analyze the asymptotic profiles of the endemic equilibrium. We show that limited movement of the individuals cannot eradicate the disease because of a varying total population, while large population mobility can cause the disease extinction due to the hostile exterior environment. [ABSTRACT FROM AUTHOR]

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.

In this paper, we study a network-based SIR epidemic model with a saturated treatment function in which a parameter α is introduced to measure the extent of the effect of the infected being delayed for treatment. Our aim is to present a global analysis and to investigate how the parameter α affects the spreading of diseases. Our main results are as follows: (1) In the case of the threshold value R 0 < 1 , there exist two values of α : α c and α 0 , such that the disease-free equilibrium is globally asymptotically stable when α ≤ α c and multiple endemic equilibria exist when α ≥ α 0 . This means that the parameter α has an essential influence on the spreading of the disease. (2) In the case of the threshold value R 0 > 1 , if the model has only one endemic equilibrium, then the unique endemic equilibrium is globally attractive. In this case, it is also proved that if α ≤ α c , then the endemic equilibrium has only one, so is globally attractive. In addition, numerical simulation is performed to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider the delay dynamic equation Based on Lyapunov’s method, we study global attractivity of the trivial solution of ( ∗ ). Our new result generalizes some well-known results in the theory of difference and differential equations to dynamic equations of the form ( ∗ ). We also present some examples on nonstandard time scales to illustrate the importance of the new result. [ABSTRACT FROM AUTHOR]

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]

This paper is concerned with threshold dynamics of a degenerate diffusive HBV infection model with DNA-containing capsids in heterogeneous environment. We firstly address the existence of global solutions, uniform and ultimate boundedness of solutions, asymptotic smoothness of semiflows and existence of a connected global attractor for the diffusive model. Then, we identify the basic reproduction number R 0 and establish a threshold-type result for the disease eradication or uniform persistence when R 0 ≤ 1 or R 0 > 1 , respectively. Especially, when R 0 > 1 and the diffusion rate of capsids or the diffusion rate of virions is zero, we further show that the model admits a unique infection steady state which is globally attractive. Our results indicate that the pathogen can be eliminated by limiting the mobility of the capsids or virions. [ABSTRACT FROM AUTHOR]

This paper provides an expansion to a recent work [L. Duan, L. Huang, Proc. Amer. Math. Soc. 149 (11) (2021) 4789-4797], where the global asymptotic stability of positive steady state remains unsolved when $ R_0>1 $. We achieve this goal by studying the limit system and applying the theory of asymptotically autonomous semiflows. [ABSTRACT FROM AUTHOR]

In recent years, the suppression of disease population in pest control has gained popularity and remains a pressing issue in agricultural production. Additionally, the mathematical modeling method is often exploited to study the dynamics of different infectious diseases. A new pest control model with multiple delays and impulsive effects is developed and investigated. Firstly, the global attractivity of infection-free periodic solution of the system is proved by applying the theory of impulsive differential equation and comparative theorem. Secondly, the persistence of the infection-free periodic solution of the proposed model is also proved. Finally, the theoretical analysis results are validated by numerical simulations and some suggestions on how to better control the pests are presented.

In this work, we propose a hybrid impulsive prey-predator-mutualist model on nonuniform time domains. We have investigated the permanence/persistence results for the proposed model using the comparison theorems of impulsive differential equations and some dynamic inequality on the nonuniform time domains. In addition, we have established certain necessary requirements for the uniform asymptotic stability of the almost periodic solution and global attractivity of the proposed model. Furthermore, we provide several numerical examples on nonuniform time domains with computer simulation to demonstrate the viability of the results of the acquired analytical work.

In the present paper, we further study the asymptotical behavior of the following higher order nonlinear difference equation \begin{equation*} x(n+1)= ax(n)+ bf( x(n)) + cf(x(n-k)), \qquad n=0, 1, \dots \end{equation*} where $a, b $ and $c$ are constants with $00$ for $x>0$, and $k$ is a positive integer, which has been recently studied in: On global attractivity of a higher order difference equation and its applications [Electron. J. Qual. Theory Diff. Equ. 2022, No. 2, 1–14]. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation, and show the applications of these results to some population models.

In this paper, the qualitative behaviors of an important four-dimensional Lorenz system with wild pseudohyperbolic attractor that proposed in (Gonchenko et al 2021 Nonlinearity 34 2018–47) are considered. Here, we prove that the four-dimensional Lorenz system with varying parameters is global bounded according to Lyapunov's direct method. Furthermore, we provide a collection of global absorbing sets, where in addition we obtain the rate of the trajectories going from the exterior to the global absorbing set. In particular, we solve the critical case k → 0 + that cannot be resolved by using the previous methods. The fundamental qualitative behaviors are analyzed theoretically and numerically. We present bifurcation diagrams to further explore the complicated dynamical behaviors of this system. The period-doubling bifurcation phenomenon is found. To illustrate the efficiency of our method, we present numerical simulations to show the validity of our research results. Finally, we present some applications of our research results in this paper. [ABSTRACT FROM AUTHOR]

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.

Although the incidence of measles has been significantly reduced through vaccination, it remains an important public health problem. In this paper, a measles model with pulse vaccination is formulated to investigate the influential pulse vaccination on the period of time for the extinction of the disease. The threshold value of the formulated model, called the control reproduction number and denoted by R∗, is derived. It is found that the disease-free periodic solution of the model exists and is globally attractivity whenever R∗1, the positive solution of the model exists and is permanent which indicates the disease persists in the community. Theoretical conditions for disease eradication under various constraints are given. The effect of pulse vaccination is explored using data from Thailand. The results obtained can guide policymakers in deciding on the optimal scheduling in order to achieve the strategic plan of measles elimination by vaccination.

Past works on partially diffusive models of diseases typically rely on a strong assumption regarding the initial data of their infection-related compartments in order to demonstrate uniform persistence in the case that the basic reproduction number $ \mathcal{R}_0 $ is above 1. Such a model for avian influenza was proposed, and its uniform persistence was proven for the case $ \mathcal{R}_0 > 1 $ when all of the infected bird population, recovered bird population and virus concentration in water do not initially vanish. Similarly, a work regarding a model of the Ebola virus disease required that the infected human population does not initially vanish to show an analogous result. We introduce a modification on the standard method of proving uniform persistence, extending both of these results by weakening their respective assumptions to requiring that only one (rather than all) infection-related compartment is initially non-vanishing. That is, we show that, given $ \mathcal{R}_0 > 1 $, if either the infected bird population or the viral concentration are initially nonzero anywhere in the case of avian influenza, or if any of the infected human population, viral concentration or population of deceased individuals who are under care are initially nonzero anywhere in the case of the Ebola virus disease, then their respective models predict uniform persistence. The difficulty which we overcome here is the lack of diffusion, and hence the inability to apply the minimum principle, in the equations of the avian influenza virus concentration in water and of the population of the individuals deceased due to the Ebola virus disease who are still in the process of caring.

This paper is devoted to the study of propagation dynamics for a large class of nonautonomous cooperative reaction-diffusion systems in a time-periodic shifting environment. We first establish the spreading properties of solutions and the existence of forced time-periodic waves for such a system by appealing to the abstract theory developed for monotone semiflows with asymptotic translation invariance. Then we prove the uniqueness of the forced wave and its attractivity under appropriate conditions. Finally, we apply our analytical results to a reaction-diffusion-advection model of malaria transmission. [ABSTRACT FROM AUTHOR]

This paper is concerned with a degenerate and time delayed virus infection model with spatial heterogeneity and general incidence. The well‐posedness of the system, including global existence, uniqueness, and ultimately boundedness of the solutions, as well as the existence of a global attractor, is discussed. The basic reproduction number R0$\mathcal {R}_0$ is defined and a characterization of R0$\mathcal {R}_0$ is presented. Without the compactness of the solution semiflow, we establish the global dynamics of the system based on R0$\mathcal {R}_0$. In addition, when the system is spatially homogeneous, the unique infection steady state is globally asymptotically stable. Simulations are presented to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

To explore the impacts of pulse vaccination on the dynamical behaviors of virus, the current paper investigates the persistence and global attractivity of a diffusive viral infection model incorporated with impulsive CTL immune response while the stronger logistic growth mechanism. Based on the comparison principle and the proper upper and lower solutions, we first discuss the existence of the infection-free steady state, and define the crucial CTL-activated viral infection reproduction number $ R_0 $ R 0 . Then, the fundamental extinction and uniform persistence behaviors of virus are distinguished by various cases of the threshold parameter $ R_0 $ R 0 . Finally, the existence and global attractivity of the positive steady state are still established by employing Brouwer's fixed point theorem and auxiliary functional. Our results indicate that high vaccination rates will stimulate the CTL response more effectively, and thus eventually can force the virus to eradicate. [ABSTRACT FROM AUTHOR]

In this work, we set up a new discrete predator–prey competitive model with time-varying delays and feedback controls. By virtue of the difference inequality knowledge, a sufficient condition which guarantees the permanence of the established discrete predator–prey competitive model with time-varying delays and feedback controls is derived. Under some appropriate parameter conditions, we have proved that the periodic solution of the system without delay exists and globally attractive. To verify the correctness of the derived theoretical fruits, we give two examples and execute computer simulations. Our obtained results are novel and complement previous known results. [ABSTRACT FROM AUTHOR]

We are first concerned with the discrete Beverton-Holt equation$ \begin{equation*} x_{n+1} = \frac{\mu_n K_n x_n}{K_n+(\mu_n-1) x_n}, \ n\in \mathbb{Z} = \{0, 1, 2, \cdots\}, \end{equation*} $where $ \{\mu_n\}_{n = 0}^{\infty} $ and $ \{K_n\}_{n = 0}^{\infty} $ are two positive $ p $-periodic sequences. We find a sufficient condition that ensures if the origin is unstable, there exists a unique positive $ p $-periodic solution that globally attracts all positive solutions. This confirms the Cushing-Henson conjecture (a) under weaker conditions. A necessary and sufficient condition on the local stability of the origin is also provided. Then, based on this discrete equation, a single-species population model is proposed to describe the sterile insect technique for mosquito suppression. Some sufficient conditions on bistability are obtained. There is a stable extinction equilibrium and exactly two periodic orbits. One orbit is a repellor, and the other is an attractor. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce a non-autonomous environmental transmission model for most respiratory and enteric infectious diseases to study the impact of periodic environmental changes on related infectious diseases. The transmission and decay rates of pathogens in the environment are set as periodic functions to summarize the influence of environmental fluctuations on diseases. The solutions of the model are qualitatively analyzed, and the equilibrium points and the reference criterion, $ R_0 $, for judging the infectivity of infectious diseases are deduced. The global stability of the disease-free equilibrium and the uniform persistence of the disease are proved by using the persistence theory. Common infectious diseases such as COVID-19, influenza, dysentery, pertussis and tuberculosis are selected to fit periodic and non-periodic models. Fitting experiments show that the periodic environmental model can respond to epidemic fluctuations more accurately than the non-periodic model. The periodic environment model is reasonable and applicable for seasonal infectious diseases. The response effects of the periodic and non-periodic models are basically the same for perennial infectious diseases. The periodic model can inform epidemiological trends in relevant emerging infectious diseases. Taking COVID-19 as an example, the sensitivity analysis results show that the virus-related parameters in the periodic model have the most significant influence on the system. It reminds us that, even late in the pandemic, we must focus on the viral load on the environment.

In this paper, we formulate a mathematical model of malaria transmission dynamics with impulsive and periodic release of Wolbachia-infected male mosquitoes. Our objective is to evaluate the impact of periodic release of Wolbachia-infected male mosquitoes on malaria transmission. Subsequently, to find out how often and in what quantities Wolbachia-infected male mosquitoes need to be released to effectively reduce malaria transmission. By using the comparison principle, Floquet theory and some of the analytical methods, we obtain the basic reproduction number. Then, we demonstrated that the disease-free periodic solution is globally stable if the reproduction number is less than unity, and we prove the uniform persistence of the disease if the reproduction number is greater than unity. Finally, we perform numerical simulations to illustrate our theoretically pertinent results. [ABSTRACT FROM AUTHOR]

This paper studies a class of the non-autonomous competition and cooperation model of two enterprises involving discrete time delays and feedback controls. The paper proposes new criteria for analyzing the permanence, periodic solution, and global attractiveness of the model. The common mathematical techniques of the Lyapunov method, the continuation theorem, and the comparison principle are used in this paper. By means of the comparison principle and inequality techniques, the concept of permanence is investigated, which refers to the long-term survival of the enterprises within the competitive and cooperative framework. Meanwhile, using the continuation theorem, we establish conditions under which the system exhibits periodic behavior. Additionally, the global attractiveness of the system is derived by constructing multiple Lyapunov functionals. Finally, an example is presented to illustrate the applicability and validity of the proposed criteria in this paper. This example serves as a demonstration that showcases the main results derived from the analysis. [ABSTRACT FROM AUTHOR]

Past works on partially diffusive models of diseases typically rely on a strong assumption regarding the initial data of their infection-related compartments in order to demonstrate uniform persistence in the case that the basic reproduction number R 0 is above 1. Such a model for avian influenza was proposed, and its uniform persistence was proven for the case R 0 > 1 when all of the infected bird population, recovered bird population and virus concentration in water do not initially vanish. Similarly, a work regarding a model of the Ebola virus disease required that the infected human population does not initially vanish to show an analogous result. We introduce a modification on the standard method of proving uniform persistence, extending both of these results by weakening their respective assumptions to requiring that only one (rather than all) infection-related compartment is initially non-vanishing. That is, we show that, given R 0 > 1 , if either the infected bird population or the viral concentration are initially nonzero anywhere in the case of avian influenza, or if any of the infected human population, viral concentration or population of deceased individuals who are under care are initially nonzero anywhere in the case of the Ebola virus disease, then their respective models predict uniform persistence. The difficulty which we overcome here is the lack of diffusion, and hence the inability to apply the minimum principle, in the equations of the avian influenza virus concentration in water and of the population of the individuals deceased due to the Ebola virus disease who are still in the process of caring. [ABSTRACT FROM AUTHOR]

In this paper, the oscillation of a Hematopoiesis model in both cases delay and non-delay are discussed. The place β(t) and δ(t) are continuous positive w-periodic functions. In the non-delay case, we will exhibit that a nonlinear differential equation of hematopoiesis model has a global attractor H(t) for all different positive solutions. Also, in the delay case, the sufficient conditions for the oscillation of all positive solutions of it about H(t)are presented and we establish sufficient conditions for the global attractive of H(t). To illustrate the obtained results some examples are given. [ABSTRACT FROM AUTHOR]

The goal of this paper is to introduce a non-autonomous environmental transmission model for most respiratory and enteric infectious diseases to study the impact of periodic environmental changes on related infectious diseases. The transmission and decay rates of pathogens in the environment are set as periodic functions to summarize the influence of environmental fluctuations on diseases. The solutions of the model are qualitatively analyzed, and the equilibrium points and the reference criterion, R 0 , for judging the infectivity of infectious diseases are deduced. The global stability of the disease-free equilibrium and the uniform persistence of the disease are proved by using the persistence theory. Common infectious diseases such as COVID-19, influenza, dysentery, pertussis and tuberculosis are selected to fit periodic and non-periodic models. Fitting experiments show that the periodic environmental model can respond to epidemic fluctuations more accurately than the non-periodic model. The periodic environment model is reasonable and applicable for seasonal infectious diseases. The response effects of the periodic and non-periodic models are basically the same for perennial infectious diseases. The periodic model can inform epidemiological trends in relevant emerging infectious diseases. Taking COVID-19 as an example, the sensitivity analysis results show that the virus-related parameters in the periodic model have the most significant influence on the system. It reminds us that, even late in the pandemic, we must focus on the viral load on the environment. [ABSTRACT FROM AUTHOR]

In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms \[x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), \quad n=0, 1, \ldots,\] where \(a\), \(b\) and \(c\) are constants with \(0\lt a\lt 1\), \(0\leq b\lt 1\), \(0\leq c \lt 1\) and \(a+b+c=1\), \(g\in C[[0, \infty), [0, \infty)]\) is decreasing, and \(k\) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given.

In this paper, well-posedness and asymptotic behaviors of a stochastic two-predator one-prey system with S-type distributed time delays are studied by using stochastic analytical techniques. First, the existence and uniqueness of global positive solution with positive initial condition is proved. Second, sufficient conditions for persistence in mean and extinction of each species are obtained. Then, sufficient conditions for global attractivity are established. Finally, some numerical simulations are provided to support the analytical results.

In this paper, a Lotka–Volterra (L-V) competitor–competitor–mutualist system with time-varying delays is studied. Some dynamical behaviors of the considered system are investigated. Firstly, we obtain the boundedness, permanence and periodic solution of the system using the comparison principle of differential equations and inequality estimation method. Then, the global attractiveness of the system is analyzed by multiple Lyapunov functionals. Meanwhile, the existence and global attractivity of positive periodic solutions is derived. In the third section, in order to validate the practicability and feasibility of the obtained theoretical results, we conducted numerical simulations using MATLAB function ddesd. Finally, the fourth section is where conclusions are drawn. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonlocal within-host viral infection model with general incidence in a spatially heterogeneous environment. We first establish the well-posedness, boundedness and asymptotic compactness of this model. The basic reproduction number R 0 is defined by using the perturbation technique. Then we study global stability of the infection-free steady state and show that the system is uniformly persistent when R 0 > 1 . In addition, for a special incidence function, the existence, uniqueness and global stability of the infection steady state of this model are established. Finally, we present some examples and numerical simulations to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

We studied a class of generalized n-species non-autonomous cooperative Lotka–Volterra (L-V) systems with time delays. We obtained new criteria on the dynamic properties of the systems. First, we obtained the boundedness and permanence of the system using the inequality analysis technique and comparison method. Then, the existence of positive periodic solutions was investigated using the coincidence degree theory. The global attractivity of the system was obtained by constructing suitable Lyapunov functionals and utilizing Barbalat's lemma. The existence and global attractivity of the periodic solutions were also obtained. Finally, we conducted two numerical simulations to validate the feasibility and practicability of our results. [ABSTRACT FROM AUTHOR]

In this paper, we analyze the global dynamics of a neoclassical growth system incorporating patch structure and multiple pairs of time-varying delays. First, we derive the global existence, positiveness and boundedness of solutions for the addressed system. Then, by employing some novel differential inequality analyses and the fluctuation lemma, both delay-independent and delay-dependent criteria are established to ensure that all solutions are convergent to a unique positive equilibrium point vector, which does not possess the same components. Our results supplement and improve some existing results. Ultimately, some numerical examples are afforded to prove the effectiveness and feasibility of the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper is intended to investigate a class of Nicholson's blowflies system with patch structure and multiple pairs of distinct time-varying delays, we are interested in finding the influence of the distinct time-varying delays in the same reproductive function on its asymptotic behavior. By using the theory of functional differential equations, the fluctuation lemma, and the technique of differential inequalities, some new delay-dependent criteria on the global attractivity of the positive equilibrium point are established. In addition, the effectiveness and feasibility of the theoretical achievements are illustrated by some numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the dynamics of a discrete commensal symbiosis model with Holling type functional response and density dependent birth rate. Sufficient conditions are obtained for the permanence, partial extinction, and global attractivity of the system. Our analysis presents that the density dependent birth rate plays a crucial role on the dynamic behaviors of the system. Finally, numerical simulations and graphical illustrations are given to indicate the analytical results. [ABSTRACT FROM AUTHOR]

In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms x(n + 1) = ax(n) + bx(n)g(x(n)) + cx(n - k)g(x(n - k)), n = 0, 1, . . ., where a, b and c are constants with 0 < a < 1, 0 ≤ b < 1, 0 ≤ c < 1 and a + b + c = 1, g ∈ C[[0,∞), [0,∞)] is decreasing, and k is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given. [ABSTRACT FROM AUTHOR]

This paper formulates and explores a nonautonomous impulsive stochastic predator-prey system with Beddington-DeAngelis (BD) functional response, where only the prey has a disease, which incorporates modified saturated incidence. The sufficient criteria of extinction and non-persistence in the mean of the target model are established, revealing that different intensities of stochastic perturbations contribute to dynamics of the system mentioned above. Stochastically ultimate boundedness is examined, and we further establish sufficient conditions for global attractivity. Our analytical findings are verified through numerical simulations. [ABSTRACT FROM AUTHOR]

This paper studies global attractivity for uncertain differential systems, which are effective tools to solve the problems with uncertainty. And They have been applied in many areas. This article presents several global attractivity concepts. Based on the knowledge of uncertainty theory, some sufficient conditions of global attractivity for linear uncertain differential systems are given. In particular, the attractivity on the solutions and α-path of uncertain differential systems is studied. Last, as an application of attractivity, an interest rate model with uncertainty is shown.

Consider the following higher order difference equation \begin{align*} x(n+1)= ax(n)+ bf( x(n)) + cf(x(n-k)), \qquad n=0, 1, \dots \end{align*} where $a, b $ and $c$ are constants with $00$ for $x>0$, and $k$ is a positive integer. Our aim in this paper is to study the global attractivity of positive solutions of this equation and its applications to some population models.

This paper deals with a class of delayed Nicholson-type systems involving patch structure. First, we prove that the solution of the initial value problem with respect to the addressed system exists globally and is bounded. Second, we employ the contraction fixed point theorem and analytical techniques to establish the existence of a positive asymptotically almost periodic solution and its global attractivity. Finally, an example is arranged to illustrate the effectiveness and feasibility of the obtained results. [ABSTRACT FROM AUTHOR]

A discrete modified Leslie–Gower prey-predator model considering the effect of fear on prey species is proposed and studied in this paper. First, we discuss the existence of equilibria and the local stability of the model. Second, we use the iterative method and comparison principle to obtain the set of conditions which ensures the global attractivity of positive equilibrium point. The results show that prey and predator can coexist stably when the intrinsic growth rates of both prey and predator are maintained within a certain range. Then, we study the global attractivity of the boundary equilibrium point. Our results suggest that when the intrinsic rate of prey is small enough or the fear factor is large enough, the prey will tend to go extinct, while the predator can survive stably due to the availability of other food sources. Subsequently, we discuss flip bifurcation, transcritical bifurcation at the equilibria of the system, by using the center manifold theorem and bifurcation theory. We find that system changes from chaotic state to four-period orbit, two-period orbit, stable state, and finally prey species will be driven to extinction, while predator species survive in a stable state for enough large birth rate of prey species with the increasing of fear effect. Finally, we verify the feasibility of the main results by numerical simulations, and discuss the influence of the fear effect. The results show that the fear effect within a certain range can enhance the stability of the system. [ABSTRACT FROM AUTHOR]

This article demonstrates the stability property of two boundary equilibria of a symbiotic model of commensalism and parasitism with harvesting in the commensal population. The model was proposed by Nurmaini Puspitasari, Wuryansari Muharini Kusumawinahyu, Trisilowati (2021). We first give two numeric examples to show that the corresponding results of the mentioned paper may be incorrect. Then, by analysis of the characteristic roots of the characteristic equations, we obtain sufficient conditions that ensure the locally asymptotic stability of the equilibria. After that, by applying the standard comparison theorem, some novel results on the global attractivity of these two equilibria are obtained respectively. Our results complement and supplement some known results. [ABSTRACT FROM AUTHOR]

This paper considers the existence, uniqueness and the asymptotic behavior of a classic heat equation coupled with a fractional ordinary differential system. By the Lumer-Phillips theorem, we obtain the existence and uniqueness of the solution. Furthermore, we use a new Lyapunov functional to show the global attractivity for the considered system, then the tractable global attractive conditions are obtained. Finally, numerical examples are provided to illustrate the applications of our results. [ABSTRACT FROM AUTHOR]

In this paper, we present a malaria transmission model with climatic factors to study the climatic transmission dynamics of malaria with the effect of the Serratia AS1 bacterium. It has been shown in controlled laboratory experiments that Serratia AS1 bacterium can rapidly disseminate throughout mosquito populations and efficiently inhibit development of malaria parasites in mosquitoes. We derive the basic reproduction ratio R 0 . We introduce the basic reproduction ratio R 0 which can be used as a threshold parameter in the global dynamical model. When R 0 ≤ 1 , the disease-free periodic solution of this model is globally asymptotically stable; and when R 0 > 1 , the disease remains persistent. We analyze the sensitivity of R 0 in terms of the vertical and horizontal transmission rates of Serratia AS1, and find that R 0 can be small when the vertical transmission rate of Serratia AS1 tends to 1 and the influence of the horizontal transmission rate performs inversely proportional to R 0 when the vertical transmission rate is less than 1. Based on the data of Luanda, we perform numerical simulations to illustrate our theoretical results, which indicate that treatment of Serratia AS1 provides us an effective measure in controlling malaria. [ABSTRACT FROM AUTHOR]