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This paper explores general coupled dynamic equations on time scales with multiple delays and investigates the existence of periodic solutions using the coincidence degree theory approach. The model studied includes the mathematical models of Nicholson, Mackey–Glass, and Lasota–Wazewska as special cases. Furthermore, we demonstrate that the solutions are not only asymptotically stable but also exponentially synchronized. The presented results significantly extend and complement existing findings in the field. The paper concludes with illustrative examples that highlight the practical implications of our analytical discoveries. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new mathematical epidemic model with the effect of vaccination. We formulate a Susceptible-High risk-Infective-Recovered-Vaccinated (SHIRV) model in which the susceptible individuals with a higher probability of being infected (H) are selected as a separate class. We study the dynamical behavior of this model and define the basic reproductive number, R0. It is proved that the disease-free equilibrium is asymptotically stable if R0 < 1, and it is unstable if R0 1. Also, we investigate the existence and stability of the endemic equilibrium point analytically. For the system of differential equations of the SHIRV model, we give an approximating solution by using the Legendre-Ritz-Galerkin method. Finally, we study the influence of vaccination on measles and smallpox, two cases of the epidemic, using the proposed method in this paper. Numerical results showed that choosing high-risk people for vaccination can prevent them from getting infected and reduce mortality in the community. [ABSTRACT FROM AUTHOR]

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of octonion‐valued stochastic shunting inhibitory cellular neural networks with delays. First, we give an estimate of the distance between two different moments of finite‐dimensional distributions of a stochastic process. Then, based on this and by using fixed point theorems and inequality techniques, we establish the existence and global exponential stability of Besicovitch almost periodic (B$$ \mathcal{B} $$‐almost periodic for short) solutions in finite‐dimensional distributions for this kind of networks. Our results are new even if the networks we consider in the paper are real‐valued ones. At the same time, the method proposed in this paper can be applied to study the existence of Besicovitch almost periodic solutions in finite‐dimensional distributions for other types of stochastic neural networks. Finally, an example is given to illustrate the effectiveness of our results. [ABSTRACT FROM AUTHOR]

In this paper, we for the first time prove local solvability and stability of the inverse Sturm‐Liouville problem with complex‐valued singular potential and with polynomials of the spectral parameter in the boundary conditions. The proof method is constructive. It is based on the reduction of the inverse problem to a linear equation in the Banach space of bounded infinite sequences. We prove that, under a small perturbation of the spectral data, the main equation of the inverse problem remains uniquely solvable. Furthermore, we derive new reconstruction formulas for obtaining the problem coefficients from the solution of the main equation and get stability estimates for the recovered coefficients. [ABSTRACT FROM AUTHOR]

Understanding and accounting for the slow–fast effect are crucial for accurately modeling and predicting the dynamics of predator–prey models, emphasizing the importance of considering the relative speeds of interacting populations in ecological research. This paper examines a predator–prey interaction to study its complex dynamics due to its slow–fast effect on predator populations. The occurrence and stability of equilibria are analyzed. The stability of positive fixed point is dependent on the slow–fast effect parameter ϵ$$ \epsilon $$, which must fall within a specific range when the generation gap is larger. The positive fixed point becomes unstable for bigger values of ϵ$$ \epsilon $$ because the growth of predators is faster, resulting in the extinction of all prey. Smaller values of ϵ$$ \epsilon $$ cause the positive fixed point to become unstable since the prey grows more quickly while the predator grows more slowly, ultimately causing the extinction of the predator. Moreover, it is shown that Leslie–Gower model experiences Neimark–Sacker and period‐doubling bifurcations at positive equilibrium point. In order to control bifurcation, hybrid control and feedback control methods are employed. Finally, analytical results are confirmed by numerical examples. [ABSTRACT FROM AUTHOR]

Atherosclerosis is a chronic inflammatory disease that poses a serious threat to human health. It starts with the buildup of plaque in the artery wall, which results from the accumulation of pro‐inflammatory factors and other substances. In this paper, we propose a mathematical model of early atherosclerosis with a free boundary and time delay. The time delay represents the time required for macrophages to transit to foam cells through cholesterol accumulation. We obtain an explicit solution and analyze the stability of the model and the effect of the time delay on plaque size. We show that in the form of perturbation cos(nθ)$$ \cos \left( n\theta \right) $$ (where n$$ n $$ represents the mode of angle), when n=0$$ n&#x0003D;0 $$ or 1, the steady‐state solution (M∗,p∗,r∗)$$ \left({M}_{\ast },{p}_{\ast },{r}_{\ast}\right) $$ is linearly stable; when n≥2$$ n\ge 2 $$, there exists a critical parameter L∗$$ {L}_{\ast } $$ such that the steady‐state solution is linearly stable for LL∗$$ L>{L}_{\ast } $$. Moreover, we find that smaller plaque are associated with the presence of time delay. [ABSTRACT FROM AUTHOR]

Recently, many scholars have discovered that fractional calculus possess infinite memory and can better reflect the memory characteristics of neurons. Therefore, this paper studies the Hopf bifurcation of a fractional‐order network with short‐cut connections structure and self‐delay feedback. Firstly, we use the Laplace transform to obtain the characteristic equation of the model, which is the transcendental equation containing four times transcendental item. Secondly, by selecting the communication delay as the bifurcation parameter and the other delay as the constant in its stability interval, the conditions for the occurrence of Hopf bifurcation are established; the bifurcation diagrams are provided to ensure that the derived bifurcation findings are accurate. Thirdly, in the case of identical neurons, the crossing curves method is exploited to the fractional‐order functional function equation to extract the Hopf bifurcation curve. Finally, two numerical examples are employed to confirm the efficiency of the developed theoretical outcomes. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of Mittag‐Leffler stability and synchronization for discrete‐time fractional‐order delayed quaternion valued neural networks (DTFODQVNN). Firstly, a criterion is achieved to ensure the existence and uniqueness of the equilibrium point (EP) of DTFODQVNN through applying Brouwer's fixed‐point theory. Secondly, based on a Lyapunov function and a new discrete fractional inequality, the stability condition is established by employing a linear matrix inequality approach. In addition, by constructing a proper controller, drive–response synchronization is investigated by means of deploying Lyapunov direct method (LDM). Finally, two examples with simulations test the correctness of the acquired results. Moreover, image encryption is considered as an application based on the chaotic DTFODQVNN. [ABSTRACT FROM AUTHOR]

The aim of this paper is to examine the stability of a class of nonlinear stochastic operators on a finite‐dimensional simplex. We identify a number of properties of the set of the class of stochastic operators, including the stability of each operator. We then use these operators to define non‐homogeneous Markov measures that rely on initial data and carry out an investigation into the absolute continuity and singularity of these measures. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two‐species generalist predator–prey model in closed advective environments. Compared with the model in open advective environments to the generalist predator–prey model, the dynamics of our system are different. It is easy to verify that the generalist predator can always invade. Meanwhile, we provide the dynamical behaviors of the model in terms of the intrinsic growth rates, the carrying capacity of both species, the predation rate, and the trophic conversion efficiency. More precisely, we establish the local/global stability of the semi‐trivial steady states and a unique positive equilibrium for the model. [ABSTRACT FROM AUTHOR]

In this paper, we study the asymptotic expansion for the homogeneous model of wind‐driven oceanic circulation with the nonslip boundary condition when both of the beta‐plane parameter and the Reynolds number go to infinity. By asymptotic analysis under certain constraints on wind tensor, we derive a formal expansion including the geophysical boundary layer in the large beta‐plane parameter and high Reynolds number limit, from which one sees that the external force, wind tensor, has an important effect on the behavior of the boundary layers. Moreover, when the external force and initial data satisfy certain constraints, to exclude the appearance of strong boundary layers, we construct approximate solutions with steady boundary layers to the homogeneous model of wind‐driven oceanic circulation in the large beta‐plane parameter and high Reynolds number limit. By using an energy method, we obtain the L∞$$ {L}&#x0005E;{\infty } $$‐stability of small perturbation around this approximate solutions, which verifies the validity of the asymptotic expansion of weak boundary layers in the large beta‐plane parameter and high Reynolds number limit. [ABSTRACT FROM AUTHOR]

This paper explores the dynamics analysis of a human immunodeficiency virus (HIV) model with saturated cytotoxic T lymphocyte (CTL) immune response and Beddington–DeAngelis infection rate. There are two time delays in the model to describe the time needed for infection of cell and CTL immune response generation, respectively. We obtain two thresholds and three possible equilibria from the model. By analyzing the corresponding characteristic equations, we study the stabilities of equilibrium and the effect of delays on CTL immune response. The results indicate that when immune delay is present, the steady state of equilibrium is disrupted and leads to a Hopf bifurcation. Finally, we use sensitivity analyses to show the effect of parameters on thresholds and numerical simulations to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a predator–prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle‐node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes an uncoupled and linearized compact finite difference scheme for the generalized dissipative symmetric regularized long‐wave (GDSRLW) equations. The unique solvability and some a priori estimates of the proposed difference scheme are rigorously proved based on the mathematical induction method. To obtain the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm estimation of numerical solutions, the discrete energy method is used to prove the convergence and stability of the difference scheme. The proposed difference scheme preserves the original energy dissipation property, and the convergence of the scheme is proved to be fourth‐order in space and second‐order in time in the ‖·‖∞$$ {\left\Vert \cdotp \right\Vert}_{\infty } $$‐norm for both u$$ u $$ and ρ$$ \rho $$. Some numerical experiments are given to verify the theoretical analysis and the reliability of the proposed difference scheme. [ABSTRACT FROM AUTHOR]

Exposure of cells to non‐optimal growth conditions or to any environment that reduces cell viability can be considered as a stress. In this paper, we are going to highlight the main factors that determine the danger of stress to a cell considered as a biochemical system. To this end, we introduce a new mathematical concept of biosystem stability, where we take into account a signal transduction by deep gene networks. Using this concept and known results on approximations by deep networks, we find asymptotic estimates of the size and the depth of gene regulation networks that define the stress response. We propose a new algorithm to find the gene network approximating a prescribed output. It allows us, with the help of Kolmogorov ϵ$$ \epsilon $$‐entropy and the deep neural network theory, to estimate the number of genes involved in regulation of responses on a stress (for example, a heat shock). We show that the main factors that increase the sensitivity of the systems with respect to a stress are the number of biochemical network parameters affected by the stress and sensitivities of kinetic rates with respect to these parameters. [ABSTRACT FROM AUTHOR]

This paper is devoted to the Schnakenberg model with crucial reversible reactions, under Neumann boundary conditions. Existence and uniqueness of the strong solution are obtained on basis of semigroup theory. It is found that the proposed system admits four possible positive constant steady states, and explicit conditions of the stability, Turing instability, steady‐state bifurcation, and Hopf bifurcation are determined. Besides, numerical simulations are given to show the theoretical results and depict the spatiotemporal patterns. [ABSTRACT FROM AUTHOR]

The present paper deals with mathematical modeling of a fishery resource system in an aquatic atmosphere consisting of two zones: a free fishing zone and a reserved zone, where fishing is strictly prohibited. The dynamics of the system is studied in the presence of bird predator. In this paper, the quadratic harvesting of the fish species in unreserved zone has been considered whereas bird predator species is subjected to linear harvesting. All the possible biological and bionomic equilibria of the system are studied extensively for their existence, local as well as global stability. We have found various ranges of harvesting parameter for maintaining the Sustainability in the proposed ecosystem. Optimal harvesting is discussed using Pontryagin's maximum principle. Numerical simulations are done to support the theoretical results obtained. [ABSTRACT FROM AUTHOR]

In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction 1≤n≤4$$ 1\le n\le 4 $$. The method we use is based on a finite‐dimensional approach by combining the Faedo‐Galerkin method with suitable energy estimates and multiplier techniques. [ABSTRACT FROM AUTHOR]

This paper studies the numerical solution of the stochastic volatility model with jumps under European options. This model can be transformed into a partial integro‐differential equation (PIDE) with spatial mixed derivative terms. Due to the nonsmoothness of the initial function, the variable step‐size extrapolated Crank–Nicolson (CN) method, which explicitly discretizes the jump term and implicitly the rest, is proposed to solve this model. The finite difference method is used to discretize the spatial differential operator, the composite trapezoidal rule to calculate the jump integral, and then a linear system with a nine‐diagonal coefficient matrix is obtained, which is easy to solve. The stability of the variable step‐size extrapolated CN method is then proved. Based on realistic regularity assumptions on the data, the consistency error and global error bounds of the variable step‐size extrapolated CN method are derived in L2$$ {L}_2 $$ norm. Compared with the variable step‐size IMEX BDF2 method and the variable step‐size IMEX MP method, the numerical results show the efficiency of the proposed variable step‐size extrapolated CN method. [ABSTRACT FROM AUTHOR]

During the recent pandemic caused by COVID‐19, media awareness played a crucial role in educating people about social distancing, wearing masks, quarantine, vaccination, and medication. Media awareness brought individual behavioral changes among the people, which in turn helped reduce the infection rate. Motivated by this, we have formulated a mathematical model introducing a media compartment to mitigate measles disease transmission. In this paper, the SEIR model is used to study measles disease in three cases: one with a delay in vaccination, the second with regular vaccination, and the third with the impact of media awareness on the spreading of measles disease. Further, the dynamical behavior of the models is studied in terms of positivity, boundedness, equilibrium, and basic reproduction number (BRN). The sensitivity analysis of the models is conducted, which verifies the importance of the BRN (R0$$ {R}_0 $$) to be less than one for disease eradication. The numerical study confirms the impact of media awareness on exposed and infected populations. [ABSTRACT FROM AUTHOR]

In the paper, we consider the penalty finite element methods (FEMs) for the stationary Smagorinsky model. Firstly, a one‐grid penalty FEM is proposed and analyzed. Since this method is nonlinear, a novel linearized iteration scheme is derived for solving it. We also derived the stability and convergence of numerical solutions for this iteration scheme. Furthermore, a two‐grid penalty FEM is developed for Smagorinsky model. Under ε<

In this paper, the global asymptotic stability and projective lag synchronization of second‐order competitive neural networks with mixed time‐varying delays and uncertainties are studied without converting the original system into the usual first‐order system. Firstly, according to the Lyapunov functional method, inequality technique and the designed adaptive control strategy, the algebraic criteria of stability, and projective lag synchronization are derived by adjusting the control gain parameters in the controller. The obtained sufficient conditions are simple and easy to verify. Different from the traditional feedback controller, the adaptive controller can adjust its characteristics according to the system model, and external disturbance makes the system have better control performance. Besides, unlike the existing ones, this stability and synchronization problem is directly analyzed by constructing some new Lyapunov functionals with state variables and state variable derivatives. Finally, the effectiveness and practicability of the results are verified by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider switched linear systems that have jumps in the state (determined by a reset rule) at each switching instant. Those systems are called switched systems with state reset. We show that it is possible to analyze the state trajectories of a switched system with state reset by considering suitable switched systems without state resets, that is, with continuous state trajectories. We establish sufficient conditions for the stability of switched systems with state reset using the associated systems without reset. Moreover, introducing some restrictions, we also derive simple algebraic stability conditions. The obtained results are used in numerical examples to show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In the present paper, we first introduce the concept of double measure r$$ r $$‐mean piecewise pseudo almost periodic for stochastic processes for r≥2$$ r\ge 2 $$. Next, we make extensive use of the exponential dichotomy techniques and a fixed point strategy with stochastic analysis theory to obtain the existence of doubly measure r$$ r $$‐mean piecewise pseudo almost periodic mild solutions for a class of impulsive non‐autonomous partial stochastic evolution equations in Hilbert spaces. In addition, we study the exponential stability of r$$ r $$‐mean piecewise pseudo almost periodic mild solutions. Finally, we give an example to confirm the reliability and feasibility of our findings. [ABSTRACT FROM AUTHOR]

Cancer caused by malignant tumor is a very common and serious disease which can lead to people towards death. However, sometimes, cancer may be controlled if the cancer cells can be infected and split by particular virus with weak pathogenicity. To give a better understanding on tumor dynamics, in this paper, we deal with a reaction–diffusion tumor model, which comes from the process of tumor therapy with oncolytic virus. Our main objective is to analyze the biological compatibility of uninfected and infected tumor cells. In addition, we are concerned with the long‐time behavior, stability, and a priori estimates for the model. The compatibility conditions show that under certain range of parameters control, the infected tumor cells can coexist with normal tumor cells or be eliminated by oncolytic virus as time goes on; in the elimination case, it implies that the organism's complete recovery is highly possible as a result of the attack of oncolytic virus. An indirect idea in calculating fixed point index is proposed; correspondingly, a new condition on coexistence is presented. [ABSTRACT FROM AUTHOR]

This paper is concerned with the nonlinear stability of forced traveling waves for a Lotka–Volterra cooperative model under climate change. Firstly, by applying the L2$$ {L}^2 $$‐weighted energy estimate method, the comparison principle, and the squeezing technique, we investigate that all forced traveling waves with the speed c>c˜$$ c>\tilde{c} $$ are exponentially stable in the form of e−νt$$ {e}^{-\nu t} $$ for some ν>0$$ \nu >0 $$. Secondly, in order to improve the former results, we take another weight function and construct the related different weighted Sobolev space. Instead of the L2$$ {L}^2 $$‐weighted energy estimate, we first establish a L1$$ {L}^1 $$‐weighted energy estimate in the weighted Sobolev space. Then, by using this crucial L1$$ {L}^1 $$‐estimate, we further obtain the desired L2$$ {L}^2 $$‐energy estimate. Finally, we obtain that all forced traveling waves with the speed c>c¯$$ c>\overline{c} $$ are exponentially stable in the form of e−μ3t$$ {e}^{-\frac{\mu }{3}t} $$ for some μ>0$$ \mu >0 $$, where c¯

The generalized time‐fractional Fisher's equation is a substantial model for illustrating the system's dynamics. Studying effective numerical methods for this equation has considerable scientific importance and application value. In that direction, this paper presents designing and analyzing a high‐order numerical scheme for the generalized time‐fractional Fisher's equation. The time‐fractional derivative is taken in the Caputo sense and approximated using Euler backward discretization. The quasilinearization technique is used to linearize the problem, and then a compact finite difference scheme is considered for discretizing the equation in space direction. Our numerical method is convergent of Ok2−α+h4$$ O\left({k}^{2-\alpha }+{h}^4\right) $$, where h$$ h $$ and k$$ k $$ are step sizes in spatial and temporal directions, respectively. Three problems are tested numerically by implementing the proposed technique, and the acquired results reveal that the proposed method is suitable for solving this problem. [ABSTRACT FROM AUTHOR]

This paper presents an epidemic model with varying population, incorporating a new vaccination strategy and time delay. It investigates the impact of vaccination with respect to vaccine efficacy and the time required to see the effects, followed by determining how to control the spread of the disease according to the basic reproduction ratio of the disease. Some numerical simulations are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of a Schrödinger system with one boundary damping, which consists of two constant coefficients Schrödinger equations coupled through zero‐order terms. First, we show that when ϱ=1,$$ \varrho =1, $$ the one‐dimensional Schrödinger system is not exponentially stable by the asymptotic expansions of eigenvalues. Then, by the frequency domain approach and the multiplier method, we show that the energy decay rate of the multidimensional Schrödinger system is t−1$$ {t}^{-1} $$ for sufficiently smooth initial data when ϱ=1,$$ \varrho =1, $$|α|$$ \mid \alpha \mid $$ is sufficiently small, and the boundary of domain satisfies suitable geometric assumption. Next, by solving the characteristic equation of unbounded operator, we show that the strong stability of the one‐dimensional Schrödinger system is completely determined by ϱ$$ \varrho $$ and α$$ \alpha $$ and give the necessary and sufficient condition that ϱ$$ \varrho $$ and α$$ \alpha $$ satisfy. Finally, by solving the resolvent equation of unbounded operator and using the frequency domain approach, we show that when ϱ≠1$$ \varrho \ne 1 $$ and |α|$$ \mid \alpha \mid $$ is small enough, the energy of the one‐dimensional Schrödinger system decays polynomially and the decay rate depends on the arithmetic property of ϱ.$$ \varrho. $$ [ABSTRACT FROM AUTHOR]

This paper is concerned with the propagation dynamics of a discrete diffusive equation with non‐local delay. First of all, by exploring the asymptotic behavior of the solution of the upper system corresponding to the perturbation equation, we obtain the global stability of semi‐wavefronts. It is noteworthy that the kernel function β(k)$$ \beta (k) $$ can be asymmetric (i.e., β(k)≠β(−k)$$ \beta (k)\ne \beta \left(-k\right) $$). Secondly, we estimate the level set of the equation on the basis of the above stability. In particular, the results show that the solution of the compact supported initial value problem is not persistent in special case. [ABSTRACT FROM AUTHOR]

We only focus on the shock waves to Burgers system by considering the porous medium diffusion and singularity in energy estimates. In this paper, the conditions u+=0$$ {u}_{&#x0002B;}&#x0003D;0 $$ and D≥μ$$ D\ge \mu $$ are considered. Moreover, we establish the existence of shock waves through the phase plane. The weighted function is employed to handle the singularity in the weighted energy estimates under small perturbations and arbitrary wave amplitudes. Finally, the weighted energy estimates are then used to prove the stability of shock waves. [ABSTRACT FROM AUTHOR]

The Boussinesq equations with partial or fractional dissipation not only naturally generalize the classical Boussinesq equations but also are physically relevant and mathematically important. Unfortunately, it is not often well‐understood for many ranges of fractional powers. This paper focuses on a system of the 3D Boussinesq equations with fractional horizontal (−Δh)αu$$ {\left(-{\Delta}_h\right)}^{\alpha }u $$ and (−Δh)βθ$$ {\left(-{\Delta}_h\right)}^{\beta}\theta $$ dissipation and proves that if the initial data (u0,θ0$$ {u}_0,{\theta}_0 $$) in the Sobolev space H3(ℝ3)$$ {H}^3\left({\mathrm{\mathbb{R}}}^3\right) $$ are close enough to the hydrostatic balance state, respectively, the equations with α,β∈12,1$$ \alpha, \beta \in \left(\frac{1}{2},1\right] $$ then always lead to a steady solution. [ABSTRACT FROM AUTHOR]

The aim of this study is to explore the global stability of Cohen–Grossberg Clifford‐valued neutral‐type neural network models with time delays. In order to achieve the aim of this paper, and to solve the non‐commutativity problem caused by Clifford numbers multiplication, the original Clifford‐valued system is first decomposed into 2mn$$ {2}^mn $$‐dimensional real‐valued systems. Some sufficient criteria for the global stability of the addressed network models are established by constructing an appropriate Lyapunov functional. The established stability conditions have not been affected by the neutral delay and time delay values. The proposed method and results of this paper are new. The feasibility of the stability criteria obtained are verified using two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the qualitative behavior of a class of fractional SEIR epidemic models with a more general incidence rate function and time delay to incorporate latent infected individuals. We first prove positivity and boundedness of solutions of the system. The basic reproduction number R0$$ {\mathcal{R}}_0 $$ of the model is computed using the method of next generation matrix, and we prove that if R0<1$$ {\mathcal{R}}_0<1 $$, the healthy equilibrium is locally asymptotically stable, and when R0>1$$ {\mathcal{R}}_0>1 $$, the system admits a unique endemic equilibrium which is locally asymptotically stable. Moreover, using a suitable Lyapunov function and some results about the theory of stability of differential equations of delayed fractional‐order type, we give a complete study of global stability for both healthy and endemic steady states. The model is used to describe the COVID‐19 outbreak in Algeria at its beginning in February 2020. A numerical scheme, based on Adams–Bashforth–Moulton method, is used to run the numerical simulations and shows that the number of new infected individuals will peak around late July 2020. Further, numerical simulations show that around 90% of the population in Algeria will be infected. Compared with the WHO data, our results are much more close to real data. Our model with fractional derivative and delay can then better fit the data of Algeria at the beginning of infection and before the lock and isolation measures. The model we propose is a generalization of several SEIR other models with fractional derivative and delay in literature. [ABSTRACT FROM AUTHOR]

In this paper, an SIR model with a nonlinear incidence rate is studied. A disease‐free equilibrium E0$$ {E}_0 $$, an endemic equilibrium E1$$ {E}_1 $$, and the basic reproduction number of the model R0$$ {R}_0 $$ are obtained. If R0<1,E0$$ {R}_0&lt;1,{E}_0 $$ is locally asymptotically stable and if R0>1,E1$$ {R}_0&gt;1,{E}_1 $$ is locally asymptotically stable. By Barbalat's lemma, we study the global stability of the model. Transcritical bifurcation analysis is investigated by using the Sotomayor theorem. As the infection rate increases, the asymptotic behavior of the system near E0$$ {E}_0 $$ approaches E1$$ {E}_1 $$ and the system has a transcritical bifurcation. Also, we check the existence of Hopf bifurcation for the given system. In addition, a sensitivity analysis is provided for the basic reproduction number. Our results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

In this paper, the fourth‐order parabolic equation with the first Neumann boundary value conditions is concerned, where the values of the first and second spatial derivatives of the unknown function at the boundary are given. A compact difference scheme is established for this kind of problem by using the weighted average and order reduction methods. The difficulty lies in the challenges of handling the boundary conditions with high accuracy. The unique solvability, convergence and stability of the proposed compact difference scheme are proved by the energy method. Some novel techniques are introduced for the analysis. Then, the extension to a more general case with the reaction term is briefly explored. Finally, two numerical examples are numerically calculated to show the efficiency of the proposed numerical schemes. [ABSTRACT FROM AUTHOR]

This paper presents reduced‐order methods to study the stability of initialized or Grünwald‐Letnikov fractional nonlinear systems. It is shown that the initialization procedure must be formalized by introducing a class of systems, and the corresponding stability analysis must be established for each element of that class. The main features obtained using this novel approach are (a) the requirements for stability are imposed directly on the equations of the system and involve only finite‐dimensional variables; (b) the conclusions are asserted on the variables of interest; (c) the method can be extended in several ways, including multi‐order systems. Illustrative examples, including an application in adaptive control, are finally presented to convey the usefulness of our approach. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the stability and large time behavior problem on the three‐dimensional Boussinesq equations for magnetohydrodynamic convection near hydrostatic equilibrium by energy method. The stability for the fluids with certain symmetries is built on the spatial domain Ω=R2×T$$ \Omega ={\mathbf{R}}^2\times \mathbf{T} $$ with T=−12,12$$ \mathbf{T}=\left[-\frac{1}{2},\frac{1}{2}\right] $$ being a 1D period box. In addition, the exponential decay of the oscillation parts is established. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of singular prey‐taxis models in a smooth bounded domain under homogeneous Neumann boundary conditions. The main challenge of analysis is the possible singularity as the prey density vanishes. Employing the technique of a priori assumption, the comparison principle of differential equations and semigroup estimates, we show that the singularity can be precluded if the intrinsic growth rate of prey is suitably large and hence obtain the existence of global classical bounded solutions. Moreover, the global stability of co‐existence and prey‐only steady states with convergence rates is established by the method of Lyapunov functionals. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability problem and large‐time behavior for the 3D Boussinesq equations with mixed partial kinematic viscosity and thermal diffusivity near one equilibrium, i.e., (U(0),Θ(0))=((0,0,0),βx3)$$ \left({U}&#x0005E;{(0)},{\Theta}&#x0005E;{(0)}\right)&#x0003D;\left(\left(0,0,0\right),\beta {x}_3\right) $$ with β=1$$ \beta &#x0003D;1 $$ (where β$$ \beta $$ is chosen to be large, e.g., satisfying the well‐known Miles‐Howard criterion). These problems can be extremely difficult due to the equations lack of full dissipation. Our results are twofold. In the first place, we establish the global H2$$ {H}&#x0005E;2 $$ and H3$$ {H}&#x0005E;3 $$ stability for the nonlinear system with seven different partially dissipated systems of Boussinesq equations. In the second place, we derive precise large‐time behavior for the linearized system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the stability of the inverse scattering problem for the damped biharmonic plate equation in a bounded domain with the Cauchy data. A sharp estimate for the mass density is established using a priori information concerning Sobolev norms and a priori information about the support of the inhomogeneity. Our results improve previous estimates and explicitly depend on the damping coefficient. The proof mainly relies on the complex geometric optics solution method and the Fourier analysis. [ABSTRACT FROM AUTHOR]

In this paper, a delayed mosquito population suppression model with stage and sex structure is constructed incorporating periodic releases of the sterile mosquito and its sexual lifespan. Sterile mosquitoes are released periodically into the field to suppress the wild mosquito population, and only sexually active sterile mosquitoes are included in the interaction system of these two populations. According to the relationship between the release period T$$ T $$ and the sexual lifespan T1$$ {T}_1 $$, we study the dynamic behaviors of the system, such as the existence of positive periodic solutions and the stability of extinction equilibrium in three cases. A series of numerical simulations are performed, which not only verify the reliability of the theoretical results but also make a numerical analysis of the open problems raised in the theoretical research. [ABSTRACT FROM AUTHOR]

In this paper, we consider a quasilinear viscoelastic wave equation that features a distributed delay as well as logarithmic nonlinearity. We combine the potential well theory, Faedo‐Galerkin's approximation, and some energy estimates to construct the global existence of the solution. With weaker conditions on the relaxation function, we establish explicit and general decay rate results by using the multiplier method and some properties of convex functions. Our results improve and generalize several earlier related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a host‐parasitoid model with harvest effort. The existence and stability of a positive fixed point are analyzed. The period‐doubling and Neimark–Sacker bifurcations are studied. These analyses are achieved by applying the normal form of the difference‐algebraic system, bifurcation theory, and center manifold theorem. Furthermore, we apply a state‐delayed feedback control strategy to control the complex dynamics of the proposed model. Numerical examples and simulations are given to verify our findings. Owing to the framework of Nicholson–Bailey host‐parasitoid system, the proposed difference‐algebraic model shows rich dynamics compared with the continuous‐time models. [ABSTRACT FROM AUTHOR]

We deal with a fractional‐order nonlinear financial model with time delay in this paper. The quantitative analysis of the model has been done in which the asymptotic stability of the equilibrium points of the model have been discussed. Furthermore, the Hopf bifurcation analysis of the model has been done under the impact of time delay. The stability of the model has been studied with the reproduction number less than or greater than 1. The analysis of the model has been done by using the Laplace transformation technique. The analysis shows that the fractional‐order model with a time delay can adequately enhance the elements and fortify the results for either stable or unstable criteria. All unstable cases in the nonlinear financial system are converted to stable cases for fractional order under neighborhood points. The reproduction ranges of all parameters have been discussed. The paper examines the impact of time delay and the importance of determining essential parameters such as saving amount by using Hopf bifurcation. Finally, numerical simulations have been carried out by MATLAB to illustrate our derived results. The financial system is verified under the theoretical outcomes from parameter estimation. [ABSTRACT FROM AUTHOR]

This paper is concerned with an efficient numerical method for a class of fourth‐order partial integro‐differential equations (PIDEs) with weakly singular kernel. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t=0$$ t=0 $$. The proposed method is constructed on the graded meshes, which can achieve the second‐order convergence in time for weakly singular solutions. The product integral rule of the Riemann–Liouville fractional integral with the generalized time‐stepping is used for the time discretization, and the standard central difference formula is used for the space discretization. The stability and convergence of the method are proved, and the optimal error estimates in the discrete L2$$ {L}^2 $$‐norm and H1$$ {H}^1 $$‐norm are obtained. Numerical results show the effectiveness of the proposed method. Further, by the extrapolation method, we improve the convergence order in space and time to four order, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we study an age‐structured reaction‐diffusion‐advection population model. First, we use a non‐densely defined operator to the linear age‐structured reaction‐diffusion‐advection population model in a patchy environment. By spectral analysis, we obtain the asynchronous exponential growth of the population model. Then we consider nonlinear death rate and birth rate, which all depend on the function related to the generalized total population, and we prove the existence of a steady state of the system. Finally, we study the age‐structured reaction‐diffusion‐advection population model in non‐autonomous situations. We give the comparison principle and prove the eventual compactness of semiflow by using integrated semigroup. We also prove the existence of compact attractors under the periodic situation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decoupled stabilized finite element method for the time–dependent Navier–Stokes/Biot problem by using the lowest equal‐order finite elements. The coupling problem is divided into two subproblems which can be solved in parallel: One is the Navier–Stokes model by treating the nonlinear term explicitly, and the other is the Biot model. In the numerical scheme, we use the implicit backward Euler method in time, while treat the coupling terms explicitly. The stability analysis and error estimates are established for the proposed fully discrete scheme. Numerical results are provided to justify the theory. [ABSTRACT FROM AUTHOR]

This paper presents a new numerical algorithm dealing with multi‐term time‐fractional Burgers‐type equation involving the Caputo derivative. The proposed method consists of temporal discretization of L2$$ L2 $$ formula and spatial discretization using the exponential B‐splines. The semi implicit approach is applied to discretize the nonlinear term u∂xu$$ u{\partial}_{\mathtt{x}}u $$. We adopt the Von–Neumann method to study stability. We also establish the convergence analysis. The proposed method is employed to solve a few numerical examples in order to test its efficiency and accuracy. Comparisons with the recent works confirm the efficiency and robustness of the proposed method. [ABSTRACT FROM AUTHOR]