American Institute of Mathematical Sciences

In this paper, we introduce the weak Galerkin (WG) finite element method for the Stokes equations with damping. We establish the WG numerical scheme on general meshes and prove the well-posedness of the scheme. Optimal error estimates for the velocity and pressure are derived. Furthermore, in order to accelerate the WG algorithm, we present a two-level method and give the corresponding error estimates. Finally, some numerical examples are reported to validate the theoretical analysis.

Common air pollutants, such as ozone (\begin{document}$ \rm{O}_{3} $\end{document}), sulfur dioxide (\begin{document}$ \rm{SO}_2 $\end{document}) and nitrogen dioxide (\begin{document}$ \rm{NO}_2 $\end{document}), can affect the spread of influenza. We propose a new non-autonomous impulsive differential equation model with the effects of ozone and vaccination in this paper. First, the basic reproduction number of the impulsive system is obtained, and the global asymptotic stability of the disease-free periodic solution is proved. Furthermore, the uniform persistence of the system is demonstrated. Second, the unknown parameters of the ozone dynamics model are obtained by fitting the ozone concentration data by the least square method and Bootstrap. The MCMC algorithm is used to fit influenza data in Gansu Province to identify the most suitable parameter values of the system. The basic reproduction number \begin{document}$ R_{0} $\end{document} is estimated to be \begin{document}$ 1.2486 $\end{document} (\begin{document}$ 95\%\rm{CI}:(1.2470, 1.2501) $\end{document}). Then, a sensitivity analysis is performed on the system parameters. We find that the average annual incidence of seasonal influenza in Gansu Province is 31.3374 per 100,000 people. Influenza cases started to surge in 2016, rising by a factor of one and a half between 2014 and 2016, further increasing in 2019 (54.6909 per 100,000 population). The average incidence rate during the post-upsurge period (2017-2019) is one and a half times more than in the pre-upsurge period (2014-2016). In particular, we find that the peak ozone concentration appears 5–8 months in Gansu Province. A moderate negative correlation is seen between influenza cases and monthly ozone concentration (Pearson correlation coefficient: \begin{document}$ r $\end{document} = -0.4427). Finally, our results show that increasing the vaccination rate and appropriately increasing the ozone concentration can effectively prevent and control the spread of influenza.

Body weight control is gaining interest since its dysregulation eventually leads to obesity and metabolic disorders. An accurate mathematical description of the behavior of physiological variables in humans after food intake may help in understanding regulation mechanisms and in finding treatments. This work proposes a multi-compartment mathematical model of food intake that accounts for glucose-insulin homeostasis and ghrelin dynamics. The model involves both food volumes and glucose amounts in the two-compartment system describing the gastro-intestinal tract. Food volumes control ghrelin dynamics, whilst glucose amounts clearly impact on the glucose-insulin system. The qualitative behavior analysis shows that the model solutions are mathematically coherent, since they stay positive and provide a unique asymptotically stable equilibrium point. Ghrelin and insulin experimental data have been exploited to fit the model on a daily horizon. The goodness of fit and the physiologically meaningful time courses of all state variables validate the efficacy of the model to capture the main features of the glucose-insulin-ghrelin interplay.

In this paper, we study the following Kirchhoff-type fractional Schrödinger system with critical exponent in \begin{document}$ \mathbb{R}^N $\end{document}:

\begin{document}$ \begin{equation*} \begin{cases} \left(a_{1}+b_{1}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u +\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u,\\ \left(a_{2}+b_{2}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx\right)(-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+ \frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\\ \end{cases} \end{equation*} $\end{document}

where \begin{document}$ (-\Delta)^{s} $\end{document} is the fractional Laplacian, \begin{document}$ 0<s<1 $\end{document}, \begin{document}$ N>2s, $\end{document} \begin{document}$ 2_{s}^{\ast} = 2N/(N-2s) $\end{document} is the fractional critical Sobolev exponent, \begin{document}$ \mu_{1},\mu_{2},\gamma, k>0 $\end{document}, \begin{document}$ \alpha+\beta = 2_{s}^{\ast},\ 1<p<2_{s}^{\ast}-1 $\end{document}, \begin{document}$ a_{i},b_{i}\geq 0, $\end{document} with \begin{document}$ a_{i}+b_{i}>0,\ \ i = 1,2 $\end{document}. By using appropriate transformation, we first get its equivalent system which may be easier to solve:

\begin{document}$ \begin{equation*} \begin{cases} (-\Delta)^{s}u+u = \mu_1|u|^{2^*_s-2}u+\frac{\alpha\gamma}{2^*_s}|u|^{\alpha-2}u|v|^{\beta}+k|u|^{p-1}u, \ \ x\in \mathbb{R}^N, \\ (-\Delta)^{s}v+v = \mu_2|v|^{2^*_s-2}v+\frac{\beta\gamma}{2^*_s}|u|^{\alpha}|v|^{\beta-2}v+k|v|^{p-1}v,\ \ x\in \mathbb{R}^N,\\ \lambda_{1}^{s}-a_{1}-b_{1}\lambda_{1}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}u|^2dx = 0, \ \ \lambda_{1}\in \mathbb{R}^+,\\ \lambda_{2}^{s}-a_{2}-b_{2}\lambda_{2}^{\frac{N-2s}{2}}\int_{\mathbb{R}^N}|(-\Delta)^{\frac{s}{2}}v|^2dx = 0, \ \ \lambda_{2}\in \mathbb{R}^+. \end{cases} \end{equation*} $\end{document}

Then, by using the mountain pass theorem, together with some classical arguments from Brézis and Nirenberg, we obtain the existence of solutions for the new system under suitable conditions. Finally, based on the equivalence of two systems, we get the existence of solutions for the original system. Our results give improvement and complement of some recent theorems in several directions.

The financial model proposed involves the liquidation process of a portfolio through sell / buy orders placed at a price \begin{document}$ x\in\mathbb{R}^n $\end{document}, with volatility. Its rigorous mathematical formulation results to an \begin{document}$ n $\end{document}-dimensional outer parabolic Stefan problem with noise. The moving boundary encloses the areas of zero trading. We will focus on a case of financial interest when one or more markets are considered. We estimate the areas of zero trading with diameter approximating the minimum of the \begin{document}$ n $\end{document} spreads for orders from the limit order books. In dimensions \begin{document}$ n = 3 $\end{document}, for zero volatility, this problem stands as a mean field model for Ostwald ripening, and has been proposed and analyzed by Niethammer in [25], and in [7]. We propose a spherical moving boundaries approach where the zero trading area consists of a union of spherical domains centered at portfolios various prices with radii representing the half of the minimum spread. We apply Itô calculus and provide second order formal asymptotics for the stochastic dynamics of the spreads that seem to disconnect the financial model from a large diffusion assumption on the liquidity coefficient of the Laplacian that would correspond to an increased trading density. Moreover, we solve the approximating systems numerically.

Bloch wave homogenization is a spectral method for obtaining effective coefficients for periodically heterogeneous media. This method hinges on the direct integral decomposition of periodic operators, which is not available in a suitable form for almost periodic operators. In particular, the notion of Bloch eigenvalues and eigenvectors does not exist for almost periodic operators. However, we are able to recover the almost periodic homogenization result by employing a sequence of periodic approximations to almost periodic operators. We also establish a rate of convergence for approximations of homogenized tensors for a class of almost periodic media. The results are supported by a numerical study.

A model of two microbial species in a chemostat competing for a single resource is considered, where one of the competitors that produces a toxin, which is lethal to the other competitor (allelopathic inhibition), is itself inhibited by the substrate. Using general growth rate functions of the species, necessary and sufficient conditions of existence and local stability of all equilibria of the four-dimensional system are determined according to the operating parameters represented by the dilution rate and the input concentration of the substrate. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. If a non monotonic growth rate is considered (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. We describe its operating diagram, which is the bifurcation diagram giving the behavior of the system with respect to the operating parameters. By means of this bifurcation diagram, we show that the general model presents a set of fifteen possible behaviors: washout, competitive exclusion of one species, coexistence, multi-stability, occurrence of stable limit cycles through a super-critical Hopf bifurcations, homoclinic bifurcations and flip bifurcation. This diagram is very useful to understand the model from both the mathematical and biological points of view.

For a sequence \begin{document}$ (a_n) $\end{document} of complex numbers we consider the cubic parabolic polynomials \begin{document}$ f_n(z) = z^3+a_n z^2+z $\end{document} and the sequence \begin{document}$ (F_n) $\end{document} of iterates \begin{document}$ F_n = f_n\circ\dots\circ f_1 $\end{document}. The Fatou set \begin{document}$ \mathcal{F}_0 $\end{document} is the set of all \begin{document}$ z\in\hat{\mathbb{C}} $\end{document} such that the sequence \begin{document}$ (F_n) $\end{document} is normal. The complement of the Fatou set is called the Julia set and denoted by \begin{document}$ \mathcal{J}_0 $\end{document}. The aim of this paper is to study some properties of \begin{document}$ \mathcal{J}_0 $\end{document}. As a particular case, when the sequence \begin{document}$ (a_n) $\end{document} is constant, \begin{document}$ a_n = a $\end{document}, then the iteration \begin{document}$ F_n $\end{document} becomes the classical iteration \begin{document}$ f^n $\end{document} where \begin{document}$ f(z) = z^3+a z^2+z $\end{document}. The connectedness locus, \begin{document}$ M $\end{document}, is the set of all \begin{document}$ a\in\mathbb{C} $\end{document} such that the Julia set is connected. In this paper we investigate some symmetric properties of \begin{document}$ M $\end{document} as well.

In this paper, we consider the following chemotaxis-consumption model with porous medium diffusion and singular sensitivity

\begin{document}$ \begin{align*} \left\{ \begin{aligned} &u_{t} = \Delta u^{m}-\chi \mathrm{div}(\frac{u}{v}\nabla v)+\mu u(1-u), \\ &v_{t} = \Delta v-u^{r}v, \end{aligned}\right. \end{align*} $\end{document}

in a bounded domain \begin{document}$ \Omega\subset\mathbb R^N $\end{document} (\begin{document}$ N\ge 2 $\end{document}) with zero-flux boundary conditions. It is shown that if \begin{document}$ r<\frac{4}{N+2} $\end{document}, for arbitrary case of fast diffusion (\begin{document}$ m\le 1 $\end{document}) and slow diffusion \begin{document}$ (m>1) $\end{document}, this problem admits a locally bounded global weak solution. It is worth mentioning that there are no smallness restrictions on the initial datum and chemotactic coefficient.

In this paper, we study an age-structured HIV model with intracellular delay, logistic growth and antiretrowviral therapy. We first rewrite the model as an abstract non-densely defined Cauchy problem and obtain the existence of the unique positive steady state. Then through the linearization arguments we investigate the asymptotic behavior of steady states by determining the distribution of eigenvalues. We obtain successfully the globally asymptotic stability for the null equilibrium and (locally) asymptotic stability for the positive equilibrium respectively. Moreover, we also prove that Hopf bifurcations occur around the positive equilibrium under some conditions. In addition, we address the persistence of the semi-flow by showing the existence of a global attractor. Finally, some numerical examples are provided to illustrate the main results.

In infinite \begin{document}$ m $\end{document}-dimensional lattices, we obtain the existence of two nontrivial solutions for a class of non-periodic Schrödinger lattice systems with perturbed terms, where the potentials are coercive and the nonlinearities are asymptotically linear at infinity. In addition, examples are given to illustrate our results.

This paper presents a singular function on the unit interval \begin{document}$ [0, 1] $\end{document} derived from the dynamics of one-dimensional elementary cellular automaton Rule \begin{document}$ 150 $\end{document}. We describe the properties of the resulting function, which is strictly increasing, uniformly continuous, and differentiable almost everywhere, and show that it is not differentiable at dyadic rational points. We also derive functional equations that this function satisfies and show that this function is the only solution of the functional equations.

The expression of individual genes into functional protein molecules is a noisy dynamical process. Here we model the protein concentration as a jump-drift process which combines discrete stochastic production bursts (jumps) with continuous deterministic decay (drift). We allow the drift rate, the jump rate, and the jump size to depend on the protein level to implement feedback in protein stability, burst frequency, and burst size. We specifically focus on positive feedback in burst size, while allowing for arbitrary autoregulation in burst frequency and protein stability. Two versions of feedback in burst size are thereby considered: in the first, newly produced molecules instantly participate in feedback, even within the same burst; in the second, within-burst regulation does not occur due to the so-called infinitesimal delay. Without infinitesimal delay, the model is explicitly solvable; with its inclusion, an exact distribution to the model is unavailable, but we are able to construct a WKB approximation that applies in the asymptotic regime of small but frequent bursts. Comparing the asymptotic behaviour of the two model versions, we report that they yield the same WKB quasi-potential but a different exponential prefactor. We illustrate the difference on the case of a bimodal protein distribution sustained by a sigmoid feedback in burst size: we show that the omission of the infinitesimal delay overestimates the weight of the upper mode of the protein distribution. The analytic results are supported by kinetic Monte-Carlo simulations.

Empirical data exhibit a common phenomenon that vegetation biomass fluctuates periodically over time in ecosystem, but the corresponding internal driving mechanism is still unclear. Simultaneously, considering that the conversion of soil water absorbed by roots of the vegetation into vegetation biomass needs a period time, we thus introduce the conversion time into Klausmeier model, then a spatiotemporal vegetation model with time delay is established. Through theoretical analysis, we not only give the occurence conditions of stability switches for system without and with diffusion at the vegetation-existence equilibrium, but also derive the existence conditions of saddle-node-Hopf bifurcation of non-spatial system and Hopf bifurcation of spatial system at the coincidence equilibrium. Our results reveal that the conversion delay induces the interaction between the vegetation and soil water in the form of periodic oscillation when conversion delay increases to the critical value. By comparing the results of system without and with diffusion, we find that the critical value decreases with the increases of spatial diffusion factors, which is more conducive to emergence of periodic oscillation phenomenon, while spatial diffusion factors have no effects on the amplitude of periodic oscillation. These results provide a theoretical basis for understanding the spatiotemporal evolution behaviors of vegetation system.

The Riemann problem is solved for a system arising in chemotaxis. The system is of mixed-type and transitions from a hyperbolic to an elliptic region. It is genuinely nonlinear in the \begin{document}$ u $\end{document}-\begin{document}$ v $\end{document} plane except on the \begin{document}$ v $\end{document}-axis, where it is linearly degenerate. We have solved the Riemann problem in the physically relevant region up to the boundary of the hyperbolic-elliptic region, which is non-strictly hyperbolic. We also solved the problem on the linearly degenerate region. While solving the Riemann problem, we found classical shock and rarefaction waves in the hyperbolic region and contact discontinuities in the linearly degenerate region.

This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region \begin{document}$ [0,h_0] $\end{document}. We first show that the problem has a unique solution defined for all \begin{document}$ t>0 $\end{document}. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria.

In this work we study from both variational and numerical points of view a thermoelastic problem which appears in the dual-phase-lag theory with two temperatures. An existence and uniqueness result is proved in the general case of different Taylor approximations for the heat flux and the inductive temperature. Then, in order to provide the numerical analysis, we restrict ourselves to the case of second-order approximations of the heat flux and first-order approximations for the inductive temperature. First, variational formulation of the corresponding problem is derived and an energy decay property is proved. Then, a fully discrete scheme is introduced by using the finite element method for the approximation of the spatial variable and the implicit Euler scheme for the discretization of the time derivatives. A discrete stability

We introduce an active swarming model on the sphere which contains additional temporal dynamics for the natural frequency, inspired from the recently introduced modified Kuramoto model, where the natural frequency has its own dynamics. For the attractive interacting particle system, we provide a sufficient framework that leads to the asymptotic aggregation, i.e., all the particles are aggregated to the single point and the natural frequencies also tend to a common value. On the other hand, for the repulsive interacting particle system, we present a sufficient condition for the disaggregation, i.e., the order parameter of the system decays to 0, which implies that the particles are uniformly distributed over the sphere asymptotically. Finally, we also provide several numerical simulation results that support the theoretical results of the paper.

A depletion-type reaction-diffusion Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme and the homogeneous Neumann boundary conditions is introduced and investigated in this paper. Firstly, the boundedness of positive solution of the parabolic system is given, and the constant steady state solutions of the model are exhibited by the Shengjin formulas. Through rigorous theoretical analysis, the stability of the corresponding positive constant steady state solution is explored. Next, a priori estimates, the properties of the nonconstant steady states, non-existence and existence of the nonconstant steady state solution for the corresponding elliptic system are investigated by some estimates and the Leray-Schauder degree theory, respectively. Then, some existence conditions are established and some properties of the Hopf bifurcation and the steady state bifurcation are presented, respectively. It is showed that the temporal and spatial bifurcation structures will appear in the reaction-diffusion model. Theoretical results are confirmed and complemented by numerical simulations.

This paper is concerned with the existence of unstable stationary solutions for nonlinear stochastic differential equations (SDEs) with additive white noise. Assume that the nonlinear term \begin{document}$ f $\end{document} is monotone (or anti-monotone) and the global Lipschitz constant of \begin{document}$ f $\end{document} is smaller than the positive real part of the principal eigenvalue of the competitive matrix \begin{document}$ A $\end{document}, the random dynamical system (RDS) generated by SDEs has an unstable \begin{document}$ \mathscr{F}_+ $\end{document}-measurable random equilibrium, which produces a stationary solution for nonlinear SDEs. Here, \begin{document}$ \mathscr{F}_+ = \sigma\{\omega\mapsto W_t(\omega):t\geq0\} $\end{document} is the future \begin{document}$ \sigma $\end{document}-algebra. In addition, we get that the \begin{document}$ \alpha $\end{document}-limit set of all pull-back trajectories starting at the initial value \begin{document}$ x(0) = x\in\mathbb{R}^n $\end{document} is a single point for all \begin{document}$ \omega\in\Omega $\end{document}, i.e., the unstable \begin{document}$ \mathscr{F}_+ $\end{document}-measurable random equilibrium. Applications to stochastic neural network models are given.

In this paper, we consider the following Schrödinger-Poisson system with double quasi-linear terms

\begin{document}$ \begin{equation*} \label{1.1} \begin{cases} -\Delta u+V(x)u+\phi u-\frac{1}{2}u\Delta u^2 = \lambda f(x,u),\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ -\triangle\phi-\varepsilon^4\Delta_4\phi = u^{2},\; &\; {\rm{in}}\; \mathbb{R}^{3},\\ \end{cases} \end{equation*} $\end{document}

where \begin{document}$ \lambda,\varepsilon $\end{document} are positive parameters. Under suitable assumptions on \begin{document}$ V $\end{document} and \begin{document}$ f $\end{document}, we prove that the above system admits at least one pair of positive solutions for \begin{document}$ \lambda $\end{document} large by using perturbation method and truncation technique. Furthermore, we research the asymptotical behavior of solutions with respect to the parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \varepsilon $\end{document} respectively. These results extend and improve some existing results in the literature.

In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution via operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.

Kawasaki disease (KD) is an acute febrile vasculitis that occurs predominantly in infants and young children. With coronary artery abnormalities (CAAs) as its most serious complications, KD has become the leading cause of acquired heart disease in developed countries. Based on some new biological findings, we propose a time-delayed dynamic model of KD pathogenesis. This model exhibits forward\begin{document}$ / $\end{document}backward bifurcation. By analyzing the characteristic equations, we completely investigate the local stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibria. Our results show that the time delay does not affect the local stability of the inflammatory factors-free equilibrium. However, the time delay as the bifurcation parameter may change the local stability of the inflammatory factors-existent equilibrium, and stability switches as well as Hopf bifurcation may occur within certain parameter ranges. Further, by skillfully constructing Lyapunov functionals and combining Barbalat's lemma and Lyapunov-LaSalle invariance principle, we establish some sufficient conditions for the global stability of the inflammatory factors-free equilibrium and the inflammatory factors-existent equilibrium. Moreover, it is shown that the model is uniformly persistent if the basic reproduction number is greater than one, and some explicit analytic expressions of eventual lower bounds of the solutions of the model are given by analyzing the properties of the solutions and the range of time delay very precisely. Finally, some numerical simulations are carried out to illustrate the theoretical results.

The main purpose of this paper is to study the homogenization problem of stochastic reaction-diffusion equations with singular perturbation term. The difficulty in studying such problems is how to get the uniform estimates of the equations under the influence of the singularity term. Firstly, we use the properties of the elliptic equation corresponding to the generator to eliminate the influence of singular terms and obtain the uniform estimates of the slow equation and thus, get the tightness. Finally, we prove that under appropriate assumptions, the slow equation converges to a homogenization equation in law.

We consider the classical Nicholson's blowflies model incorporating two distinctive time-varying delays. One of the delays corresponds to the length of the individual's life cycle, and another corresponds to the specific physiological stage when self-limitation feedback takes place. Unlike the classical formulation of Nicholson's blowflies equation where self-regulation appears due to the competition of the productive adults for resources, the self-limitation of our considered model can occur at any developmental stage of an individual during the entire life cycle. We aim to find sharp conditions for the global asymptotic stability of a positive equilibrium. This is a significant challenge even when both delays are held at constant values. Here, we develop an approach to obtain a sharp and explicit criterion in an important situation where the two delays are asymptotically apart. Our approach can be also applied to the non-autonomous Mackey-Glass equation to provide a partial solution to an open problem about the global dynamics.