American Institute of Mathematical Sciences

In this paper, we study consensus problem in a discrete-time multi-agent system with uncertain topologies and random delays governed by a Markov chain. The communication topology is assumed to be directed but interrupted by system uncertainties. Furthermore, the system delays are modeled by a Markov chain. We first use a reduced-order system featuring the error dynamics to transform the consensus problem of the original one into the stabilization of the error dynamic system. By using the linear matrix inequality method and the stability theory in stochastic systems with time-delay, several sufficient conditions are established for the mean square stability of the error dynamics which guarantees consensus. By redesigning its adjacency matrices, we develop a switching control scheme which is delay-dependent. Finally, simulation results are worked out to illustrate the theoretical results.

We investigate the long-time solvability in Besov spaces of the initial value problem for the inviscid 3D-Boussinesq equations with Coriolis force. First we prove a local existence and uniqueness result with critical and supercritical regularity and existence-time \begin{document}$ T $\end{document} uniform with respect to the rotation speed \begin{document}$ \Omega $\end{document}. Afterwards, we show a blow-up criterion of BKM type, estimates for arbitrarily large \begin{document}$ T $\end{document}, and then obtain the long-time existence and uniqueness of solutions for arbitrary initial data, provided that \begin{document}$ \Omega $\end{document} is large enough.

The Feigenbaum-Kadanoff-Shenker equation for universal scaling in circle maps characterizes the quasiperiodic route to chaos. In this paper, using two different iterative methods, we construct all strictly decreasing continuous solutions. Furthermore, we present respectively the corresponding conditions to guarantee \begin{document}$ C^1 $\end{document} smoothness of those continuous solutions.

This paper deals with the following competitive two-species chemotaxis system with two chemicals

\begin{document}$ \left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - {\chi _1}\nabla \cdot (u\nabla v) + {\mu _1}u\left( {1 - u - {a_1}w} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta v - v + w,}&{x \in \Omega ,t > 0,}\\{{w_t} = \Delta w - {\chi _2}\nabla \cdot (w\nabla z) + {\mu _2}w\left( {1 - w - {a_2}u} \right),}&{x \in \Omega ,t > 0,}\\{0 = \Delta z - z + u,}&{x \in \Omega ,t > 0}\end{array}} \right. $\end{document}

under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} (\begin{document}$ n\geq1 $\end{document}), where the parameters \begin{document}$ \chi_i>0 $\end{document}, \begin{document}$ \mu_i>0 $\end{document} and \begin{document}$ a_i>0 $\end{document} (\begin{document}$ i = 1, 2 $\end{document}). It is proved that the corresponding initial-boundary value problem possesses a unique global bounded classical solution if one of the following cases holds:

(ⅰ) \begin{document}$ q_1\leq a_1; $\end{document} (ⅱ) \begin{document}$ q_2\leq a_2 $\end{document};

(ⅲ) \begin{document}$ q_1>a_1 $\end{document} and \begin{document}$ q_2> a_2 $\end{document} as well as \begin{document}$ (q_1-a_1)(q_2-a_2)<1 $\end{document},

where \begin{document}$ q_1: = \frac{\chi_1}{\mu_1} $\end{document} and \begin{document}$ q_2: = \frac{\chi_2}{\mu_2} $\end{document}, which partially improves the results of Zhang et al. [53] and Tu et al. [34].

Moreover, it is proved that when \begin{document}$ a_1, a_2\in(0, 1) $\end{document} and \begin{document}$ \mu_1 $\end{document} and \begin{document}$ \mu_2 $\end{document} are sufficiently large, then any global bounded solution exponentially converges to \begin{document}$ \left(\frac{1-a_1}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_2}{1-a_1a_2}, \frac{1-a_1}{1-a_1a_2}\right) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document}; When \begin{document}$ a_1>1>a_2>0 $\end{document} and \begin{document}$ \mu_2 $\end{document} is sufficiently large, then any global bounded solution exponentially converges to \begin{document}$ (0, 1, 1, 0) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document}; When \begin{document}$ a_1 = 1>a_2>0 $\end{document} and \begin{document}$ \mu_2 $\end{document} is sufficiently large, then any global bounded solution algebraically converges to \begin{document}$ (0, 1, 1, 0) $\end{document} as \begin{document}$ t\rightarrow\infty $\end{document}. This result improves the conditions assumed in [34] for asymptotic behavior.

We deal with the Cauchy problem of nonhomogeneous micropolar fluid equations with zero density at infinity in the entire space \begin{document}$ \mathbb{R}^2 $\end{document}. We show that for the initial density allowing vacuum, the strong solution exists globally if a weighted density is bounded from above. It should be noted that our blow-up criterion is independent of micro-rotational velocity.

We first show that the stochastic two-compartment Gray-Scott system generates a non-autonomous random dynamical system. Then we establish some uniform estimates of solutions for stochastic two-compartment Gray-Scott system with multiplicative noise. Finally, the existence of uniform and cocycle attractors is proved.

In this paper we study the protection zone problem to a predator-prey model subject to Beddington-DeAngelis functional responses and small prey growth rate. This is a successive work to a previous paper of the authors [X. He, S. N. Zheng, Protection zone in a diffusive predator-prey model with Beddington-DeAngelis functional response, J. Math. Biol. 75 (2017) 239-257], where the model with large prey growth rate was considered. At first we establish the existence and multiplicity of positive steady state solutions, and then give the dynamic behavior of the evolution problem. It is proved that there may be no positive steady state, or may have at leat one, two, or even three positive steady states, depending on the parameters involved such as the growth rate, the predation rate, and the food handling time of the predators, the growth rate and the refuge ability of the preys, and the sizes of the habitat with protection zone. In addition, it is shown that the dynamics of the solutions rely on the initial state as well, e.g., though there could be multiple positive steady states, the prey will go to extinction as time tends to infinity if its initial value is small.

The technique of sterile mosquitoes plays an important role in the control of mosquito-borne diseases such as malaria, dengue, yellow fever, west Nile, and Zika. To explore the interactive dynamics between the wild and sterile mosquitoes, we formulate a delayed mosquito population suppression model with constant releases of sterile mosquitoes. Through the analysis of global dynamics of solutions of the model, we determine a threshold value of the release rate such that if the release threshold is exceeded, then the wild mosquito population will be eventually suppressed, whereas when the release rate is less than the threshold, the wild and sterile mosquitoes coexist and the model exhibits a complicated feature. We also obtain theoretical results including a sufficient and necessary condition for the global asymptotic stability of the zero solution. We provide numerical examples to demonstrate our results and give brief discussions about our findings.

Travel frequency of people varies widely with occupation, age, gender, ethnicity, income, climate and other factors. Meanwhile, the distribution of the numbers of times people in different regions or with different travel behaviors bitten by mosquitoes may be nonuniform. To reflect these two heterogeneities, we develop a multipatch model to study the impact of travel frequency and human biting rate on the spatial spread of mosquito-borne diseases. The human population in each patch is divided into four classes: susceptible unfrequent, infectious unfrequent, susceptible frequent, and infectious frequent. The basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} is defined. It is shown that the disease-free equilibrium is globally asymptotically stable if \begin{document}$ \mathcal{R}_0\leq 1 $\end{document}, and there is a unique endemic equilibrium that is globally asymptotically stable if \begin{document}$ \mathcal{R}_0>1 $\end{document}. A more detailed study is conducted on the single patch model. We use analytical and numerical methods to demonstrate that the model without considering the difference of humans in travel frequency mostly underestimates the risk of infection. Numerical simulations suggest that the greater the difference in travel frequency, the larger the underestimate of the transmission potential. In addition, the basic reproduction number \begin{document}$ \mathcal{R}_0 $\end{document} may decreasingly, or increasingly, or nonmonotonically vary when more people travel frequently.

We study the chemotaxis–Navier–Stokes system

\begin{document}$ \left\{\;\; \begin{aligned} n_t + u \cdot \nabla n &\;\; = \;\; \Delta n - \nabla \cdot (nS(x,n,c) \nabla c), \\ c_t + u\cdot \nabla c &\;\; = \;\; \Delta c - n f(c), \\ u_t + (u\cdot \nabla) u &\;\; = \;\; \Delta u + \nabla P + n \nabla \phi, \;\;\;\;\;\; \nabla \cdot u = 0, \;\;\;\;\;\; \end{aligned} \right. \tag{$\star$} $\end{document}

with no-flux boundary conditions for \begin{document}$ n $\end{document}, \begin{document}$ c $\end{document} and Dirichlet boundary conditions for \begin{document}$ u $\end{document} in a bounded, convex, smooth domain \begin{document}$ \Omega \subseteq \mathbb{R}^2 $\end{document}, which is motivated by recent modeling approaches from biology for aerobic bacteria suspended in a sessile water drop. We further do not assume the chemotactic sensitivity \begin{document}$ S $\end{document} to be scalar as is common, but to be able to attain values in \begin{document}$ \mathbb{R}^{2\times2} $\end{document}, which allows for more complex modeling of bacterial behavior.

While there have been various results for scalar \begin{document}$ S $\end{document} and some for the non-scalar case with only a Stokes fluid equation simplifying the analysis of the third equation in (\begin{document}$ \star $\end{document}), we consider the fully combined case giving us very little to go on in terms of a priori estimates. We nonetheless manage to still achieve sufficient estimates using Trudinger–Moser type inequalities to extend the existence results seen in a recent work by Winkler for the Stokes case with non-scalar \begin{document}$ S $\end{document} to the full Navier–Stokes case. Namely, we construct a similar global mass-preserving solution for (\begin{document}$ \star $\end{document}) in planar convex domains under fairly weak assumptions on the parameter functions.

This is part Ⅱ of our study on the free boundary problems with nonlocal and local diffusions. In part Ⅰ, we obtained the existence, uniqueness, regularity and estimates of global solution. In part Ⅱ here, we show a spreading-vanishing dichotomy, and provide the criteria of spreading and vanishing, as well as the long time behavior of solution when spreading happens.

We study a kind of better recurrence than Kolmogorov's one: periodicity recurrence, which corresponds periodic solutions in distribution for stochastic differential equations. On the basis of technique of upper and lower solutions and comparison principle, we obtain the existence of periodic solutions in distribution for stochastic differential equations (SDEs). Hence this provides an effective method how to study the periodicity of stochastic systems by analyzing deterministic ones. We also illustrate our results.

Mosquito-borne diseases pose a great threat to humans' health. Wolbachia is a promising biological weapon to control the mosquito population, and is not harmful to humans' health, environment, and ecology. In this work, we present a stage-structure model to investigate the effective releasing strategies for Wolbachia-infected mosquitoes. Besides some key factors for Wolbachia infection, the CI effect suffering by uninfected ova produced by infected females, which is often neglected, is also incorporated. We analyze the conditions under which Wolbachia infection still can be established even if the basic reproduction number is less than unity. Numerical simulations manifest that the threshold value of infected mosquitoes required to be released at the beginning can be evaluated by the stable manifold of a saddle equilibrium, and low levels of MK effect, fitness costs, as well as high levels of CI effect and maternal inheritance all contribute to the establishment of Wolbachia-infection. Moreover, our results suggest that ignoring the CI effect suffering by uninfected ova produced by infected females may result in the overestimation of the threshold infection level for the Wolbachia invasion.

We consider three matrix models of order 2 with one random entry \begin{document}$ \epsilon $\end{document} and the other three entries being deterministic. In the first model, we let \begin{document}$ \epsilon\sim \rm{Bernoulli}\left(\frac{1}{2}\right) $\end{document}. For this model we develop a new technique to obtain estimates for the top Lyapunov exponent in terms of a multi-level recursion involving Fibonacci-like sequences. This in turn gives a new characterization for the Lyapunov exponent in terms of these sequences. In the second model, we give similar estimates when \begin{document}$ \epsilon\sim \rm{Bernoulli}\left(p\right) $\end{document} and \begin{document}$ p\in [0, 1] $\end{document} is a parameter. Both of these models are related to random Fibonacci sequences. In the last model, we compute the Lyapunov exponent exactly when the random entry is replaced with \begin{document}$\xi\epsilon$\end{document} where \begin{document}$\epsilon$\end{document} is a standard Cauchy random variable and \begin{document}$\xi$\end{document} is a real parameter. We then use Monte Carlo simulations to approximate the variance in the CLT for both parameter models.

This work aims to prove the small time large deviation principle (LDP) for a class of stochastic partial differential equations (SPDEs) with locally monotone coefficients in generalized variational framework. The main result could be applied to demonstrate the small time LDP for various quasilinear and semilinear SPDEs such as stochastic porous medium equations, stochastic \begin{document}$ p $\end{document}-Laplace equations, stochastic Burgers type equation, stochastic 2D Navier-Stokes equation, stochastic power law fluid equation and stochastic Ladyzhenskaya model. In particular, our small time LDP result seems to be new in the case of general quasilinear SPDEs with multiplicative noise.

In this paper, we propose a new weak solution to an optimal stopping problem in finance and economics. The main advantage of this new definition is that we do not need the Dynamic Programming Principle, which is critical for both classical verification argument and modern viscosity approach. Additionally, the classical methods in differential equations, e.g. penalty method, can be used to derive some useful results.

We consider a force-based model for cell motion which models cell forces using Hooke's law and a random outreach from the cell center. In previous work this model was simplified to track the centroid by setting the relaxation time to zero, and a formula for the expected velocity of the centroid was derived. Here we extend that formula to allow for chemotaxis of the cell by allowing the outreach distribution to depend on the spatial location of the centroid.

In this paper we focus on the computation of periodic solutions of Delay Differential Equations (DDEs) with constant delays. The method is based on defining a Poincaré section in a suitable functional space and looking for a fixed point of the flow in this section. This is done by applying a Newton method on a suitable discretisation of the section. To avoid computing and storing large matrices we use a GMRES method to solve the linear system because in this case GMRES converges very fast due to the compactness of the flow of the DDE. The derivatives of the Poincaré map are obtained in a simple way, by applying Automatic Differentiation to the numerical integration. The stability of the periodic orbit is also obtained in a very efficient way by means of Arnoldi methods. The examples considered include temporal and spatial Poincaré sections.

This paper proposes an efficient computational framework for longitudinal velocity control of a large number of autonomous vehicles (AVs) and develops a traffic flow theory for AVs. Instead of hypothesizing explicitly how AVs drive, our goal is to design future AVs as rational, utility-optimizing agents that continuously select optimal velocity over a period of planning horizon. With a large number of interacting AVs, this design problem can become computationally intractable. This paper aims to tackle such a challenge by employing mean field approximation and deriving a mean field game (MFG) as the limiting differential game with an infinite number of agents. The proposed micro-macro model allows one to define individuals on a microscopic level as utility-optimizing agents while translating rich microscopic behaviors to macroscopic models. Different from existing studies on the application of MFG to traffic flow models, the present study offers a systematic framework to apply MFG to autonomous vehicle velocity control. The MFG-based AV controller is shown to mitigate traffic jam faster than the LWR-based controller. MFG also embodies classical traffic flow models with behavioral interpretation, thereby providing a new traffic flow theory for AVs.

The ensemble method has been developed for accelerating a sequence of numerical simulations of evolution problems. Its main idea is, by manipulating the time stepping and grouping discrete problems, to make all members in the same group share a common coefficient matrix. Thus, at each time step, instead of solving a sequence of linear systems each of which contains only one right-hand-side vector, the ensemble method simultaneously solves a single linear system with multiple right-hand-side vectors for each group. Such a system could be solved efficiently by using direct linear solvers when the problems are of small scale, as the same LU factorization would work for the entire group members. However, for large-scale problems, iterative linear solvers often have to be used and then this appealing advantage becomes not obvious. In this paper we systematically investigate numerical performance of the ensemble method with block iterative solvers for two typical evolution problems: the heat equation and the incompressible Navier-Stokes equations. In particular, the block conjugate gradient (CG) solver is considered for the former and the block generalized minimal residual (GMRES) solver for the latter. Our numerical results demonstrate the effectiveness and efficiency of the ensemble method when working together with these block iterative solvers.