# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

September 2019 , Volume 24 , Issue 9

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2019, 24(9): 4665-4684 doi: 10.3934/dcdsb.2018328 +[Abstract](1302) +[HTML](474) +[PDF](485.06KB)
Abstract:

In this paper we develop a new and convenient technique, with fractional Gagliardo-Nirenberg type inequalities inter alia involved, to treat the quasilinear fully parabolic chemotaxis system with indirect signal production: \begin{document}$u_t = \nabla\cdot(D(u)\nabla u-S(u)\nabla v)$\end{document}, \begin{document}$\tau_1v_t = \Delta v-a_1v+b_1w$\end{document}, \begin{document}$\tau_2w_t = \Delta w-a_2w+b_2u$\end{document}, under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$\Omega\subset\Bbb{R}^{n}$\end{document} (\begin{document}$n\geq 1$\end{document}), where \begin{document}$\tau_i,a_i,b_i>0$\end{document} (\begin{document}$i = 1,2$\end{document}) are constants, and the diffusivity \begin{document}$D$\end{document} and the density-dependent sensitivity \begin{document}$S$\end{document} satisfy \begin{document}$D(s)\geq a_0(s+1)^{-\alpha}$\end{document} and \begin{document}$0\leq S(s)\leq b_0(s+1)^{\beta}$\end{document} for all \begin{document}$s\geq 0$\end{document} with \begin{document}$a_0,b_0>0$\end{document} and \begin{document}$\alpha,\beta\in\Bbb R$\end{document}. It is proved that if \begin{document}$\alpha+\beta<3$\end{document} and \begin{document}$n = 1$\end{document}, or \begin{document}$\alpha+\beta<4/n$\end{document} with \begin{document}$n\geq 2$\end{document}, for any properly regular initial data, this problem has a globally bounded and classical solution. Furthermore, consider the quasilinear attraction-repulsion chemotaxis model: \begin{document}$u_t = \nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla z)+\xi\nabla\cdot(u\nabla w)$\end{document}, \begin{document}$z_t = \Delta z-\rho z+\mu u$\end{document}, \begin{document}$w_t = \Delta w-\delta w+\gamma u$\end{document}, where \begin{document}$\chi,\mu,\xi,\gamma,\rho,\delta>0$\end{document}, and the diffusivity \begin{document}$D$\end{document} fulfills \begin{document}$D(s)\geq c_0(s+1)^{M-1}$\end{document} for any \begin{document}$s\geq 0$\end{document} with \begin{document}$c_0>0$\end{document} and \begin{document}$M\in\Bbb R$\end{document}. As a corollary of the aforementioned assertion, it is shown that when the repulsion cancels the attraction (i.e. \begin{document}$\chi\mu = \xi\gamma$\end{document}), the solution is globally bounded if \begin{document}$M>-1$\end{document} and \begin{document}$n = 1$\end{document}, or \begin{document}$M>2-4/n$\end{document} with \begin{document}$n\geq 2$\end{document}. This seems to be the first result for this quasilinear fully parabolic problem that genuinely concerns the contribution of repulsion.

2019, 24(9): 4685-4702 doi: 10.3934/dcdsb.2018329 +[Abstract](1230) +[HTML](494) +[PDF](467.91KB)
Abstract:

This paper is concerned with the following planar Schrödinger-Poisson system

where \begin{document}$V(x)$\end{document} and \begin{document}$f(x, u)$\end{document} are axially symmetric in \begin{document}$x$\end{document}, and \begin{document}$f(x, u)$\end{document} is asymptotically cubic or super-cubic in \begin{document}$u$\end{document}. With a different variational approach used in [S. Cingolani, T. Weth, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 33 (2016) 169-197], we obtain the existence of an axially symmetric Nehari-type ground state solution and a nontrivial solution for the above system. The axial symmetry is more general than radial symmetry, but less used in the literature, since the embedding from the space of axially symmetric functions to \begin{document}$L^s( \mathbb{R} ^N)$\end{document} is not compact. Our results generalize previous ones in the literature, and some of new phenomena do not occur in the corresponding problem for higher space dimensions.

2019, 24(9): 4703-4720 doi: 10.3934/dcdsb.2018330 +[Abstract](1070) +[HTML](590) +[PDF](786.11KB)
Abstract:

This paper deals with a reaction-diffusion model on a periodically and isotropically evolving domain in order to explore the diffusive dynamics of Aedes aegypti mosquito, where we divide it into two sub-populations: the winged population and an aquatic form. The spatial-temporal risk index \begin{document}$R_0(\rho)$\end{document} depending on the domain evolution rate \begin{document}$\rho(t)$\end{document} as well as its analytical properties is investigated. The long-time behaviors of the periodic solutions under the condition \begin{document}$R_0(\rho)>1$\end{document} and \begin{document}$R_0(\rho)\leq1$\end{document} are explored, respectively. Moreover, we consider the specific case where \begin{document}$\rho(t)\equiv1$\end{document} to better understand the impact of the periodic evolution rate on the persistence and extinction of Aedes aegypti mosquito. Numerical simulations further verify our analytical results that the periodic domain evolution has a significant impact on the dispersal of Aedes aegypti mosquito.

2019, 24(9): 4721-4737 doi: 10.3934/dcdsb.2019027 +[Abstract](1331) +[HTML](414) +[PDF](455.3KB)
Abstract:

In this paper, we consider the initial boundary problem for the Kirchhoff type wave equation. We prove that the Kirchhoff wave model is globally well-posed in the sufficiently regular space \begin{document}$(H^2(\Omega)\cap H^1_0(\Omega))\times H^1_0(\Omega)$\end{document}, then, we also obtain that the semigroup generated by the equation has a global attractor in the corresponding phase space, in the presence of a quite general nonlinearity of supercritical growth.

2019, 24(9): 4739-4753 doi: 10.3934/dcdsb.2019028 +[Abstract](1005) +[HTML](394) +[PDF](1051.13KB)
Abstract:

Vegetation and plateau pika are two key species in alpine meadow ecosystems on the Tibetan Plateau. It is frequently observed on the field that plateau pika reduces the carrying capacity of vegetation and the mortality of plateau pika increases along with the increasing height of vegetation. This motivates us to propose and study a predator-prey model with state-dependent carrying capacity. Theoretical analysis and numerical simulations show that the model exhibits complex dynamics including the occurrence of saddle-node bifurcation, transcritical bifurcation and Hopf bifurcation, and the coexistence of two stable equilibria.

2019, 24(9): 4755-4782 doi: 10.3934/dcdsb.2019029 +[Abstract](1054) +[HTML](375) +[PDF](9860.41KB)
Abstract:

The existence of weak solutions and upper bounds for the blow-up time for time-discrete parabolic-elliptic Keller-Segel models for chemotaxis in the two-dimensional whole space are proved. For various time discretizations, including the implicit Euler, BDF, and Runge-Kutta methods, the same bounds for the blow-up time as in the continuous case are derived by discrete versions of the virial argument. The theoretical results are illustrated by numerical simulations using an upwind finite-element method combined with second-order time discretizations.

Hui li and
2019, 24(9): 4783-4797 doi: 10.3934/dcdsb.2019030 +[Abstract](1115) +[HTML](400) +[PDF](440.26KB)
Abstract:

This paper is devoted to investigate a virus infection model with a spatially heterogeneous structure and nonlinear diffusion. First we establish the properties of the basic reproduction number \begin{document}$R_0$\end{document} for infected cells and free virus particles. Then we prove that the comparison principle can be applied to an auxiliary system with quasilinear diffusion under appropriate conditions. Then the sufficient conditions for the globally asymptotical stability of infection-free steady state are obtained, which indicates that \begin{document}$R_0<1$\end{document} is necessary for infected cells and free virus particles to be extinct. Next we prove the existence of positive non-constant steady states and the persistence of infected cells and free virion where \begin{document}$R_0>1$\end{document} is required. Finally, it is shown that, for the spatially homogeneous case when the infected cells rate of change of the repulsive effect is small enough, \begin{document}$R_0$\end{document} is the only determinant of the global dynamics of the underlying virus infection system. The obtained results give an insight into the optimal control of the virion.

2019, 24(9): 4799-4813 doi: 10.3934/dcdsb.2019031 +[Abstract](903) +[HTML](372) +[PDF](421.08KB)
Abstract:

In this work, we develop an efficient spectral method to solve the Helmholtz transmission eigenvalue problem in polar geometries. An essential difficulty is that the polar coordinate transformation introduces the polar singularities. In order to overcome this difficulty, we introduce some pole conditions and the corresponding weighted Sobolev space. The polar coordinate transformation and variable separation techniques are presented to transform the original problem into a series of equivalent one-dimensional eigenvalue problem, and error estimate for the approximate eigenvalues and corresponding eigenfunctions are obtained. Finally, numerical simulations are performed to confirm the validity of the numerical method.

2019, 24(9): 4815-4826 doi: 10.3934/dcdsb.2019032 +[Abstract](1081) +[HTML](480) +[PDF](1696.87KB)
Abstract:

This article aims to contribute to the understanding of the curvature flow of curves in a higher-dimensional space. Evolution of curves in \begin{document}$\mathbb{R}^m$\end{document} by their curvature is compared to the motion of hypersurfaces with constrained normal velocity. The special case of shrinking hyperspheres is further analyzed both theoretically and numerically by means of a semi-discrete scheme with discretization based on osculating circles. Computational examples of evolving spherical curves are provided along with the measurement of the experimental order of convergence.

2019, 24(9): 4827-4849 doi: 10.3934/dcdsb.2019033 +[Abstract](909) +[HTML](423) +[PDF](3729.43KB)
Abstract:

Many studies of mathematical epidemiology assume that model parameters are precisely known. However, they can be imprecise due to various uncertainties. Deterministic epidemic models are also subjected to stochastic perturbations. In this paper, we analyze a stochastic SIRS model that includes interval parameters and environmental noises. We define the stochastic basic reproduction number, which is shown to govern disease extinction or persistence. When it is less than one, the disease is predicted to die out with probability one. When it is greater than one, the model admits a stationary distribution. Thus, larger stochastic noises (resulting in a smaller stochastic basic reproduction number) are able to suppress the emergence of disease outbreaks. Using numerical simulations, we also investigate the influence of parameter imprecision and susceptible response to the disease information that may change individual behavior and protect the susceptible from infection. These parameters can greatly affect the long-term behavior of the system, highlighting the importance of incorporating parameter imprecision into epidemic models and the role of information intervention in the control of infectious diseases.

2019, 24(9): 4851-4861 doi: 10.3934/dcdsb.2019034 +[Abstract](891) +[HTML](387) +[PDF](841.39KB)
Abstract:

The T-singularity (invisible two-fold singularity) is one of the most intriguing objects in the study of 3D piecewise smooth vector fields. The occurrence of just one T-singularity already arouses the curiosity of experts in the area due to the wealth of behaviors that may arise in its neighborhood. In this work we show the birth of an arbitrary number, including infinite, of such singularities. Moreover, we are able to show the existence of an arbitrary number of limit cycles, hyperbolic or not, surrounding each one of these singularities.

2019, 24(9): 4863-4897 doi: 10.3934/dcdsb.2019035 +[Abstract](898) +[HTML](391) +[PDF](593.18KB)
Abstract:

This paper is concerned with the mathematical analysis of a coupled elliptic-parabolic system modeling the interaction between the propagation of electric potential coupled with general physiological ionic models and subsequent deformation of the cardiac tissue. A prototype system belonging to this class is provided by the electromechanical bidomain model, which is frequently used to study and simulate electrophysiological waves in cardiac tissue. The coupling between muscle contraction, biochemical reactions and electric activity is introduced with a so-called active strain decomposition framework, where the material gradient of deformation is split into an active (electrophysiology-dependent) part and an elastic (passive) one. We prove existence of weak solutions to the underlying coupled electromechanical bidomain model under the assumption of linearized elastic behavior and a truncation of the updated nonlinear diffusivities. The proof of the existence result, which constitutes the main thrust of this paper, is proved by means of a non-degenerate approximation system, the Faedo-Galerkin method, and the compactness method.

2019, 24(9): 4899-4912 doi: 10.3934/dcdsb.2019036 +[Abstract](1000) +[HTML](353) +[PDF](433.74KB)
Abstract:

The paper investigates the upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with structural damping: \begin{document}$u_{tt}-M(\|\nabla u\|^2)\Delta u+(-\Delta)^\alpha u_t+f(u) = g(x,t)$\end{document}, where \begin{document}$\alpha\in(1/2, 1)$\end{document} is said to be a dissipative index. It shows that when the nonlinearity \begin{document}$f(u)$\end{document} is of supercritical growth \begin{document}$p: 1 \leq p< p_{\alpha}\equiv\frac{N+4\alpha}{(N-4\alpha)^+}$\end{document}, the related evolution process has a pullback attractor for each \begin{document}$\alpha\in(1/2, 1)$\end{document}, and the family of pullback attractors is upper semicontinuous with respect to \begin{document}$\alpha$\end{document}. These results extend those in [27] for autonomous Kirchhoff wave models.

2019, 24(9): 4913-4928 doi: 10.3934/dcdsb.2019037 +[Abstract](966) +[HTML](374) +[PDF](460.42KB)
Abstract:

We consider a two-species Lotka-Volterra weak competition model in a one-dimensional advective homogeneous environment, where individuals are exposed to unidirectional flow. It is assumed that two species have the same population dynamics but different diffusion rates, advection rates and intensities of competition. We study the following useful scenarios: (1) if one species disperses by random diffusion only and the other assumes both random and unidirectional movements, two species will coexist; (2) if two species are drifting along the different direction, two species will coexist; (3) if the intensities of inter-specific competition are small enough, two species will coexist; (4) if the intensities of inter-specific competition are close to 1, the competitive exclusion principle holds. These results provide a new mechanism for the coexistence of competing species. Finally, we apply a perturbation argument to illustrate that two species will converge to the unique coexistence steady state.

2019, 24(9): 4929-4936 doi: 10.3934/dcdsb.2019038 +[Abstract](1037) +[HTML](340) +[PDF](329.05KB)
Abstract:

In this paper, we investigate the effects of nonlocal dispersal and spatial heterogeneity on the total biomass of species via nonlocal dispersal logistic equations. In order to make the model more relevant for real biological systems, we consider a logistic reaction term, with two parameters, \begin{document}$r(x)$\end{document} for intrinsic growth rate and \begin{document}$K(x)$\end{document} for carrying capacity. We first establish the existence, uniqueness and asymptotic stability of the positive steady state solution for this equation. And then we study the continuous property and asymptotic limit of the positive steady state solution with respect to the dispersal rate. Finally, the function about the total biomass of species is defined by the positive steady state solution. Our results show in a heterogeneous environment, the total biomass is always strictly greater than the total carrying capacity in the special case when the nonlocal dispersal is allowed.

2019, 24(9): 4937-4954 doi: 10.3934/dcdsb.2019039 +[Abstract](1453) +[HTML](446) +[PDF](433.24KB)
Abstract:

In this paper, we study the averaging principle for multivalued SDEs with jumps and non-Lipschitz coefficients. By the Bihari's inequality and the properties of the concave function, we prove that the solution of averaged multivalued SDE with jumps converges to that of the standard one in the sense of mean square and also in probability. Finally, two examples are presented to illustrate our theory.

2019, 24(9): 4955-4981 doi: 10.3934/dcdsb.2019040 +[Abstract](982) +[HTML](372) +[PDF](1850.13KB)
Abstract:

Lyapunov functions are functions with negative derivative along solutions of a given ordinary differential equation. Moreover, sub-level sets of a Lyapunov function are subsets of the domain of attraction of the equilibrium. One of the numerical construction methods for Lyapunov functions uses meshfree collocation with radial basis functions (RBF). In this paper, we propose two verification estimates combined with this RBF construction method to ensure that the constructed function is a Lyapunov function. We show that this combination of the RBF construction method and the verification estimates always succeeds in constructing and verifying a Lyapunov function for nonlinear ODEs in \begin{document}$\mathbb{R}^d$\end{document} with an exponentially stable equilibrium.

2019, 24(9): 4983-5001 doi: 10.3934/dcdsb.2019041 +[Abstract](1087) +[HTML](465) +[PDF](2735.89KB)
Abstract:

SIR models with directed diffusions are important in describing the population movement. However, efficient numerical simulations of such systems of fully nonlinear second order partial differential equations (PDEs) are challenging. They are often mixed type PDEs with ill-posed or degenerate components. The solutions may develop singularities along with time evolution. Stiffness due to nonlinear diffusions in the system gives strict constraints in time step sizes for numerical methods. In this paper, we design efficient Krylov implicit integration factor (IIF) Weighted Essentially Non-Oscillatory (WENO) method to solve SIR models with directed diffusions. Numerical experiments are performed to show the good accuracy and stability of the method. Singularities in the solutions are resolved stably and sharply by the WENO approximations in the scheme. Unlike a usual implicit method for solving stiff nonlinear PDEs, the Krylov IIF WENO method avoids solving large coupled nonlinear algebraic systems at every time step. Large time step size computations are achieved for solving the fully nonlinear second-order PDEs, namely, the time step size is proportional to the spatial grid size as that for solving a pure hyperbolic PDE. Two biologically interesting cases are simulated by the developed scheme to study the finite-time blow-up time and location or discontinuity locations in the solution of the SIR model.

2019, 24(9): 5003-5039 doi: 10.3934/dcdsb.2019042 +[Abstract](1074) +[HTML](376) +[PDF](1170.5KB)
Abstract:

We study a modified version of a prey-predator system with modified Leslie-Gower and Holling type Ⅱ functional responses studied by M.A. Aziz-Alaoui and M. Daher-Okiye. The modification consists in incorporating a refuge for preys, and substantially complicates the dynamics of the system. We study the local and global dynamics and the existence of cycles. We also investigate conditions for extinction or existence of a stationary distribution, in the case of a stochastic perturbation of the system.

2019, 24(9): 5041-5081 doi: 10.3934/dcdsb.2019043 +[Abstract](703) +[HTML](348) +[PDF](1176.3KB)
Abstract:

A number of empirical and theoretical studies shows that the exploitation of fish sources has benefitted a lot from artificial floating objects (abbr. FOBs) on the surface of ocean. In this paper we investigate the dynamical distribution in aggregations of tuna around two FOBs. We abandon the effort of precise computation for steady states and eigenvalues but utilize the monotonic intervals to determine the location of zeros and signs of eigenvalues qualitatively and use the symmetry of AS steady states to simplify the system. Our method enables us to find two more steady states than known results and complete the analysis of all steady states effectively. Furthermore, we display all bifurcations at steady states, including six bifurcations of co-dimension 1 and two bifurcations of co-dimension 2. One of bifurcations is a degenerate pitchfork bifurcation of co-dimension 4 but only a part of co-dimension 2 can be unfolded within the system. We construct sectorial regions to prove the nonexistence of closed orbits. Those results provide long-time prediction of steady numbers of tuna around the two FOBs and critical conditions for transitions of cases.

2019, 24(9): 5083-5105 doi: 10.3934/dcdsb.2019044 +[Abstract](791) +[HTML](415) +[PDF](468.99KB)
Abstract:

In this paper we consider an optimal dividend problem for an insurance company whose surplus process evolves a classical \begin{document}${\rm Cram\acute{e}r}$\end{document}-Lundberg process. We impose a varied bound over the dividend rate to raise the dividend payment at a acceptable survival probability. Our objective is to find a strategy consisting of both investment and dividend payment which maximizes the cumulative expected discounted dividend payment until the ruin time. We show that the optimal value function is a unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with a given boundary condition. We characterize the optimal value function as the smallest viscosity supersolution of the HJB equation. We introduce a method to construct the potential solution of our problem and give a verification theorem to check its optimality. Finally we show some numerical results.

2019, 24(9): 5107-5120 doi: 10.3934/dcdsb.2019045 +[Abstract](891) +[HTML](324) +[PDF](478.96KB)
Abstract:

The main objective of this paper is to investigate the superfluidity phase transition theory-modeling and analysis-for liquid \begin{document}$^{4}$\end{document}He system. Based on the new Gibbs free energy and the potential-descending principle proposed recently in [18,25], the dynamic equations describing the \begin{document}$\lambda$\end{document}-transition and solid-liquid transition of liquid \begin{document}$^{4}$\end{document}He system are derived. Further, by the dynamical transition theory, the two obtained models are proven to exhibit Ehrenfest second-order transition and first-order transition, respectively, which are well consistent with the physical experimental results.

2019, 24(9): 5121-5148 doi: 10.3934/dcdsb.2019046 +[Abstract](959) +[HTML](359) +[PDF](524.82KB)
Abstract:

In this paper, we investigate the long term behavior of the solutions to a class of stochastic discrete complex Ginzburg-Landau equations with time-varying delays and driven by multiplicative white noise. We first prove the existence and uniqueness of random attractor in a weighted space for these equations. Then, we analyze the upper semicontinuity of the random attractors as the time delay approaches to zero.

Xueke Pu and
2019, 24(9): 5149-5181 doi: 10.3934/dcdsb.2019055 +[Abstract](548) +[HTML](267) +[PDF](567.08KB)
Abstract:

In this paper, we study the asymptotic behaviors for the quantum Navier-Stokes-Maxwell equations with general initial data in a torus \begin{document}$\mathbb{T}^{3}$\end{document}. Based on the local existence theory, we prove the convergence of strong solutions for the full compressible quantum Navier-Stokes-Maxwell equations towards those for the incompressible e-MHD equations plus the fast singular oscillating in time of the sequence of solutions as the Debye length goes to zero. We also mention that similar arguments can be applied to the Euler-Maxwell system. Remarkably, we eliminate the highly oscillating terms produced by the general initial data by using the formal two-timing method. Moreover, using the curl-div decomposition and elaborate energy estimates, we derive uniform (in the Debye length) estimates for the remainder system.

2019, 24(9): 5183-5201 doi: 10.3934/dcdsb.2019056 +[Abstract](671) +[HTML](258) +[PDF](729.81KB)
Abstract:

We study the synchronization of fully-connected and totally excitatory integrate and fire neural networks in presence of Gaussian white noises. Using a large deviation principle, we prove the stability of the synchronized state under stochastic perturbations. Then, we give a lower bound on the probability of synchronization for networks which are not initially synchronized. This bound shows the robustness of the emergence of synchronization in presence of small stochastic perturbations.

2019, 24(9): 5203-5224 doi: 10.3934/dcdsb.2019129 +[Abstract](256) +[HTML](123) +[PDF](563.26KB)
Abstract:

In this paper, a glycolysis model subject to no-flux boundary condition is considered. First, by discussing the corresponding characteristic equation, the stability of constant steady state solution is discussed, and the Turing's instability is shown. Next, based on Lyapunov-Schmidt reduction method and singularity theory, the multiple stationary bifurcations with singularity are analyzed. In particular, under no-flux boundary condition we show the existence of nonconstant steady state solution bifurcating from a double zero eigenvalue, which is always excluded in most existing works. Also, the stability, bifurcation direction and multiplicity of the bifurcation steady state solutions are investigated by the singularity theory. Finally, the theoretical results are confirmed by numerical simulations. It is also shown that there is no Hopf bifurcation on basis of the condition \begin{document}$(C)$\end{document}.

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