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1531-3492

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1553-524X

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## Discrete & Continuous Dynamical Systems - B

October 2016 , Volume 21 , Issue 8

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2016, 21(8): 2379-2407
doi: 10.3934/dcdsb.2016052

*+*[Abstract](1052)*+*[PDF](632.2KB)**Abstract:**

In this work, we present some new Carleman inequalities for Stokes and Oseen equations with non-homogeneous boundary conditions. These estimates lead to log type stability inequalities for the problem of recovering the solution of the Stokes and Navier-Stokes equations from both boundary and distributed observations. These inequalities fit the well-known unique continuation result of Fabre and Lebeau [18]: the distributed observation only depends on interior measurement of the velocity, and the boundary observation only depends on the trace of the velocity and of the Cauchy stress tensor measurements. Finally, we present two applications for such inequalities. First, we apply these estimates to obtain stability inequalities for the inverse problem of recovering Navier or Robin boundary coefficients from boundary measurements. Next, we use these estimates to deduce the rate of convergence of two reconstruction methods of the Stokes solution from the measurement of Cauchy data: a quasi-reversibility method and a penalized Kohn-Vogelius method.

2016, 21(8): 2409-2422
doi: 10.3934/dcdsb.2016053

*+*[Abstract](938)*+*[PDF](1188.0KB)**Abstract:**

We consider the class of two-lag linear delay differential equations and develop a series expansion to solve for the roots of the nonlinear characteristic equation. The expansion draws on results from complex analysis, combinatorics, special functions, and classical analysis for differential equations. Supporting numerical results are presented along with application of our method to study the stability of a two-lag model from ecology.

2016, 21(8): 2423-2449
doi: 10.3934/dcdsb.2016054

*+*[Abstract](1130)*+*[PDF](1382.7KB)**Abstract:**

West Nile virus (WNV) was first detected in the United States (U.S.) during an outbreak in New York City in 1999 with 62 human cases including seven deaths. In 2001, the first human case in Florida was identified, and in Texas and California it was 2002 and 2004, respectively. WNV has now been spread to almost all states in the US. In 2015, the Center for Disease Control and Prevention (CDC) reported 2,175 human cases, including 146 deaths, from 45 states. WNV is maintained in a cycle between mosquitoes and animal hosts in which birds are the predominant and preferred reservoirs while most mammals, including humans, are considered dead-end hosts, as they do not appear to develop high enough titers of WNV in the blood to infect mosquitoes. In this article, we propose a deterministic model by including interactions among mosquitoes, birds, and humans to study the local transmission dynamics of WNV. To validate the model, it is used to simulate the WNV human data of infected cases and accumulative deaths from 1999 to 2013 in the states of New York, Florida, Texas, and California as reported to the CDC. These simulations demonstrate that the epidemic of WNV in New York, Texas, and California (and thus in the U.S.) has not reached its equilibrium yet and may be expected to get worse if the current control strategies are not enhanced. Mathematical and numerical analyses of the model are carried out to understand the transmission dynamics of WNV and explore effective control measures for the local outbreaks of the disease. Our studies suggest that the larval mosquito control measure should be taken as early as possible in a season to control the mosquito population size and the adult mosquito control measure is necessary to prevent the transmission of WNV from mosquitoes to birds and humans.

2016, 21(8): 2451-2472
doi: 10.3934/dcdsb.2016055

*+*[Abstract](1015)*+*[PDF](523.2KB)**Abstract:**

We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.

2016, 21(8): 2473-2489
doi: 10.3934/dcdsb.2016056

*+*[Abstract](932)*+*[PDF](404.7KB)**Abstract:**

Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are a highly tumorigenic cell type found in developmentally diverse tumors that are believed to be resistant to standard chemotherapeutic drugs and responsible for tumor recurrence. Thus understanding the tumor growth kinetics is critical for development of novel strategies for cancer treatment. In this paper, the moment stability of nonlinear stochastic systems of breast cancer stem cells with time-delays has been investigated. First, based on the technique of the variation- of-constants formula, we obtain the second order moment equations for the nonlinear stochastic systems of breast cancer stem cells with time-delays. By the comparison principle along with the established moment equations, we can get the comparative systems of the nonlinear stochastic systems of breast cancer stem cells with time-delays. Then moment stability theorems have been established for the systems with the stability properties for the comparative systems. Based on the linear matrix inequality (LMI) technique, we next obtain a criteria for the exponential stability in mean square of the nonlinear stochastic systems for the dynamics of breast cancer stem cells with time-delays. Finally, some numerical examples are presented to illustrate the efficiency of the results.

2016, 21(8): 2491-2507
doi: 10.3934/dcdsb.2016057

*+*[Abstract](840)*+*[PDF](425.2KB)**Abstract:**

This paper considers the stabilization of a wave equation with interior input delay: $\mu_1u(x,t)+\mu_2u(x,t-\tau)$, where $u(x,t)$ is the control input. A new dynamic feedback control law is obtained to stabilize the closed-loop system exponentially for any time delay $\tau>0$ provided that $|\mu_1|\neq|\mu_2|$. Moreover, some sufficient conditions are given for discriminating the asymptotic stability and instability of the closed-loop system.

2016, 21(8): 2509-2530
doi: 10.3934/dcdsb.2016058

*+*[Abstract](919)*+*[PDF](395.8KB)**Abstract:**

Fractional diffusion equations are used for mass spreading in inhomogeneous media. They are applied to model anomalous diffusion, where a cloud of particles spreads in a different manner than the classical diffusion equation predicts. Thus, they involve fractional derivatives. Here we present a continuous variant of Grünwald-Letnikov's formula, which is useful to compute the flux of particles performing random walks, allowing for heavy-tailed jump distributions. In fact, we set a definition of fractional derivatives yielding the operators which enable us to retrieve the space fractional variant of Fick's law, for enhanced diffusion in disordered media, without passing through any partial differential equation for the space and time evolution of the concentration.

2016, 21(8): 2531-2550
doi: 10.3934/dcdsb.2016059

*+*[Abstract](953)*+*[PDF](525.7KB)**Abstract:**

We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature.

2016, 21(8): 2551-2566
doi: 10.3934/dcdsb.2016060

*+*[Abstract](1060)*+*[PDF](499.9KB)**Abstract:**

The present paper provides a mathematical analysis of the model of intracellular protein dynamics proposed in [14]. The model describes protein and mRNA transport inside a cell and takes into account diffusion in the nucleus and cytoplasm as well as active transport of protein molecules along microtubules in the cytoplasm. The model is a complex system of nonlinear PDEs with appropriate boundary conditions. The model reproduces, at least in numerical simulations, the oscillatory changes in protein concentration observed in the experimental data. To our knowledge this is the first paper that, in the multidimensional case, deals with a rigorous mathematical analysis of a model of intracellular dynamics with active transport on microtubules. In particular, in the present paper, we prove the existence and uniqueness result for the model in arbitrary space dimension. The model may be adapted to other signaling pathways.

2016, 21(8): 2567-2585
doi: 10.3934/dcdsb.2016061

*+*[Abstract](828)*+*[PDF](913.1KB)**Abstract:**

In this paper, we study the effect of chemotactic movement of CTLs on HIV-1 infection dynamics by a reaction diffusion system with chemotaxis. Choosing a typical chemosensitive function, we find that

*chemoattractive*movement of CTLs due to HIV infection does not change stability of a positive steady state of the model, meaning that the stability/instability of the model remains the same as in the model without spatial effect. However,

*chemorepulsion*movement of CTLs can destabilize the positive steady state as the strength of the chemotactic sensitivity increases. In this case, Turing instability occurs, which may result in Hopf bifurcation or steady state bifurcation, and spatial inhomogeneous pattern forms.

2016, 21(8): 2587-2599
doi: 10.3934/dcdsb.2016062

*+*[Abstract](1114)*+*[PDF](1822.6KB)**Abstract:**

This paper studies the flocking behavior in a new discrete-time Cucker-Smale model under hierarchical leadership. The features of this model are that each individual has its own intrinsic nonlinear dynamics and the interaction between individuals follows a hierarchical leadership structure. Based on a specific matrix norm, we prove that the conditional flocking indeed occurs. Numerical experiments are given to confirm the theoretical results.

2016, 21(8): 2601-2614
doi: 10.3934/dcdsb.2016063

*+*[Abstract](851)*+*[PDF](555.4KB)**Abstract:**

The main purpose of this article is to investigate the phase transition of oscillation solutions and travelling wave solutions in a class of relaxation systems as follows \begin{eqnarray} \left\{ \begin{array}{ll} \frac{\partial u}{\partial t}=\pm u(u-a)(u-b)-v+D \frac{\partial ^{2}u}{\partial^{2} x},~~~a\neq b, \\\frac{\partial v}{\partial t}=\varepsilon( mu + nv + p ), ~~~~0<\varepsilon\ll 1,\nonumber \end{array} \right. \end{eqnarray} where $a,b,m,n,p$ are parameters in this system. By using the orbit analysis method of planar dynamical system and the homoclinic bifurcation theory, the phase transitions of the solitary oscillators, kink oscillators, periodic oscillators and travelling waves in the relaxation system above are studied. Various critical parameters of the phase transition are obtained under different parametric conditions, while various sufficient conditions to guarantee the existence of the above oscillation solutions and travelling waves are given. As some applications, this paper studied the FitzHugh-Nagumo equation, the van der Pol-equation and the Winfree generic system.

2016, 21(8): 2615-2630
doi: 10.3934/dcdsb.2016064

*+*[Abstract](951)*+*[PDF](426.2KB)**Abstract:**

The aim of this paper is to investigate the threshold dynamics of a heroin epidemic in heterogeneous populations. The model is described by a delayed multi-group model, which allows us to model interactions both within-group and inter-group separately. Here we are able to prove the existence of heroin-spread equilibrium and the uniform persistence of the model. The proofs of main results come from suitable applications of graph-theoretic approach to the method of Lyapunov functionals and Krichhoff's matrix tree theorem. Numerical simulations are performed to support the results of the model for the case where $n=2$.

2016, 21(8): 2631-2648
doi: 10.3934/dcdsb.2016065

*+*[Abstract](918)*+*[PDF](481.6KB)**Abstract:**

In this paper, we first establish the global well-posedness of strong solutions of the simplified Ericksen-Leslie model for nonhomogeneous incompressible nematic liquid crystal flows in dimensions two, if the initial data satisfies some smallness condition. It is worth pointing out that the initial density is allowed to contain vacuum states and the initial velocity can be arbitrarily large. Next, we present a Serrin's type criterion, depending only on $\nabla d$, for the breakdown of local strong solutions. As a byproduct, the global strong solutions with large initial data are obtained, provided the macroscopic molecular orientation of the liquid crystal materials satisfies a natural geometric angle condition (cf. [19]).

2016, 21(8): 2649-2662
doi: 10.3934/dcdsb.2016066

*+*[Abstract](802)*+*[PDF](379.2KB)**Abstract:**

Let $B_1$ be the unit ball in $\mathbb{R}^N$ with $N \geq 2$. Let $f\in C^1([0, \infty), \mathbb{R})$, $f(0)=0$, $f(\beta) = \beta, \ f(s) < s$ for $s\in (0,\beta), \ f(s) > s$ for $s \in (\beta, \infty)$ and $f'(\beta)>\lambda^{r}_k$. D. Bonheure, B. Noris and T. Weth [Ann. Inst. H. Poincaré Anal. Non Linéaire 29(4) (2012)] proved the existence of nondecreasing, radial positive solutions of the semilinear Neumann problem $$ -\Delta u+u=f(u) \ \text{in}\ B_1,\ \ \ \ \partial_\nu u=0 \ \text{on}\ \partial B_1 $$ for $k=2$, and they conjectured that there exists a radial solution with $k$ intersections with $\beta$ provided that $f'(\beta) >\lambda^r_k$ for $k>2$, where $\lambda^r_k$ is the $k$-th radial eigenvalue of $\Delta + I$ in the unit ball with Neumann boundary conditions. In this paper, we show that the answer is yes in the case of linearly bounded nonlinearities.

2016, 21(8): 2663-2685
doi: 10.3934/dcdsb.2016067

*+*[Abstract](1216)*+*[PDF](438.7KB)**Abstract:**

In this article, we study a coupled Cahn-Hilliard-Navier-Stokes model with delays in a two-dimensional domain. The model consists of the Navier-Stokes equations for the velocity, coupled with an Cahn-Hilliard model for the order (phase) parameter. We prove the existence and uniqueness of the weak and strong solution when the external force contains some delays. We also discuss the asymptotic behavior of the weak solutions and the stability of the stationary solutions.

2016, 21(8): 2687-2702
doi: 10.3934/dcdsb.2016068

*+*[Abstract](819)*+*[PDF](372.6KB)**Abstract:**

In this paper, we prove the existence of a pullback attractor in higher regularity space for the multi-valued process associated with the $2D$-non-autonomous incompressible non-Newtonian fluid with delays and without the uniqueness of solutions.

2016, 21(8): 2703-2728
doi: 10.3934/dcdsb.2016069

*+*[Abstract](951)*+*[PDF](979.3KB)**Abstract:**

Understanding the plankton dynamics can help us take effective measures to settle the critical issue on how to keep plankton ecosystem balance. In this paper, a nutrient-phytoplankton-zooplankton (NPZ) model is formulated to understand the mechanism of plankton dynamics. To account for the harmful effect of the phytoplankton allelopathy, a prototype for a non-monotone response function is used to model zooplankton grazing, and nonlinear phytoplankton mortality is also included in the NPZ model. Using the model, we will focus on understanding how the phytoplankton allelopathy and nonlinear phytoplankton mortality affect the plankton population dynamics. We first examine the existence of multiple equilibria and provide a detailed classification for the equilibria, then stability and local bifurcation analysis are also studied. Sufficient conditions for Hopf bifurcation and zero-Hopf bifurcation are given respectively. Numerical simulations are finally conducted to confirm and extend the analytic results. Both theoretical and numerical findings imply that the phytoplankton allelopathy and nonlinear phytoplankton mortality may lead to a rich variety of complex dynamics of the nutrient-plankton system. The results of this study suggest that the effects of the phytoplankton allelopathy and nonlinear phytoplankton mortality should receive more attention to understand the plankton dynamics.

2016, 21(8): 2729-2744
doi: 10.3934/dcdsb.2016070

*+*[Abstract](912)*+*[PDF](792.8KB)**Abstract:**

We propose and analyze a variation of the Euler scheme for state constrained ordinary differential inclusions under weak assumptions on the right-hand side and the state constraints. Convergence results are given for the space-continuous and the space-discrete versions of this scheme, and a numerical example illustrates in which sense these limits have to be interpreted.

2016, 21(8): 2745-2766
doi: 10.3934/dcdsb.2016071

*+*[Abstract](926)*+*[PDF](430.7KB)**Abstract:**

In this paper, we consider compressible Navier-Stokes-Korteweg (N-S-K) equations with more general pressure laws, that is the pressure $P$ is non-monotone. We prove the stability of weak solutions in the periodic domain $\Omega=\mathbb{T}^{N}$, when $N = 2,3$. Utilizing an interesting Sobolev inequality to tackle the complicated Korteweg term, we obtain the global existence of weak solutions in one dimensional case. Moreover, when the initial data is compactly supported in the whole space $\mathbb{R}$, we prove the compressible N-S-K equations will blow-up in finite time.

2016, 21(8): 2767-2784
doi: 10.3934/dcdsb.2016072

*+*[Abstract](907)*+*[PDF](419.6KB)**Abstract:**

In this paper, we study the Rayleigh-Taylor instability phenomena for two compressible, immiscible, inviscid, ideal polytropic fluids. Such two kind of fluids always evolve together with a free interface due to the uniform gravitation. We construct the steady-state solutions for the denser fluid lying above the light one. With an assumption on the steady-state temperature function, we find some growing solutions to the related linearized problem, which in turn demonstrates the linearized problem is ill-posed in the sense of Hadamard. By such an ill-posedness result, we can finally prove the solutions to the original nonlinear problem does not have the property EE(k). Precisely, the $H^3$ solutions to the original nonlinear problem can not Lipschitz continuously depend on their initial data.

2016, 21(8): 2785-2809
doi: 10.3934/dcdsb.2016073

*+*[Abstract](1077)*+*[PDF](2323.1KB)**Abstract:**

In this paper, we propose a general partial differential equation (PDE) model of cholera epidemics that extends previous mathematical cholera studies. Our new formation concerns the impact of the bacterial and human diffusion, bacterial convection, and their interaction with the intrinsic bacterial growth and multiple disease transmission pathways. A sensitivity analysis for a few key model parameters indicates the significance of diffusion and convection in shaping cholera epidemics. We then investigate the traveling wave solutions of our PDE model based on analytical derivation and numerical simulation, with a focus on the interplay of different biological, environmental and physical factors that determines the spatial spreading speeds of cholera. In addition, disease threshold dynamics are studied by computing the basic reproduction number associated with the PDE model, using both asymptotic analysis and numerical calculation.

2016, 21(8): 2811-2837
doi: 10.3934/dcdsb.2016074

*+*[Abstract](871)*+*[PDF](633.4KB)**Abstract:**

In this paper, we are concerned with a predator-prey model with Beddington-DeAngelis functional response in heterogeneous environment. By the bifurcation theory and some estimates, the global bifurcation of positive stationary solution is shown. Our result shows that new stationary patterns are produced by the spatial degeneracy and the Beddington-DeAngelis functional response. Essentially different from the known results, the two factors generate two critical values for the prey growth rate $\lambda.$ As $\lambda$ crosses each critical value, the positive stationary solution set undergoes a drastic change. In particular, when $\lambda$ is suitably large, the interaction between the two factors yields nonexistence of positive stationary solutions for any $\mu,$ which is in strong contrast to the existence for suitable ranges of $\mu$ corresponding to the Lotka-Volterra or Holling-II functional response. Moreover, which one of the two factors plays a dominating role in the stationary patterns is shown. In addition, we give the asymptotic behavior of the positive stationary solutions as $\mu\rightarrow\infty$. Finally, both uniqueness and multiplicity of the positive stationary solutions are shown as well as their stability.

2016, 21(8): 2839-2850
doi: 10.3934/dcdsb.2016075

*+*[Abstract](843)*+*[PDF](357.2KB)**Abstract:**

In this paper, we consider the derivation of the modified Wenzel's and Cassie's equations for wetting phenomena on rough surfaces from a three-dimensional phase field model. We derive an effective boundary condition by asymptotic two-scale homogenization technique when the size of the roughness is small. The modified Wenzel's and Cassie's equations for the apparent contact angles on the rough surfaces are then derived from the effective boundary condition. The homogenization results are proved rigorously by the $\Gamma$-convergence theory.

2016, 21(8): 2851-2866
doi: 10.3934/dcdsb.2016076

*+*[Abstract](1105)*+*[PDF](804.9KB)**Abstract:**

In this paper, a two-strain epidemic model on a complex network is proposed. The two strains are the drug-sensitive strain and the drug-resistant strain. The related basic reproduction numbers $R_s$ and $R_r$ are obtained. If $R_0=\max\{R_s,R_r\}<1$, then the disease-free equilibrium is globally asymptotically stable. If $R_r>1$, then there is a unique drug-resistant strain dominated equilibrium $E_r$, which is locally asymptotically stable if the invasion reproduction number $R_r^s<1$. If $R_s>1$ and $R_s>R_r$, then there is a unique coexistence equilibrium $E^*$. The persistence of the model is also proved. The theoretical results are supported with numerical simulations.

2016, 21(8): 2867-2881
doi: 10.3934/dcdsb.2016077

*+*[Abstract](1007)*+*[PDF](388.8KB)**Abstract:**

In this work, we consider the community of three species food web model with Lotka-Volterra type predator-prey interaction. In the absence of other species, each species follows the traditional logistical growth model and the top predator is an omnivore which is defined as feeding on the other two species. It can be seen as a model with one basal resource and two generalist predators, and pairwise interactions of all species are predator-prey type. It is well known that the omnivory module blends the attributes of several well-studied community modules, such as food chains (food chain models), exploitative competition (two predators-one prey models), and apparent competition (one predator-two preys models). With a mild biological restriction, we completely classify all parameters. All local dynamics and most parts of global dynamics are established corresponding to the classification. Moreover, the whole system is uniformly persistent when the unique coexistence appears. Finally, we conclude by discussing the strategy of inferior species to survive and the mechanism of uniform persistence for the three species ecosystem.

2016, 21(8): 2883-2903
doi: 10.3934/dcdsb.2016078

*+*[Abstract](889)*+*[PDF](663.5KB)**Abstract:**

This paper studies the bounded traveling wave solutions of MKdV-Burgers equation with the negative dispersive coefficient by the theory of planar dynamical systems, undetermined coefficients method. The global phase portraits under the different parameter conditions, as well as the existent number and conditions of the bounded traveling wave solutions are obtained for the dynamical system corresponding to the traveling wave solutions of MKdV-Burgers equation. The relation is investigated between the profiles of the bounded traveling wave solutions and dissipative coefficient. And a critical value characterizing the scale of dissipative effect, is given, which is different from the one proposed by R.F. Bikbaev in his article. Focusing on the open issue proposed by R.F. Bikbaev, based on the bell and kink profile solitary wave solutions of MKdV-Burgers equation we presented, approximate damped oscillatory solutions of MKdV-Burgers equation are obtained according to the evolution relation of orbits corresponding to the approximate damped oscillatory solutions in the global phase portraits.

2016, 21(8): 2905-2926
doi: 10.3934/dcdsb.2016079

*+*[Abstract](1226)*+*[PDF](1700.4KB)**Abstract:**

We study numerically the three-dimensional droplets spreading on physically flat chemically patterned surfaces with periodic squares separated by channels. Our model consists of the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions. Stick-slip behavior and contact angle hysteresis are observed. Moreover, we also study the relationship between the effective advancing/receding angle and the two intrinsic angles of the surface patterns. By increasing the volume of droplet gradually, we find that the advancing contact line tends gradually to an equiangular octagon with the length ratio of the two adjacent sides equal to a fixed value that depends on the geometry of the pattern.

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