All Issues

Volume 23, 2018

Volume 22, 2017

Volume 21, 2016

Volume 20, 2015

Volume 19, 2014

Volume 18, 2013

Volume 17, 2012

Volume 16, 2011

Volume 15, 2011

Volume 14, 2010

Volume 13, 2010

Volume 12, 2009

Volume 11, 2009

Volume 10, 2008

Volume 9, 2008

Volume 8, 2007

Volume 7, 2007

Volume 6, 2006

Volume 5, 2005

Volume 4, 2004

Volume 3, 2003

Volume 2, 2002

Volume 1, 2001

Centered around dynamics, DCDS-B is an interdisciplinary journal focusing on the interactions between mathematical modeling, analysis and scientific computations. The mission of the Journal is to bridge mathematics and sciences by publishing research papers that augment the fundamental ways we interpret, model and predict scientific phenomena. The Journal covers a broad range of areas including chemical, engineering, physical and life sciences. A more detailed indication is given by the subject interests of the members of the Editorial Board.

DCDS-B is edited by a global community of leading scientists to guarantee its high standards and a close link to the scientific and engineering communities. A unique feature of this journal is its streamlined review process and rapid publication. Authors are kept informed at all times throughout the process through the rapid, direct and personal communication between the authors and editors.

  • AIMS is a member of COPE. All AIMS journals adhere to the publication ethics and malpractice policies outlined by COPE.
  • Publishes 10 issues a year in January, March, May, June, July, August, September, October, November and December.
  • Publishes both online and in print.
  • Indexed in Science Citation Index, ISI Alerting Services, CompuMath Citation Index, Current Contents/Physics, Chemical, & Earth Sciences, INSPEC, Mathematical Reviews, MathSciNet, PASCAL/CNRS, Scopus, Web of Science and Zentralblatt MATH.
  • Archived in Portico and CLOCKSS.
  • DCDS-B is a publication of the American Institute of Mathematical Sciences. All rights reserved.

Note: “Most Cited” is by Cross-Ref , and “Most Downloaded” is based on available data in the new website.

Select all articles


Preface: "Structural Dynamical Systems: Computational aspects"
Nicoletta Del Buono, Cinzia Elia, Roberto Garrappa and Alessandro Pugliese
2018, 23(7) : i-i doi: 10.3934/dcdsb.201807i +[Abstract](67) +[HTML](22) +[PDF](92.02KB)
In memory of Timo Eirola, 1951-2016
Luca Dieci
2018, 23(7) : ⅲ-ⅲ doi: 10.3934/dcdsb.20187iii +[Abstract](107) +[HTML](41) +[PDF](63.5KB)
A comparison of boundary correction methods for Strang splitting
Lukas Einkemmer and Alexander Ostermann
2018, 23(7) : 2641-2660 doi: 10.3934/dcdsb.2018081 +[Abstract](435) +[HTML](329) +[PDF](571.3KB)

In this paper we investigate splitting methods in the presence of non-homogeneous boundary conditions. In particular, we consider the corrections that have been described and analyzed in Einkemmer, Ostermann 2015 and Alonso-Mallo, Cano, Reguera 2016. The latter method is extended to the non-linear case, and a rigorous convergence analysis is provided. We perform numerical simulations for diffusion-reaction, advection-reaction, and dispersion-reaction equations in order to evaluate the relative performance of these two corrections. Furthermore, we introduce an extension of both methods to obtain order three locally and evaluate under what circumstances this is beneficial.

Positive symplectic integrators for predator-prey dynamics
Fasma Diele and Carmela Marangi
2018, 23(7) : 2661-2678 doi: 10.3934/dcdsb.2017185 +[Abstract](1354) +[HTML](577) +[PDF](940.26KB)

We propose novel positive numerical integrators for approximating predator-prey models. The schemes are based on suitable symplectic procedures applied to the dynamical system written in terms of the log transformation of the original variables. Even if this approach is not new when dealing with Hamiltonian systems, it is of particular interest in population dynamics since the positivity of the approximation is ensured without any restriction on the temporal step size. When applied to separable M-systems, the resulting schemes are proved to be explicit, positive, Poisson maps. The approach is generalized to predator-prey dynamics which do not exhibit an M-system structure and successively to reaction-diffusion equations describing spatially extended dynamics. A classical polynomial Krylov approximation for the diffusive term joint with the proposed schemes for the reaction, allows us to propose numerical schemes which are explicit when applied to well established ecological models for predator-prey dynamics. Numerical simulations show that the considered approach provides results which outperform the numerical approximations found in recent literature.

Effect of perturbation in the numerical solution of fractional differential equations
Roberto Garrappa, Eleonora Messina and Antonia Vecchio
2018, 23(7) : 2679-2694 doi: 10.3934/dcdsb.2017188 +[Abstract](1153) +[HTML](596) +[PDF](446.48KB)

The equations describing engineering and real-life models are usually derived in an approximated way. Thus, in most cases it is necessary to deal with equations containing some kind of perturbation. In this paper we consider fractional differential equations and study the effects on the continuous and numerical solution, of perturbations on the given function, over long-time intervals. Some bounds on the global error are also determined.

On the stability of $\vartheta$-methods for stochastic Volterra integral equations
Dajana Conte, Raffaele D'Ambrosio and Beatrice Paternoster
2018, 23(7) : 2695-2708 doi: 10.3934/dcdsb.2018087 +[Abstract](430) +[HTML](331) +[PDF](584.98KB)

The paper is focused on the analysis of stability properties of a family of numerical methods designed for the numerical solution of stochastic Volterra integral equations. Stability properties are provided with respect to the basic test equation, as well as to the convolution test equation. For each equation, stability properties are intended both in the mean-square and in the asymptotic sense. Stability regions are also provided for a selection of methods. Numerical experiments confirming the theoretical study are also given.

Two-step collocation methods for fractional differential equations
Angelamaria Cardone, Dajana Conte and Beatrice Paternoster
2018, 23(7) : 2709-2725 doi: 10.3934/dcdsb.2018088 +[Abstract](503) +[HTML](297) +[PDF](465.33KB)

We propose two-step collocation methods for the numerical solution of fractional differential equations. These methods increase the order of convergence of one-step collocation methods, with the same number of collocation points. Moreover, they are continuous methods, i.e. they furnish an approximation of the solution at each point of the time interval. We describe the derivation of two-step collocation methods and analyse convergence. Some numerical experiments confirm theoretical expectations.

Pseudospectral reduction to compute Lyapunov exponents of delay differential equations
Dimitri Breda and Sara Della Schiava
2018, 23(7) : 2727-2741 doi: 10.3934/dcdsb.2018092 +[Abstract](452) +[HTML](366) +[PDF](1987.12KB)

A recent pseudospectral collocation is used to reduce a nonlinear delay differential equation to a system of ordinary differential equations. Standard methods are then applied to compute Lyapunov exponents. The validity of this simple approach is shown experimentally. Matlab codes are also included.

Periodic orbits of planar discontinuous system under discretization
Luca Dieci, Timo Eirola and Cinzia Elia
2018, 23(7) : 2743-2762 doi: 10.3934/dcdsb.2018103 +[Abstract](413) +[HTML](286) +[PDF](1253.06KB)

We consider a model planar system with discontinuous right-hand side possessing an attracting periodic orbit, and we investigate what happens to a Euler discretization with stepsize \begin{document}$ τ$\end{document} of this system. We show that, in general, the resulting discrete dynamical system does not possess an invariant curve, in sharp contrast to what happens for smooth problems. In our context, we show that the numerical trajectories are forced to remain inside a band, whose width is proportional to the discretization stepsize \begin{document}$ τ$\end{document}. We further show that if we consider an event-driven discretization of the model problem, whereby the solution is forced to step exactly on the discontinuity line, then there is a discrete periodic solution near the one of the original problem (for sufficiently small \begin{document}$ τ$\end{document}). Finally, we consider what happens to the Euler discretization of the regularized system rewritten in polar coordinates, and give numerical evidence that the discrete solution now undergoes a period doubling cascade with respect to the regularization parameter \begin{document}$ \epsilon$\end{document}, for fixed \begin{document}$ τ$\end{document}.

Numerical preservation of long-term dynamics by stochastic two-step methods
Raffaele D'Ambrosio, Martina Moccaldi and Beatrice Paternoster
2018, 23(7) : 2763-2773 doi: 10.3934/dcdsb.2018105 +[Abstract](420) +[HTML](276) +[PDF](328.86KB)

The paper aims to explore the long-term behaviour of stochastic two-step methods applied to a class of second order stochastic differential equations. In particular, the treatment focuses on preserving long-term statistics related to the dynamics of a linear stochastic damped oscillator whose velocity, in the stationary regime, is distributed as a Gaussian variable and uncorrelated with the position. By computing the solution of a very simple matrix equality, we a-priori determine the long-term statistics characterizing the numerical dynamics and analyze the behaviour of a selection of methods.

Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion
Anotida Madzvamuse and Raquel Barreira
2018, 23(7) : 2775-2801 doi: 10.3934/dcdsb.2018163 +[Abstract](231) +[HTML](146) +[PDF](4828.36KB)

In this article we present, for the first time, domain-growth induced pattern formation for reaction-diffusion systems with linear cross-diffusion on evolving domains and surfaces. Our major contribution is that by selecting parameter values from spaces induced by domain and surface evolution, patterns emerge only when domain growth is present. Such patterns do not exist in the absence of domain and surface evolution. In order to compute these domain-induced parameter spaces, linear stability theory is employed to establish the necessary conditions for domain-growth induced cross-diffusion-driven instability for reaction-diffusion systems with linear cross-diffusion. Model reaction-kinetic parameter values are then identified from parameter spaces induced by domain-growth only; these exist outside the classical standard Turing space on stationary domains and surfaces. To exhibit these patterns we employ the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces.

Partitioned second order method for magnetohydrodynamics in Elsässer variables
Yong Li and Catalin Trenchea
2018, 23(7) : 2803-2823 doi: 10.3934/dcdsb.2018106 +[Abstract](343) +[HTML](328) +[PDF](428.11KB)

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes equations coupled with Maxwell equations via Lorentz force and Ohm's law. Monolithic methods, which solve fully coupled MHD systems, are computationally expensive. Partitioned methods, on the other hand, decouple the full system and solve subproblems in parallel, and thus reduce the computational cost.

This paper is devoted to the design and analysis of a partitioned method for the MHD system in the Elsässer variables. The stability analysis shows that for magnetic Prandtl number of order unity, the method is unconditionally stable. We prove the error estimates and present computational tests that support the theory.

Polynomial interpolation and a priori bootstrap for computer-assisted proofs in nonlinear ODEs
Maxime Breden and Jean-Philippe Lessard
2018, 23(7) : 2825-2858 doi: 10.3934/dcdsb.2018164 +[Abstract](264) +[HTML](135) +[PDF](2931.51KB)

In this work, we introduce a method based on piecewise polynomial interpolation to enclose rigorously solutions of nonlinear ODEs. Using a technique which we call a priori bootstrap, we transform the problem of solving the ODE into one of looking for a fixed point of a high order smoothing Picard-like operator. We then develop a rigorous computational method based on a Newton-Kantorovich type argument (the radii polynomial approach) to prove existence of a fixed point of the Picard-like operator. We present all necessary estimates in full generality and for any nonlinearities. With our approach, we study two systems of nonlinear equations: the Lorenz system and the ABC flow. For the Lorenz system, we solve Cauchy problems and prove existence of periodic and connecting orbits at the classical parameters, and for ABC flows, we prove existence of ballistic spiral orbits.

Underlying one-step methods and nonautonomous stability of general linear methods
Andrew J. Steyer and Erik S. Van Vleck
2018, 23(7) : 2859-2877 doi: 10.3934/dcdsb.2018108 +[Abstract](503) +[HTML](320) +[PDF](473.14KB)

We generalize the theory of underlying one-step methods to strictly stable general linear methods (GLMs) solving nonautonomous ordinary differential equations (ODEs) that satisfy a global Lipschitz condition. We combine this theory with the Lyapunov and Sacker-Sell spectral stability theory for one-step methods developed in [34,35,36] to analyze the stability of a strictly stable GLM solving a nonautonomous linear ODE. These results are applied to develop a stability diagnostic for the solution of nonautonomous linear ODEs by strictly stable GLMs.

Conditioning and relative error propagation in linear autonomous ordinary differential equations
Stefano Maset
2018, 23(7) : 2879-2909 doi: 10.3934/dcdsb.2018165 +[Abstract](174) +[HTML](139) +[PDF](1044.93KB)

In this paper, we study the relative error propagation in the solution of linear autonomous ordinary differential equations with respect to perturbations in the initial value. We also consider equations with a constant forcing term and a nonzero equilibrium. The study is carried out for equations defined by normal matrices.

Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems
Alessandro Colombo, Nicoletta Del Buono, Luciano Lopez and Alessandro Pugliese
2018, 23(7) : 2911-2934 doi: 10.3934/dcdsb.2018166 +[Abstract](259) +[HTML](140) +[PDF](3759.2KB)

Several models in the applied sciences are characterized by instantaneous changes in the solutions or discontinuities in the vector field. Knowledge of the geometry of interaction of the flow with the discontinuities can give new insights on the behaviour of such systems. Here, we focus on the class of the piecewise smooth systems of Filippov type. We describe some numerical techniques to locate crossing and sliding regions on the discontinuity surface, to compute the sets of attraction of these regions together with the mathematical form of the separatrices of such sets. Some numerical tests will illustrate our approach.

Smooth to discontinuous systems: A geometric and numerical method for slow-fast dynamics
Luca Dieci and Cinzia Elia
2018, 23(7) : 2935-2950 doi: 10.3934/dcdsb.2018112 +[Abstract](503) +[HTML](307) +[PDF](6323.89KB)

We consider a smooth planar system having slow-fast motion, where the slow motion takes place near a curve γ. We explore the idea of replacing the original smooth system with a system with discontinuous right-hand side (DRHS system for short), whereby the DRHS system coincides with the smooth one away from a neighborhood of γ. After this reformulation, in the region of phase-space where γ is attracting for the DRHS system, we will obtain sliding motion on γ and numerical methods apt at integrating for sliding motion can be applied. Moreover, we further bypass resolving the sliding motion and monitor entries (transversal) and exits (tangential) on the curve γ, a fact that can be done independently of resolving for the motion itself. The end result is a method free from the need to adopt stiff integrators or to worry about resolving sliding motion for the DRHS system. We illustrate the performance of our method on a few problems, highlighting the feasibility of using simple explicit Runge-Kutta schemes, and that we obtain much the same orbits of the original smooth system.

Positive solutions to the unstirred chemostat model with Crowley-Martin functional response
Hai-Xia Li, Jian-Hua Wu, Yan-Ling Li and Chun-An Liu
2017doi: 10.3934/dcdsb.2017128 +[Abstract](1183) +[HTML](582) +[PDF](463.5KB)
Predicting and estimating probability density functions of chaotic systems
Noah H. Rhee, Paweł Góra and Majid Bani-Yaghoub
2017doi: 10.3934/dcdsb.2017144 +[Abstract](982) +[HTML](537) +[PDF](463.5KB)
Fractional Navier-Stokes equations
Jan W. Cholewa and Tomasz Dlotko
2017doi: 10.3934/dcdsb.2017149 +[Abstract](1658) +[HTML](890) +[PDF](547.3KB)
Stability of dislocation networks of low angle grain boundaries using a continuum energy formulation
Yang Xiang and Xiaodong Yan
2017doi: 10.3934/dcdsb.2017183 +[Abstract](885) +[HTML](578) +[PDF](634.8KB)
On the scale dynamics of the tropical cyclone intensity
Chanh Kieu and Quan Wang
2017doi: 10.3934/dcdsb.2017196 +[Abstract](1015) +[HTML](519) +[PDF](1342.2KB)
Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions
Hai-Yang Jin and Tian Xiang
2017doi: 10.3934/dcdsb.2017197 +[Abstract](927) +[HTML](548) +[PDF](443.0KB)
Eventual smoothness and asymptotic behaviour of solutions to a chemotaxis system perturbed by a logistic growth
Giuseppe Viglialoro and Thomas E. Woolley
2017doi: 10.3934/dcdsb.2017199 +[Abstract](1307) +[HTML](528) +[PDF](1131.4KB)
Positive steady states of a density-dependent predator-prey model with diffusion
Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu and Weiming Wang
2017doi: 10.3934/dcdsb.2017209 +[Abstract](930) +[HTML](551) +[PDF](583.6KB)
Continuous and discrete one dimensional autonomous fractional ODEs
Yuanyuan Feng, Lei Li, Jian-Guo Liu and Xiaoqian Xu
2017doi: 10.3934/dcdsb.2017210 +[Abstract](1006) +[HTML](517) +[PDF](598.6KB)
Cumulative and maximum epidemic sizes for a nonlinear SEIR stochastic model with limited resources
Julia Amador and Mariajesus Lopez-Herrero
2017doi: 10.3934/dcdsb.2017211 +[Abstract](1056) +[HTML](531) +[PDF](361.2KB)
Method of sub-super solutions for fractional elliptic equations
Yanqin Fang and De Tang
2017doi: 10.3934/dcdsb.2017212 +[Abstract](988) +[HTML](493) +[PDF](367.5KB)
Analysis of a free boundary problem for tumor growth with Gibbs-Thomson relation and time delays
Shihe Xu, Meng Bai and Fangwei Zhang
2017doi: 10.3934/dcdsb.2017213 +[Abstract](893) +[HTML](589) +[PDF](390.7KB)
Pullback attractors for a class of non-autonomous thermoelastic plate systems
Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento and Karina Schiabel
2017doi: 10.3934/dcdsb.2017214 +[Abstract](780) +[HTML](495) +[PDF](442.6KB)
A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential
Stefano Pasquali
2017doi: 10.3934/dcdsb.2017215 +[Abstract](761) +[HTML](484) +[PDF](488.0KB)
A space-time discontinuous Galerkin spectral element method for the Stefan problem
Chaoxu Pei, Mark Sussman and M. Yousuff Hussaini
2017doi: 10.3934/dcdsb.2017216 +[Abstract](1001) +[HTML](527) +[PDF](935.4KB)
A new criterion to a two-chemical substances chemotaxis system with critical dimension
Xueli Bai and Suying Liu
2018doi: 10.3934/dcdsb.2018074 +[Abstract](695) +[HTML](604) +[PDF](331.56KB)
Topological instabilities in families of semilinear parabolic problems subject to nonlinear perturbations
Mickaël D. Chekroun
2018doi: 10.3934/dcdsb.2018075 +[Abstract](707) +[HTML](517) +[PDF](707.16KB)
Extinction and coexistence of species for a diffusive intraguild predation model with B-D functional response
Guohong Zhang and Xiaoli Wang
2018doi: 10.3934/dcdsb.2018076 +[Abstract](363) +[HTML](336) +[PDF](695.95KB)
Global attractor of complex networks of reaction-diffusion systems of Fitzhugh-Nagumo type
B. Ambrosio, M. A. Aziz-Alaoui and V. L. E. Phan
2018doi: 10.3934/dcdsb.2018077 +[Abstract](466) +[HTML](302) +[PDF](340.38KB)
Coexistence and extinction in Time-Periodic Volterra-Lotka type systems with nonlocal dispersal
Tung Nguyen and Nar Rawal
2018doi: 10.3934/dcdsb.2018080 +[Abstract](348) +[HTML](307) +[PDF](468.39KB)
A non-autonomous predator-prey model with infected prey
Yang Lu, Xia Wang and Shengqiang Liu
2018doi: 10.3934/dcdsb.2018082 +[Abstract](428) +[HTML](308) +[PDF](766.39KB)
Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains
Xinguang Yang, Baowei Feng, Thales Maier de Souza and Taige Wang
2018doi: 10.3934/dcdsb.2018084 +[Abstract](409) +[HTML](314) +[PDF](467.51KB)
A locking free Reissner-Mindlin element with weak Galerkin rotations
Ruishu Wang, Lin Mu and Xiu Ye
2018doi: 10.3934/dcdsb.2018086 +[Abstract](362) +[HTML](271) +[PDF](367.32KB)
A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations
Wenbin Chen, Wenqiang Feng, Yuan Liu, Cheng Wang and Steven M. Wise
2018doi: 10.3934/dcdsb.2018090 +[Abstract](446) +[HTML](307) +[PDF](4796.31KB)
Lower and upper bounds of Laplacian eigenvalue problem by weak Galerkin method on triangular meshes
Qilong Zhai and Ran Zhang
2018doi: 10.3934/dcdsb.2018091 +[Abstract](386) +[HTML](278) +[PDF](351.95KB)
Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces
Chao Deng and Tong Li
2018doi: 10.3934/dcdsb.2018093 +[Abstract](424) +[HTML](300) +[PDF](408.35KB)
A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection
Iuliana Oprea and Gerhard Dangelmayr
2018doi: 10.3934/dcdsb.2018095 +[Abstract](390) +[HTML](304) +[PDF](8917.37KB)
Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation
Aslihan Demirkaya and Milena Stanislavova
2018doi: 10.3934/dcdsb.2018097 +[Abstract](395) +[HTML](326) +[PDF](2559.58KB)
Impact of spatially heterogeneous temperature on the dynamics of dengue epidemics
Naveen K. Vaidya, Xianping Li and Feng-Bin Wang
2018doi: 10.3934/dcdsb.2018099 +[Abstract](455) +[HTML](316) +[PDF](992.67KB)
A Comparison of some numerical conformal mapping methods for simply and multiply connected domains
Mohamed Badreddine, Thomas K. DeLillo and Saman Sahraei
2018doi: 10.3934/dcdsb.2018100 +[Abstract](391) +[HTML](384) +[PDF](2817.98KB)
Global regularity results for the climate model with fractional dissipation
Boqing Dong, Wenjuan Wang, Jiahong Wu and Hui Zhang
2018doi: 10.3934/dcdsb.2018102 +[Abstract](373) +[HTML](281) +[PDF](447.41KB)
Superconvergence of the semi-discrete local discontinuous Galerkin method for nonlinear KdV-type problems
Mahboub Baccouch
2018doi: 10.3934/dcdsb.2018104 +[Abstract](453) +[HTML](298) +[PDF](16469.35KB)
An FEM-MLMC algorithm for a moving shutter diffraction in time stochastic model
Mahadevan Ganesh, Brandon C. Reyes and Avi Purkayastha
2018doi: 10.3934/dcdsb.2018107 +[Abstract](403) +[HTML](313) +[PDF](727.13KB)
Two-grid finite element method for the stabilization of mixed Stokes-Darcy model
Jiaping Yu, Haibiao Zheng, Feng Shi and Ren Zhao
2018doi: 10.3934/dcdsb.2018109 +[Abstract](566) +[HTML](323) +[PDF](4434.43KB)
A comparative study on nonlocal diffusion operators related to the fractional Laplacian
Siwei Duo, Hong Wang and Yanzhi Zhang
2018doi: 10.3934/dcdsb.2018110 +[Abstract](396) +[HTML](461) +[PDF](1838.96KB)
A dimension splitting and characteristic projection method for three-dimensional incompressible flow
Hao Chen, Kaitai Li, Yuchuan Chu, Zhiqiang Chen and Yiren Yang
2018doi: 10.3934/dcdsb.2018111 +[Abstract](492) +[HTML](313) +[PDF](4837.74KB)
The impact of releasing sterile mosquitoes on malaria transmission
Hongyan Yin, Cuihong Yang, Xin'an Zhang and Jia Li
2018doi: 10.3934/dcdsb.2018113 +[Abstract](388) +[HTML](273) +[PDF](487.02KB)
On one problem of viscoelastic fluid dynamics with memory on an infinite time interval
Victor Zvyagin and Vladimir Orlov
2018doi: 10.3934/dcdsb.2018114 +[Abstract](329) +[HTML](251) +[PDF](411.15KB)
On the initial boundary value problem of a Navier-Stokes/$Q$-tensor model for liquid crystals
Yuning Liu and Wei Wang
2018doi: 10.3934/dcdsb.2018115 +[Abstract](339) +[HTML](253) +[PDF](477.75KB)
Asymptotic spreading of time periodic competition diffusion systems
Wei-Jian Bo and Guo Lin
2018doi: 10.3934/dcdsb.2018116 +[Abstract](364) +[HTML](244) +[PDF](385.48KB)
Stationary solutions of neutral stochastic partial differential equations with delays in the highest-order derivatives
Kai Liu
2018doi: 10.3934/dcdsb.2018117 +[Abstract](289) +[HTML](271) +[PDF](448.16KB)
Dynamic transitions of the Fitzhugh-Nagumo equations on a finite domain
Yiqiu Mao
2018doi: 10.3934/dcdsb.2018118 +[Abstract](343) +[HTML](253) +[PDF](301.91KB)
Improved extensibility criteria and global well-posedness of a coupled chemotaxis-fluid model on bounded domains
Jishan Fan and Kun Zhao
2018doi: 10.3934/dcdsb.2018119 +[Abstract](280) +[HTML](249) +[PDF](440.38KB)
A multiscale model of the CD8 T cell immune response structured by intracellular content
Loïc Barbarroux, Philippe Michel, Mostafa Adimy and Fabien Crauste
2018doi: 10.3934/dcdsb.2018120 +[Abstract](349) +[HTML](362) +[PDF](1723.06KB)
Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions
Xi Wang, Zuhan Liu and Ling Zhou
2018doi: 10.3934/dcdsb.2018121 +[Abstract](349) +[HTML](271) +[PDF](471.09KB)
Pullback attractors and invariant measures for discrete Klein-Gordon-Schrödinger equations
Caidi Zhao, Gang Xue and Grzegorz Łukaszewicz
2018doi: 10.3934/dcdsb.2018122 +[Abstract](370) +[HTML](242) +[PDF](562.27KB)
Analysis of a stage-structured dengue model
Jinping Fang, Guang Lin and Hui Wan
2018doi: 10.3934/dcdsb.2018125 +[Abstract](339) +[HTML](377) +[PDF](751.82KB)
Spreading speed and traveling waves for a non-local delayed reaction-diffusion system without quasi-monotonicity
Zhenguo Bai and Tingting Zhao
2018doi: 10.3934/dcdsb.2018126 +[Abstract](396) +[HTML](295) +[PDF](507.84KB)
Time asymptotics of structured populations with diffusion and dynamic boundary conditions
Mustapha Mokhtar-Kharroubi and Quentin Richard
2018doi: 10.3934/dcdsb.2018127 +[Abstract](329) +[HTML](251) +[PDF](448.24KB)
On a free boundary problem for a nonlocal reaction-diffusion model
Jia-Feng Cao, Wan-Tong Li and Meng Zhao
2018doi: 10.3934/dcdsb.2018128 +[Abstract](350) +[HTML](244) +[PDF](449.31KB)
Global phase portraits of a degenerate Bogdanov-Takens system with symmetry (Ⅱ)
Hebai Chen and Xingwu Chen
2018doi: 10.3934/dcdsb.2018130 +[Abstract](357) +[HTML](323) +[PDF](1358.71KB)
On the Cauchy problem for the XFEL Schrödinger equation
Binhua Feng and Dun Zhao
2018doi: 10.3934/dcdsb.2018131 +[Abstract](334) +[HTML](256) +[PDF](393.67KB)
A perturbed fourth order elliptic equation with negative exponent
Zongming Guo and Long Wei
2018doi: 10.3934/dcdsb.2018132 +[Abstract](344) +[HTML](241) +[PDF](396.0KB)
Carleman estimate for solutions to a degenerate convection-diffusion equation
Chunpeng Wang, Yanan Zhou, Runmei Du and Qiang Liu
2018doi: 10.3934/dcdsb.2018133 +[Abstract](318) +[HTML](258) +[PDF](431.08KB)
Global stability of a diffusive and delayed HBV infection model with HBV DNA-containing capsids and general incidence rate
Ting Guo, Haihong Liu, Chenglin Xu and Fang Yan
2018doi: 10.3934/dcdsb.2018134 +[Abstract](378) +[HTML](292) +[PDF](1059.21KB)
Upper and lower bounds for the blow-up time in quasilinear reaction diffusion problems
Juntang Ding and Xuhui Shen
2018doi: 10.3934/dcdsb.2018135 +[Abstract](329) +[HTML](286) +[PDF](378.73KB)
Invasion and coexistence of competition-diffusion-advection system with heterogeneous vs homogeneous resources
Benlong Xu and Hongyan Jiang
2018doi: 10.3934/dcdsb.2018136 +[Abstract](325) +[HTML](248) +[PDF](353.5KB)
Dynamics of weak solutions for the three dimensional Navier-Stokes equations with nonlinear damping
Fang Li, Bo You and Yao Xu
2018doi: 10.3934/dcdsb.2018137 +[Abstract](339) +[HTML](273) +[PDF](470.16KB)
Exponential stability of an incompressible non-Newtonian fluid with delay
Linfang Liu, Tomás Caraballo and Xianlong Fu
2018doi: 10.3934/dcdsb.2018138 +[Abstract](360) +[HTML](304) +[PDF](543.28KB)
Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere
Boling Guo and Guoli Zhou
2018doi: 10.3934/dcdsb.2018160 +[Abstract](140) +[HTML](129) +[PDF](438.46KB)
Modulus metrics on networks
Nathan Albin, Nethali Fernando and Pietro Poggi-Corradini
2018doi: 10.3934/dcdsb.2018161 +[Abstract](147) +[HTML](112) +[PDF](564.28KB)
Balanced truncation model reduction of a nonlinear cable-mass PDE system with interior damping
Belinda A. Batten, Hesam Shoori, John R. Singler and Madhuka H. Weerasinghe
2018doi: 10.3934/dcdsb.2018162 +[Abstract](175) +[HTML](120) +[PDF](994.3KB)
Stabilization of turning processes using spindle feedback with state-dependent delay
Qingwen Hu and Huan Zhang
2018doi: 10.3934/dcdsb.2018167 +[Abstract](186) +[HTML](113) +[PDF](370.05KB)
Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type
Abbes Benaissa and Abderrahmane Kasmi
2018doi: 10.3934/dcdsb.2018168 +[Abstract](215) +[HTML](133) +[PDF](519.94KB)
Existence and uniqueness of global classical solutions to a two dimensional two species cancer invasion haptotaxis model
Jan Giesselmann, Niklas Kolbe, Mária Lukáčová-Medvi${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over d} }}$ová and Nikolaos Sfakianakis
2018doi: 10.3934/dcdsb.2018169 +[Abstract](157) +[HTML](134) +[PDF](977.73KB)
A vicinal surface model for epitaxial growth with logarithmic free energy
Yuan Gao, Hangjie Ji, Jian-Guo Liu and Thomas P. Witelski
2018doi: 10.3934/dcdsb.2018170 +[Abstract](147) +[HTML](124) +[PDF](1158.55KB)
Convergence rate of strong approximations of compound random maps, application to SPDEs
Emmanuel Gobet and Mohamed Mrad
2018doi: 10.3934/dcdsb.2018171 +[Abstract](143) +[HTML](158) +[PDF](638.0KB)
Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities
Stanisław Migórski and Biao Zeng
2018doi: 10.3934/dcdsb.2018172 +[Abstract](195) +[HTML](111) +[PDF](448.8KB)
Theoretical analysis on a diffusive SIR epidemic model with nonlinear incidence in a heterogeneous environment
Chengxia Lei, Fujun Li and Jiang Liu
2018doi: 10.3934/dcdsb.2018173 +[Abstract](177) +[HTML](166) +[PDF](436.9KB)
Numerical study of phase transition in van der Waals fluid
Qiaolin He, Chang Liu and Xiaoding Shi
2018doi: 10.3934/dcdsb.2018174 +[Abstract](161) +[HTML](126) +[PDF](8967.21KB)
Identification of generic stable dynamical systems taking a nonlinear differential approach
Mahdi Khajeh Salehani
2018doi: 10.3934/dcdsb.2018175 +[Abstract](320) +[HTML](204) +[PDF](449.36KB)
A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment
Danhua Jiang, Zhi-Cheng Wang and Liang Zhang
2018doi: 10.3934/dcdsb.2018176 +[Abstract](201) +[HTML](109) +[PDF](392.12KB)
Prevalence of stable periodic solutions in the forced relativistic pendulum equation
Feng Wang, Jifeng Chu and Zaitao Liang
2018doi: 10.3934/dcdsb.2018177 +[Abstract](195) +[HTML](133) +[PDF](374.95KB)
Time-dependent asymptotic behavior of the solution for plate equations with linear memory
Tingting Liu and Qiaozhen Ma
2018doi: 10.3934/dcdsb.2018178 +[Abstract](216) +[HTML](134) +[PDF](490.86KB)
Existence and uniqueness of solutions of free boundary problems in heterogeneous environments
Mingxin Wang
2018doi: 10.3934/dcdsb.2018179 +[Abstract](190) +[HTML](120) +[PDF](302.43KB)
Global existence for an attraction-repulsion chemotaxis fluid model with logistic source
Abelardo Duarte-Rodríguez, Lucas C. F. Ferreira and Élder J. Villamizar-Roa
2018doi: 10.3934/dcdsb.2018180 +[Abstract](191) +[HTML](132) +[PDF](564.89KB)
Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains
Dingshi Li and Xiaohu Wang
2018doi: 10.3934/dcdsb.2018181 +[Abstract](175) +[HTML](107) +[PDF](412.42KB)
Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity
Qingyan Shi, Junping Shi and Yongli Song
2018doi: 10.3934/dcdsb.2018182 +[Abstract](249) +[HTML](130) +[PDF](1057.62KB)
Spatiotemporal attractors generated by the Turing-Hopf bifurcation in a time-delayed reaction-diffusion system
Qi An and Weihua Jiang
2018doi: 10.3934/dcdsb.2018183 +[Abstract](200) +[HTML](165) +[PDF](2451.29KB)
Bistable waves of a recursive system arising from seasonal age-structured population models
Yingli Pan, Ying Su and Junjie Wei
2018doi: 10.3934/dcdsb.2018184 +[Abstract](160) +[HTML](117) +[PDF](422.62KB)
The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition
Wen Tan
2017doi: 10.3934/dcdsb.2018194 +[Abstract](182) +[HTML](128) +[PDF](451.81KB)
Boundedness and persistence of populations in advective Lotka-Volterra competition system
Qi Wang, Yang Song and Lingjie Shao
2017doi: 10.3934/dcdsb.2018195 +[Abstract](121) +[HTML](81) +[PDF](438.3KB)
Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat
Alexander V. Budyansky, Kurt Frischmuth and Vyacheslav G. Tsybulin
2017doi: 10.3934/dcdsb.2018196 +[Abstract](116) +[HTML](59) +[PDF](757.88KB)
Efficient representation of invariant manifolds of periodic orbits in the CRTBP
Roberto Castelli
2017doi: 10.3934/dcdsb.2018197 +[Abstract](142) +[HTML](132) +[PDF](10671.43KB)
Asymptotic boundedness and stability of solutions to hybrid stochastic differential equations with jumps and the Euler-Maruyama approximation
Wei Mao, Liangjian Hu and Xuerong Mao
2017doi: 10.3934/dcdsb.2018198 +[Abstract](148) +[HTML](77) +[PDF](460.52KB)
Optimal error estimates for fractional stochastic partial differential equation with fractional Brownian motion
Litan Yan and Xiuwei Yin
2017doi: 10.3934/dcdsb.2018199 +[Abstract](116) +[HTML](73) +[PDF](436.11KB)
Advection-diffusion equation on a half-line with boundary Lévy noise
Lena-Susanne Hartmann and Ilya Pavlyukevich
2017doi: 10.3934/dcdsb.2018200 +[Abstract](142) +[HTML](76) +[PDF](517.31KB)
Persistent two-dimensional strange attractors for a two-parameter family of Expanding Baker Maps
Antonio Pumariño, José Ángel Rodríguez and Enrique Vigil
2017doi: 10.3934/dcdsb.2018201 +[Abstract](88) +[HTML](64) +[PDF](401.71KB)
Invasion fronts on graphs: The Fisher-KPP equation on homogeneous trees and Erdős-Réyni graphs
Aaron Hoffman and Matt Holzer
2017doi: 10.3934/dcdsb.2018202 +[Abstract](100) +[HTML](53) +[PDF](973.77KB)
Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments
Yulan Lu, Minghui Song and Mingzhu Liu
2017doi: 10.3934/dcdsb.2018203 +[Abstract](139) +[HTML](60) +[PDF](550.13KB)
The Rothe method for multi-term time fractional integral diffusion equations
Stanisław Migórski and Shengda Zeng
2017doi: 10.3934/dcdsb.2018204 +[Abstract](119) +[HTML](109) +[PDF](379.33KB)
Dirac-concentrations in an integro-pde model from evolutionary game theory
King-Yeung Lam
2017doi: 10.3934/dcdsb.2018205 +[Abstract](117) +[HTML](66) +[PDF](589.14KB)
Valuation of American strangle option: Variational inequality approach
Junkee Jeon and Jehan Oh
2017doi: 10.3934/dcdsb.2018206 +[Abstract](178) +[HTML](78) +[PDF](548.5KB)
Global dynamics of a latent HIV infection model with general incidence function and multiple delays
Yu Yang, Yueping Dong and Yasuhiro Takeuchi
2017doi: 10.3934/dcdsb.2018207 +[Abstract](178) +[HTML](76) +[PDF](1116.75KB)
A two-species weak competition system of reaction-diffusion-advection with double free boundaries
Bo Duan and Zhengce Zhang
2017doi: 10.3934/dcdsb.2018208 +[Abstract](111) +[HTML](79) +[PDF](517.73KB)
Boundedness in a three-dimensional Keller-Segel- Stokes system involving tensor-valued sensitivity with saturation
Dan Li, Chunlai Mu, Pan Zheng and Ke Lin
2017doi: 10.3934/dcdsb.2018209 +[Abstract](126) +[HTML](68) +[PDF](520.67KB)
Mean field model for collective motion bistability
Josselin Garnier, George Papanicolaou and Tzu-Wei Yang
2017doi: 10.3934/dcdsb.2018210 +[Abstract](97) +[HTML](62) +[PDF](5851.55KB)
Limit cycles for regularized piecewise smooth systems with a switching manifold of codimension two
Dingheng Pi
2017doi: 10.3934/dcdsb.2018211 +[Abstract](145) +[HTML](60) +[PDF](357.85KB)
Global weak solutions for a coupled chemotaxis non-Newtonian fluid
Hafedh Bousbih
2017doi: 10.3934/dcdsb.2018212 +[Abstract](137) +[HTML](95) +[PDF](515.27KB)
Traveling wave solutions for a bacteria system with density-suppressed motility
Roger Lui and Hirokazu Ninomiya
2017doi: 10.3934/dcdsb.2018213 +[Abstract](213) +[HTML](105) +[PDF](216.77KB)
Thermodynamical potentials of classical and quantum systems
Ruikuan Liu, Tian Ma, Shouhong Wang and Jiayan Yang
2017doi: 10.3934/dcdsb.2018214 +[Abstract](86) +[HTML](47) +[PDF](637.0KB)
On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces
Huijie Qiao and Jiang-Lun Wu
2017doi: 10.3934/dcdsb.2018215 +[Abstract](100) +[HTML](66) +[PDF](423.33KB)
Dynamic behavior and optimal scheduling for mixed vaccination strategy with temporary immunity
Siyu Liu, Xue Yang, Yingjie Bi and Yong Li
2017doi: 10.3934/dcdsb.2018216 +[Abstract](109) +[HTML](58) +[PDF](653.24KB)
Non-autonomous reaction-diffusion equations with variable exponents and large diffusion
Antonio Carlos Fernandes, Marcela Carvalho Gonçcalves and Jacson Simsen
2017doi: 10.3934/dcdsb.2018217 +[Abstract](118) +[HTML](62) +[PDF](3152.45KB)
Uniqueness and stability of traveling waves for a three-species competition system with nonlocal dispersal
Guo-Bao Zhang, Fang-Di Dong and Wan-Tong Li
2017doi: 10.3934/dcdsb.2018218 +[Abstract](150) +[HTML](62) +[PDF](472.18KB)
Fluctuations of mRNA distributions in multiple pathway activated transcription
Genghong Lin, Jianshe Yu, Zhan Zhou, Qiwen Sun and Feng Jiao
2017doi: 10.3934/dcdsb.2018219 +[Abstract](120) +[HTML](160) +[PDF](722.11KB)
Global existence and stability in a two-species chemotaxis system
Huanhuan Qiu and Shangjiang Guo
2017doi: 10.3934/dcdsb.2018220 +[Abstract](103) +[HTML](141) +[PDF](1061.01KB)
Novel spectral methods for Schrödinger equations with an inverse square potential on the whole space
Suna Ma, Huiyuan Li and Zhimin Zhang
2017doi: 10.3934/dcdsb.2018221 +[Abstract](104) +[HTML](82) +[PDF](7354.7KB)
Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems
Liang Ding, Rongrong Tian and Jinlong Wei
2017doi: 10.3934/dcdsb.2018222 +[Abstract](126) +[HTML](72) +[PDF](325.95KB)
A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients
Yizhuo Wang and Shangjiang Guo
2017doi: 10.3934/dcdsb.2018223 +[Abstract](116) +[HTML](103) +[PDF](483.02KB)
The dynamics of gene transcription in random environments
Jian Ren, Feng Jiao, Qiwen Sun, Moxun Tang and Jianshe Yu
2017doi: 10.3934/dcdsb.2018224 +[Abstract](157) +[HTML](71) +[PDF](4494.65KB)
Derivation of viscous Saint-Venant system for laminar shallow water; Numerical validation
Jean-Frédéric Gerbeau and Benoit Perthame
2001, 1(1) : 89-102 doi: 10.3934/dcdsb.2001.1.89 +[Abstract](1354) +[PDF](239.9KB) Cited By(102)
Optimal control of treatments in a two-strain tuberculosis model
E. Jung, Suzanne Lenhart and Z. Feng
2002, 2(4) : 473-482 doi: 10.3934/dcdsb.2002.2.473 +[Abstract](1120) +[PDF](139.6KB) Cited By(92)
Analysis of upscaling absolute permeability
X.H. Wu, Y. Efendiev and Thomas Y. Hou
2002, 2(2) : 185-204 doi: 10.3934/dcdsb.2002.2.185 +[Abstract](726) +[PDF](226.2KB) Cited By(71)
Fisher waves in an epidemic model
Xiao-Qiang Zhao and Wendi Wang
2004, 4(4) : 1117-1128 doi: 10.3934/dcdsb.2004.4.1117 +[Abstract](815) +[PDF](197.7KB) Cited By(62)
Optimal control of vector-borne diseases: Treatment and prevention
Kbenesh Blayneh, Yanzhao Cao and Hee-Dae Kwon
2009, 11(3) : 587-611 doi: 10.3934/dcdsb.2009.11.587 +[Abstract](1108) +[PDF](596.7KB) Cited By(59)
Dynamics of a HIV-1 Infection model with cell-mediated immune response and intracellular delay
Huiyan Zhu and Xingfu Zou
2009, 12(2) : 511-524 doi: 10.3934/dcdsb.2009.12.511 +[Abstract](820) +[PDF](264.3KB) Cited By(59)
Modelling and analysis of integrated pest management strategy
Sanyi Tang and Lansun Chen
2004, 4(3) : 759-768 doi: 10.3934/dcdsb.2004.4.759 +[Abstract](1056) +[PDF](161.3KB) Cited By(49)
Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian
Adam M. Oberman
2008, 10(1) : 221-238 doi: 10.3934/dcdsb.2008.10.221 +[Abstract](794) +[PDF](2040.6KB) Cited By(48)
A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms
Àlex Haro and Rafael de la Llave
2006, 6(6) : 1261-1300 doi: 10.3934/dcdsb.2006.6.1261 +[Abstract](399) +[PDF](479.5KB) Cited By(46)
Infinite propagation speed for a two component Camassa-Holm equation
David Henry
2009, 12(3) : 597-606 doi: 10.3934/dcdsb.2009.12.597 +[Abstract](747) +[PDF](181.0KB) Cited By(46)
Positive solutions to the unstirred chemostat model with Crowley-Martin functional response
Hai-Xia Li, Jian-Hua Wu, Yan-Ling Li and Chun-An Liu
2017, 22(11) : 1-16 doi: 10.3934/dcdsb.2017128 +[Abstract](1183) +[HTML](582) +[PDF](463.5KB) PDF Downloads(219)
Fractional Navier-Stokes equations
Jan W. Cholewa and Tomasz Dlotko
2017, 22(11) : 1-22 doi: 10.3934/dcdsb.2017149 +[Abstract](1658) +[HTML](890) +[PDF](547.3KB) PDF Downloads(195)
Asymptotic behaviour of the solutions to a virus dynamics model with diffusion
Toru Sasaki and Takashi Suzuki
2018, 23(2) : 525-541 doi: 10.3934/dcdsb.2017206 +[Abstract](1012) +[HTML](333) +[PDF](690.66KB) PDF Downloads(151)
Dynamics of a diffusive prey-predator system with strong Allee effect growth rate and a protection zone for the prey
Na Min and Mingxin Wang
2018, 23(4) : 1721-1737 doi: 10.3934/dcdsb.2018073 +[Abstract](935) +[HTML](492) +[PDF](391.76KB) PDF Downloads(144)
Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay
Igor Chueshov, Peter E. Kloeden and Meihua Yang
2018, 23(3) : 991-1009 doi: 10.3934/dcdsb.2018139 +[Abstract](667) +[HTML](365) +[PDF](479.0KB) PDF Downloads(135)
In memoriam Igor D. Chueshov September 23, 1951 - April 23, 2016
Ludwig Arnold
2018, 23(3) : ⅰ-ⅸ doi: 10.3934/dcdsb.201803i +[Abstract](1295) +[HTML](668) +[PDF](6684.8KB) PDF Downloads(126)
Long-time behavior of a class of nonlocal partial differential equations
Chang Zhang, Fang Li and Jinqiao Duan
2018, 23(2) : 749-763 doi: 10.3934/dcdsb.2018041 +[Abstract](1023) +[HTML](317) +[PDF](420.27KB) PDF Downloads(123)
A stochastic SIRI epidemic model with Lévy noise
Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo and Roger Pettersson
2018, 23(6) : 2415-2431 doi: 10.3934/dcdsb.2018057 +[Abstract](961) +[HTML](537) +[PDF](2371.9KB) PDF Downloads(115)
Hopf bifurcation of an age-structured virus infection model
Hossein Mohebbi, Azim Aminataei, Cameron J. Browne and Mohammad Reza Razvan
2018, 23(2) : 861-885 doi: 10.3934/dcdsb.2018046 +[Abstract](974) +[HTML](342) +[PDF](794.93KB) PDF Downloads(115)
How seasonal forcing influences the complexity of a predator-prey system
Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz and Huaiping Zhu
2018, 23(2) : 785-807 doi: 10.3934/dcdsb.2018043 +[Abstract](1334) +[HTML](348) +[PDF](4589.35KB) PDF Downloads(114)

2017  Impact Factor: 0.972




Email Alert

[Back to Top]