DCDS
Radial stability of periodic solutions of the Gylden-Meshcherskii-type problem
Jifeng Chu Pedro J. Torres Feng Wang
Discrete & Continuous Dynamical Systems - A 2015, 35(5): 1921-1932 doi: 10.3934/dcds.2015.35.1921
For the Gylden-Meshcherskii-type problem with a periodically cha-nging gravitational parameter, we prove the existence of radially periodic solutions with high angular momentum, which are Lyapunov stable in the radial direction.
keywords: Gylden-Meshcherskii-type problem Radial stability twist. periodic solutions
DCDS
On the Cauchy problem for a higher-order μ-Camassa-Holm equation
Feng Wang Fengquan Li Zhijun Qiao
Discrete & Continuous Dynamical Systems - A 2018, 38(8): 4163-4187 doi: 10.3934/dcds.2018181

In this paper, we study the Cauchy problem of a higher-order μ-Camassa-Holm equation. We first establish the Green's function of $(μ-\partial_{x}^{2}+\partial_{x}^{4})^{-1}$ and local well-posedness for the equation in Sobolev spaces $H^{s}(\mathbb{S})$, $s>\frac{7}{2}$. Then we provide the global existence results for strong solutions and weak solutions. Moreover, we show that the solution map is non-uniformly continuous in $H^{s}(\mathbb{S})$, $s≥ 4$. Finally, we prove that the equation admits single peakon solutions which have continuous second derivatives and jump discontinuities in the third derivatives.

keywords: Higher-order μ-Camassa-Holm equation global existence weak solutions non-uniformly continuous peakon solutions
DCDS-B
Prevalence of stable periodic solutions in the forced relativistic pendulum equation
Feng Wang Jifeng Chu Zaitao Liang
Discrete & Continuous Dynamical Systems - B 2018, 22(11): 1-16 doi: 10.3934/dcdsb.2018177

We study the prevalence of stable periodic solutions of the forced relativistic pendulum equation for external force which guarantees the existence of periodic solutions. We prove the results for a general planar system.

keywords: Prevalence stable periodic solutions forced relativistic pendulum equation planar system
DCDS
Lyapunov stability for regular equations and applications to the Liebau phenomenon
Feng Wang José Ángel Cid Mirosława Zima
Discrete & Continuous Dynamical Systems - A 2018, 38(9): 4657-4674 doi: 10.3934/dcds.2018204

We study the existence and stability of periodic solutions of two kinds of regular equations by means of classical topological techniques like the Kolmogorov-Arnold-Moser (KAM) theory, the Moser twist theorem, the averaging method and the method of upper and lower solutions in the reversed order. As an application, we present some results on the existence and stability of $ T$-periodic solutions of a Liebau-type equation.

keywords: Lyapunov stability regular equation Liebau phenomenon KAM theory Moser twist theorem
NACO
Primal-dual approximation algorithms for submodular cost set cover problems with linear/submodular penalties
Fengmin Wang Dachuan Xu Donglei Du Chenchen Wu
Numerical Algebra, Control & Optimization 2015, 5(2): 91-100 doi: 10.3934/naco.2015.5.91
We introduce two set cover problems with submodular costs and linear/submodular penalties and offer two approximation algorithms of ratios $\eta$ and $2\eta$ respectively via the primal-dual technique, where $\eta$ is the largest number of sets that each element belongs to.
keywords: primal-dual. penalties submodular cost Set cover approximation algorithm
DCDS-B
Partial differential equations with Robin boundary condition in online social networks
Guowei Dai Ruyun Ma Haiyan Wang Feng Wang Kuai Xu
Discrete & Continuous Dynamical Systems - B 2015, 20(6): 1609-1624 doi: 10.3934/dcdsb.2015.20.1609
In recent years, online social networks such as Twitter, have become a major source of information exchange and research on information diffusion in social networks has been accelerated. Partial differential equations are proposed to characterize temporal and spatial patterns of information diffusion over online social networks. The new modeling approach presents a new analytic framework towards quantifying information diffusion through the interplay of structural and topical influences. In this paper we develop a non-autonomous diffusive logistic model with indefinite weight and the Robin boundary condition to describe information diffusion in online social networks. It is validated with a real dataset from an online social network, Digg.com. The simulation shows that the logistic model with the Robin boundary condition is able to more accurately predict the density of influenced users. We study the bifurcation, stability of the diffusive logistic model with heterogeneity in distance. The bifurcation and stability results of the model information describe either information spreading or vanishing in online social networks.
keywords: diffusive logistic model indefinite weight Robin boundary condition. Bifurcation online social networks stability
DCDS-B
Avian influenza dynamics in wild birds with bird mobility and spatial heterogeneous environment
Naveen K. Vaidya Feng-Bin Wang Xingfu Zou
Discrete & Continuous Dynamical Systems - B 2012, 17(8): 2829-2848 doi: 10.3934/dcdsb.2012.17.2829
In this paper, we propose a mathematical model to describe the avian influenza dynamics in wild birds with bird mobility and heterogeneous environment incorporated. In addition to establishing the basic properties of solutions to the model, we also prove the threshold dynamics which can be expressed either by the basic reproductive number or by the principal eigenvalue of the linearization at the disease free equilibrium. When the environment factor in the model becomes a constant (homogeneous environment), we are able to find explicit formulas for the basic reproductive number and the principal eigenvalue. We also perform numerical simulation to explore the impact of the heterogeneous environment on the disease dynamics. Our analytical and numerical results reveal that the avian influenza dynamics in wild birds is highly affected by both bird mobility and environmental heterogeneity.
keywords: basic reproductive number threshold dynamics. diffusion principal eigenvalue heterogeneous environment Avian influenza spectral radius
CPAA
On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis
Sze-Bi Hsu Feng-Bin Wang
Communications on Pure & Applied Analysis 2011, 10(5): 1479-1501 doi: 10.3934/cpaa.2011.10.1479
In this paper we construct a mathematical model of two microbial populations competing for a single-limited nutrient with internal storage in an unstirred chemostat. First we establish the existence and uniqueness of steady-state solutions for the single population. The conditions for the coexistence of steady states are determined. Techniques include the maximum principle, theory of bifurcation and degree theory in cones.
keywords: degree theory. Maximum principle coexistence global bifurcation Chemostat
DCDS-B
Growth of single phytoplankton species with internal storage in a water column
Linfeng Mei Sze-Bi Hsu Feng-Bin Wang
Discrete & Continuous Dynamical Systems - B 2016, 21(2): 607-620 doi: 10.3934/dcdsb.2016.21.607
In this paper, we analyze a system modeling the growth of single phytoplankton populations in a water column, where population growth increases monotonically with the nutrient quota stored within individuals. We establish a threshold result on the global extinction and persistence of phytoplankton. Condition for persistence is shown to depend on the principal eigenvalue of a boundary value problem, which is related to the physical transport properties of the water column (i.e. the diffusivity and the sinking speed), nutrient uptake rate, and growth rate.
keywords: threshold dynamics spatial variations internal storage Steady states a water column.
DCDS-B
Dynamics of harmful algae with seasonal temperature variations in the cove-main lake
Feng-Bin Wang Sze-Bi Hsu Wendi Wang
Discrete & Continuous Dynamical Systems - B 2016, 21(1): 313-335 doi: 10.3934/dcdsb.2016.21.313
In this paper, we investigate two-vessel gradostat models describing the dynamics of harmful algae with seasonal temperature variations, in which one vessel represents a small cove connected to a larger lake. We first define the basic reproduction number for the model system, and then show that the trivial periodic state is globally asymptotically stable, and algae is washed out eventually if the basic reproduction number is less than unity, while there exists at least one positive periodic state and algal blooms occur when it is greater than unity. There are several types of productions for dissolved toxins, related to the algal growth rate, and nutrient limitation, respectively. For the system with a specific toxin production, the global attractivity of positive periodic steady-state solution can be established. Numerical simulations from the basic reproduction number show that the factor of seasonality plays an important role in the persistence of harmful algae.
keywords: threshold dynamics. global attractivity Harmful algae seasonal variations

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