## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
- Networks & Heterogeneous Media
- Numerical Algebra, Control & Optimization
- Electronic Research Announcements
- Conference Publications
- AIMS Mathematics

DCDS

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.

NACO

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.

DCDS-B

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.

DCDS-B

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.

CPAA

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.

DCDS-B

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

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.

DCDS-B

In this paper, we study a PDE model of two species competing for a single limiting
nutrient resource in a chemostat in which one microbial species excretes a
toxin that increases the mortality of another. Our goal is to understand the role of spatial heterogeneity and allelopathy in blooms of
harmful algae. We first demonstrate that the two-species system and its single species subsystem satisfy a mass conservation law that plays an important role in our analysis.
We investigate the possibilities of bistability and coexistence for the two-species system by appealing to the method of topological degree in cones and the theory of uniform persistence. Numerical simulations confirm the theoretical results.

keywords:
allelopathy
,
unstirred chemostat.
,
Harmful algae
,
spatial heterogeneity
,
nutrient recycling

DCDS

For a Markov semigroup $P_t$ with invariant probability measure $\mu$, a constant $\lambda>0$ is called a lower bound of the ultra-exponential convergence rate of $P_t$ to $\mu$, if there exists a constant $C\in (0,\infty)$ such that
$$ \sup_{\mu(f^2)\le 1}||P_tf-\mu(f)||_\infty \le C e^{-\lambda t},\ \ t\ge 1.$$ By using the coupling by change of measure in the line of [F.-Y. Wang, Ann. Probab. 35(2007), 1333--1350], explicit lower bounds of the ultra-exponential convergence rate are derived for a class of non-linear monotone stochastic partial differential equations. The main result is illustrated by the stochastic porous medium equation and the stochastic $p$-Laplace equation respectively.
Finally, the $V$-uniformly exponential convergence is investigated for stochastic fast-diffusion equations.

DCDS-B

Dengue fever is a virus-caused disease in the world. Since the high
infection rate of dengue fever and high death rate of its severe
form dengue hemorrhagic fever, the control of the spread of the
disease is an important issue in the public health. In an
effort to understand the dynamics of the spread of the disease,
Esteva and Vargas [2] proposed a SIR v.s. SI
epidemiological model without crowding effect and spatial
heterogeneity. They found a threshold parameter $R_0,$ if $R_0<1,$
then the disease will die out; if $R_0>1,$ then the disease will
always exist.

To investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease, we modify the autonomous system provided in [2] to obtain a reaction-diffusion system. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system. Basically, we show that the basic reproduction number is a threshold parameter that predicts whether the disease will die out or persist. Further, we demonstrate the basic reproduction number in an explicit way and construct suitable Lyapunov functionals to determine the global stability for the special case where coefficients are all constant.

To investigate how the spatial heterogeneity and crowding effect influence the dynamics of the spread of the disease, we modify the autonomous system provided in [2] to obtain a reaction-diffusion system. We first define the basic reproduction number in an abstract way and then employ the comparison theorem and the theory of uniform persistence to study the global dynamics of the modified system. Basically, we show that the basic reproduction number is a threshold parameter that predicts whether the disease will die out or persist. Further, we demonstrate the basic reproduction number in an explicit way and construct suitable Lyapunov functionals to determine the global stability for the special case where coefficients are all constant.

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