## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
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- 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
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- Mathematical Foundations of Computing
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- AIMS Mathematics
- Conference Publications
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- Mathematics in Engineering

### Open Access Journals

MBE

In this paper, we consider the spatial dynamics for a non-cooperative diffusion system arising from epidermal wound healing.
We shall establish the spreading speed and existence of traveling waves and characterize the spreading speed as the slowest speed
of a family of non-constant traveling wave solutions. We also construct some new types of entire solutions which are different from the traveling wave solutions and spatial variable independent solutions. The traveling wave solutions provide the healing speed and describe how wound healing process spreads from one side of the wound. The entire solution exhibits the interaction of several waves originated from different locations of the wound. To the best of knowledge of the authors, it is the first time that it is shown that there is an entire solution in the model for epidermal wound healing.

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.

PROC

The paper deals with the existence of
positive radial solutions for the quasilinear system $\textrm{
div} \left ( | \nabla u_i|^{p-2}\nabla u_i \right ) +
f^i(u_1,...,u_n)=0,\; p>1, R_1 <|x| < R_2,\;u_i(x)=0,$ on
$|x|=R_1$ and $R_2$, $i=1,...,n$, $x \in \mathbb{R}^N.$ $f^i$,
$i=1,...,n,$ are continuous and nonnegative functions. Let
$\vect{u}=(u_1,...,u_n),$ $\varphi(t)=|t|^{p-2}t,$ $f_0^i
=\lim_{\norm{\vect{u}} \to 0}
\frac{f^i(\vect{u})}{\var(\norm{\vect{u}})},$ $f_{\infty}^i
=\lim_{\norm{\vect{u}} \to \infty}
\frac{f^i(\vect{u})}{\var(\norm{\vect{u}})}$, $i=1,...,n,$
$\vect{f}=(f^1,...,f^n),$ $\vect{f}_0=\sum_{i=1}^n f_0^i$ and
$\vect{f}_{\infty}=\sum_{i=1}^n f_{\infty}^i$. We prove that
$\vect{f}_0 =0$ and $\vect{f}_{\infty}=\infty$ (superlinear)
guarantee the existence of positive radial solutions for the
system. We shall use fixed point theorems in a cone.

DCDS

In this paper we study an eigenvalue boundary value problem which
arises when seeking radial convex solutions of the Monge-Ampère
equations. We shall establish several criteria for the existence,
multiplicity and nonexistence of strictly convex solutions for the
boundary value problem with or without an eigenvalue parameter.

DCDS-B

The study of spatially explicit integro-difference systems when the local population dynamics are given in terms of discrete-time generations models
has gained considerable attention over the past two decades. These nonlinear systems arise naturally in the study of the spatial dispersal of organisms. The brunt of the

*mathematical*research on these systems, particularly, when dealing with*cooperative*systems, has focused on the study of the existence of traveling wave solutions and the characterization of their spreading speed. Here, we characterize the minimum propagation (spreading) speed, via the convergence of initial data to wave solutions, for a large class of*non cooperative*nonlinear systems of integro-difference equations. The spreading speed turns out to be the slowest speed from a family of non-constant traveling wave solutions. The applicability of these theoretical results is illustrated through the explicit study of an integro-difference system with*local*population dynamics governed by Hassell and Comins' non-cooperative competition model (1976). The corresponding integro-difference nonlinear systems that results from the redistribution of individuals via a dispersal kernel is shown to satisfy conditions that guarantee the existence of minimum speeds and traveling waves. This paper is dedicated to Avner Friedman as we celebrate his immense contributions to the fields of partial differential equations, integral equations, mathematical biology, industrial mathematics and applied mathematics in general. His leadership in the mathematical sciences and his mentorship of students and friends over several decades has made a huge difference in the personal and professional lives of many, including both of us.
keywords:
biological invasion
,
traveling wave
,
Minimum speed
,
dispersal
,
integro-difference systems.

DCDS-B

We study, both experimentally and through mathematical modeling, the response of wild type and mutant yeast strains to systematic variations of extracellular calcium abundance. We extend a previously developed mathematical model (Cui and Kaandorp,

*Cell Calcium*, 39, 337 (2006))[3], that explicitly considers the population and activity of proteins with key roles in calcium homeostasis. Modifications of the model can directly address the responses of mutants lacking these proteins. We present experimental results for the response of yeast cells to sharp, step-like variations in external $Ca^{++}$ concentrations. We analyze the properties of the model and use it to simulate the experimental conditions investigated. The model and experiments diverge more markedly in the case of mutants laking the Pmc1 protein. We discuss possible extensions of the model to address these findings.
PROC

The paper deals with the existence and nonexistence of positive radial solutions for the weakly coupled quasilinear system
div$( | \nabla u|^{p-2}\nabla u ) + \lambda f(v)=0$, div $( | \nabla v|^{p-2}\nabla v ) + \lambda g(u)=0$ in $\B$, and $\u =v=0$ on
$\partial B,$ where $p>1$, $B$ is a finite ball, $f$ and $g$ are continuous and nonnegative functions. We prove that there is a positive radial solution for
the problem for various intevals of $\lambda$ in sublinear cases. In addition, a nonexistence result is given. We shall use fixed point theorems in a cone.

MBE

The process of invasion is fundamental to the study of the dynamics of ecological and epidemiological systems. Quantitatively, a crucial measure of species' invasiveness is given by the rate at which it spreads into new open environments. The so-called
``linear determinacy'' conjecture equates full nonlinear
model spread rates with the spread rates computed from linearized systems with the linearization carried out around the leading
edge of the invasion. A survey that accounts for
recent developments in the identification of conditions under which linear determinacy gives the ``right" answer, particularly in the context of non-compact and non-cooperative systems, is the thrust of this contribution. Novel results that extend some of the research linked to some the contributions
covered in this survey are also discussed.

CPAA

In this paper, we establish a unilateral global bifurcation result for a class of quasilinear periodic boundary problems with a sign-changing weight.
By the Ljusternik-Schnirelmann theory, we first study the spectrum of the periodic $p$-Laplacian with the sign-changing weight.
In particular, we show that there exist two simple, isolated, principal eigenvalues $\lambda_0^+$ and $\lambda_0^-$.
Furthermore, under some natural hypotheses on perturbation function,
we show that $(\lambda_0^\nu,0)$ is a bifurcation
point of the above problems and there are two distinct unbounded sub-continua
$C_\nu^{+}$ and $C_\nu^{-}$,
consisting of the continuum $C_\nu$ emanating from $(\lambda_0^\nu, 0)$,
where $\nu\in\{+,-\}$. As an application of the above result, we study the existence of one-sign solutions
for a class of quasilinear periodic boundary problems with the sign-changing weight.
Moreover, the uniqueness of one-sign solutions and the dependence of solutions on the parameter $\lambda$ are
also studied.

DCDS

We study the traveling waves of reaction-diffusion equations for a diffusive SIR model.
The existence of traveling waves is determined by the basic reproduction number of the corresponding ordinary differential equations and the minimal wave speed.
Our proof is based on Schauder fixed point theorem and Laplace transform.

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