## 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
- AIMS Mathematics
- Conference Publications
- Electronic Research Announcements
- Mathematics in Engineering

### Open Access Journals

MBE

In this study, we expand on the susceptible-infected-susceptible
(SIS) heterosexual mixing setting by including the movement of
individuals of both genders in a spatial domain in order to more
comprehensively address the transmission dynamics of competing
strains of sexually-transmitted pathogens. In prior models, these
transmission dynamics have only been studied in the context of
nonexplicitly mobile heterosexually active populations at the
demographic steady state, or, explicitly in the simplest context
of SIS frameworks whose limiting systems are order preserving.
We introduce reaction-diffusion equations to study the dynamics of
sexually-transmitted diseases (STDs) in spatially mobile
heterosexually active populations. To accomplish this, we study a
single-strain STD model, and discuss in what forms and at what
speed the disease spreads to noninfected regions as it expands its
spatial range. The dynamics of two competing distinct strains of
the same pathogen on this population are then considered. The
focus is on the investigation of
the spatial
transition dynamics between the two endemic equilibria supported
by the nonspatial corresponding model. We establish conditions for
the successful invasion of a population living in endemic
conditions by introducing a strain with higher fitness. It is
shown that there exists a unique spreading speed (where the
spreading speed is characterized as the slowest speed of a class
of traveling waves connecting two endemic equilibria) at which the
infectious population carrying the invading stronger strain
spreads into the space where an equilibrium distribution has been
established by the population with the weaker strain. Finally, we
give sufficient conditions under which an explicit formula for the
spreading speed can be found.

DCDS-B

We propose a reaction-advection-diffusion model to study
competition between two species in a stream. We divide each
species into two compartments, individuals inhabiting the benthos
and individuals drifting in the stream. We assume that the growth
of and competitive interactions between the populations take place
on the benthos and that dispersal occurs in the stream. Our system
consists of two linear reaction-advection-diffusion equations and
two ordinary differential equations. Here, we provide a thorough
study for the corresponding single species model, which has been
previously proposed. We next give formulas for the rightward
spreading and leftward spreading speed for the model. We show that
rightward spreading speed can be characterized as is the slowest
speed of a class of traveling wave speeds. We provide sharp
conditions for the spreading speeds to be positive.
For the two species competition model, we
investigate how a species spreads into its competitor's
environment. Formulas for the spreading speeds are provided under
linear determinacy conditions. We demonstrate that under
certain conditions, the invading species can spread upstream.
Lastly, we study the existence of traveling wave solutions for the
two species competition model.

DCDS-B

We study the existence of traveling wave solutions
for the two-species Lotka-Volterra competition model in the form
of integro-differential equations. The model incorporates
asymmetric dispersal kernels that describe long distance dispersal
processes of competing species in space. Using lower and upper
traveling wave solutions, we show that the model has traveling
wave solutions that connect the origin and the coexistence
equilibrium with speeds greater than the spreading speed of each
species in the absence of its rival.

DCDS-B

We study the existence of traveling wave solutions for competition
models in the form of integro-difference equations. We show that
for a two-species competition model it is possible that two
species spread at different speeds, and there exists a traveling
wave solution. For an $m$-species competition model, under the
assumption that species have the same dispersal and growth
properties but have different competition abilities, we establish
the existence of traveling wave solutions.

DCDS-B

We study a cooperative system of integro-differential equations.
It is shown that the system in general has multiple spreading
speeds, and when the linear determinacy conditions are satisfied
all the spreading speeds are the same and equal to the spreading
speed of the linearized system. The existence of traveling wave
solutions is established via integral systems. It is shown that
when the linear determinacy conditions are satisfied, if the
unique spreading speed is not zero then it may be characterized as
the slowest speed of a class of traveling wave solutions. Some
examples are presented to illustrate the theoretical results.

DCDS-B

This special issue of Discrete and Continuous Dynamical Systems, Series B (DCDS-B),
is based on the timely special session on dynamical systems in biology and medicine
in the 7th AIMS Conference on Dynamical Systems and Differential Equations,
which took place at University of Texas at Arlington, Texas, USA, in the period of
May 18 - 21, 2008. All papers are carefully refereed and selected based on the
mathematical originality and biological relevance of the presented research work.

The bi-annual AIMS international Conference on Dynamical Systems and Differential Equations has grown steadily in size, quality and scope. Indeed, in a short period of 12 years, it has become the largest, most popular and well organized international meeting of its kind, featuring many impressive keynote speakers, dynamic and engaging special sessions, and most importantly, effective and economical conference management. A total of 867 researchers participated in this well organized meeting at Arlington.

For more information please click the “Full Text” above.

The bi-annual AIMS international Conference on Dynamical Systems and Differential Equations has grown steadily in size, quality and scope. Indeed, in a short period of 12 years, it has become the largest, most popular and well organized international meeting of its kind, featuring many impressive keynote speakers, dynamic and engaging special sessions, and most importantly, effective and economical conference management. A total of 867 researchers participated in this well organized meeting at Arlington.

For more information please click the “Full Text” above.

keywords:

DCDS-B

A ratio-dependent predator-prey model with a strong Allee effect
in prey is studied. We show that the model has a Bogdanov-Takens
bifurcation that is associated with a catastrophic crash of the
predator population. Our analysis indicates that an unstable limit
cycle bifurcates from a Hopf bifurcation, and it disappears due to
a homoclinic bifurcation which can lead to different patterns of
global population dynamics in the model. We study the heteroclinic
orbits and determine all possible phase portraits when the
Bogdanov-Takens bifurcation occurs. We also provide the conditions
for nonexistence of limit cycle under which the global dynamics of
the model can be determined.

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.

DCDS-B

In the last three decades, several models on the interaction of glucose and
insulin have appeared in the literature, the mostly used one is generally
known as the "minimal model" which was first published in 1979 and
modified in 1986. Recently, this minimal model has been questioned by De Gaetano and
Arino [4] from both physiological and modeling aspects. Instead, they proposed
a new and mathematically more reasonable model, called "dynamic model".
Their model makes use of certain simple and specific functions and introduces time delay in
a particular way. The outcome is that the model always admits a globally
asymptotically stable steady state. The objective of this paper
is to find out if and how this outcome depends on the specific
choice of functions and the way delay is incorporated. To this end,
we generalize the dynamical model to allow more general functions
and an alternative way of incorporating time delay. Our findings show that in
theory, such models can possess unstable positive steady states.
However, for all conceivable realistic data, such unstable steady
states do not exist. Hence, our work indicates
that the dynamic model does provide qualitatively robust dynamics for the purpose
of clinic application. We also perform simulations based on data from a clinic study
and point out some plausible but important implications.

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