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### Open Access Journals

DCDS-S

Various motions of camphor boats in the water channel exhibit
both a homogeneous and an inhomogeneous state, depending on
the number of boats,
when unidirectional motion along an annular water channel can be
observed even with only one camphor boat.
In a theoretical research, the unidirectional motion is represented by a
traveling wave solution in a model.
Since the experimental results described above are thought of as
a kind of bifurcation phenomena, we would like to investigate a
linearized eigenvalue problem in order to prove the destabilization of
a traveling wave solution.
However, the eigenvalue problem is too difficult to analyze even if the
number of camphor boats is 2.
Hence we need to make a reduction on the model.
In the present paper, we apply the center manifold theory
and reduce the model to an ordinary differential system.

DCDS

In this paper we study the dynamics of a single transition layer of a solution to a spatially
inhomogeneous bistable reaction diffusion equation in one space dimension.
The spatial inhomogeneity is given by a function $a(x)$. In particular,
we consider the case where $a(x)$ is identically zero on an interval $I$ and study the dynamics of the transition layer on $I$.
In this case the dynamics of the transition layer on $I$ becomes so-called very slow dynamics.
In order to analyze such a dynamics, we construct an attractive local invariant manifold
giving the dynamics of the transition layer and we derive an equation describing the flow on the manifold.
We also give applications of our results to two well known nonlinearities of bistable type.

NHM

We consider pulse-like localized solutions for reaction-diffusion systems on
a half line and impose various boundary conditions at one end of it.
It is shown that the movement of a pulse solution
with the homogeneous Neumann boundary condition is completely opposite
from that with the Dirichlet boundary condition. As general cases, Robin type boundary
conditions are also considered. Introducing one parameter connecting the
Neumann and the Dirichlet boundary conditions, we clarify the transition
of motions of solutions with respect to boundary conditions.

CPAA

Our aim in this paper is to prove the instability of multi-spot
patterns in a shadow system,
which is obtained as a limiting system of a reaction-diffusion model as one of the
diffusion coefficients goes to infinity.
Instead of investigating each eigenfunction for a linearized operator,
we characterize the eigenspace spanned by unstable eigenfunctions.

NHM

In the present paper, a model describing the self-motion of a
camphor disc on water is proposed. The stability of a standing
camphor disc is investigated by analyzing the model equation, and
a pitchfork type bifurcation diagram of a traveling spot is shown.
Multiple camphor discs are also treated by the model equations,
and the repulsive interaction of spots is discussed.

DCDS-B

Two types of aggregation systems with Fisher-KPP growth are proposed.
One is described by a normal reaction-diffusion system, and the other is
described by a cross-diffusion system. If the growth effect is dominant, a
spatially constant equilibrium solution is stable. When the growth effect
becomes weaker and the aggregation effect become dominant, the solution
is destabilized so that spatially non-constant equilibrium solutions, which
exhibit Turing's patterns, appear. When the growth effect weakens further,
the spatially non-constant equilibrium solutions are destabilized through
Hopf bifurcation, so that oscillatory Turing's patterns appear. Finally, when
the growth effect is extremely weak, there appear spatio-temporal periodic
solutions exhibiting infinite dimensional relaxation oscillation.

CPAA

The Gierer-Meinhardt system is a mathematical
model describing the process of hydra regeneration.
The authors of [3] showed that
if an initial value is close to a spiky pattern and
its peak is far away from the boundary,
the solution of the shadow Gierer-Meinhardt system,
called a

*interior spike solution*, moves towards a point on boundary which is the closest to the peak. However it has not been studied how a solution close to a spiky pattern with the peak on the boundary, called a*boundary spike solution*moves along the boundary. In this paper, we consider the shadow Gierer-Meinhardt system and dynamics of a boundary spike solution. Our results state that a boundary spike moves towards a critical point of the curvature of the boundary and approaches a stable stationary solution.
DCDS-S

We consider the mixed ODE-PDE system called a hybrid system,
in which the two interfaces interact with each other through a
continuous medium and their equations of motion are derived
in a weak interaction framework.
We study the bifurcation property of the resulting hybrid system
and construct an unstable standing pulse solution, which plays
the role of a separator
for dynamic transition from standing breather to annihilation behavior
between two interfaces.

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