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### Open Access Journals

DCDS-B

We consider a class of one-dimensional reaction-diffusion systems,
\[
\left\{
\begin{array}
[
u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\
\tau w_{t}=Dw_{xx}+g(u,w)
\end{array}
\right.
\]
with homogeneous Neumann boundary conditions on a one dimensional interval.
Under some generic conditions on the nonlinearities $f,g$ and in the singular
limit $\varepsilon\rightarrow0,$ such a system admits a steady state for which
$u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$
and for sufficiently small $\tau$, such a steady state is known to be stable
in time. On the other hand, it is also known that in the so-called shadow
limit $D\rightarrow\infty,$ patterns having more than one interface are
unstable. In this paper we analyse in detail the transition between the stable
patterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. We
show that this transition occurs when $D$ is

*exponentially large in*$\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\gg D_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if $D < D_{K}$ and is unstable when $D > D_{K}$. We also study the dynamics of the interfaces when $D$ is exponentially large; this allows us to describe in detail the mechanism leading to the instability. Direct numerical computations of stability and dynamics are performed, and these results are in excellent agreement with corresponding results as predicted by the asymptotic theory.
DCDS-B

In the limit of small activator diffusivity $\varepsilon$, and in a bounded
domain in $\mathbb{R}^{N}$ with $N=1$ or $N=2$ under homogeneous Neumann
boundary conditions, the bifurcation behavior of an equilibrium
one-spike solution to the Gierer-Meinhardt activator-inhibitor system is
analyzed for different ranges of the inhibitor diffusivity $D$. When
$D=\infty$, it is well-known that a one-spike solution for the resulting
shadow Gierer-Meinhardt system is unstable, and the locations of
unstable equilibria coincide with the points in the domain that are
furthest away from the boundary. For a unit disk domain it is shown
that as $D$ is decreased below a critical bifurcation value $D_{c}$,
with $D_{c}=O(\varepsilon^2 e^{2/\varepsilon})$, the spike at the origin becomes
stable, and unstable spike solutions bifurcate from the origin. The
locations of these bifurcating spikes tend to the boundary of the domain
as $D$ is decreased further. Similar bifurcation behavior is studied in
a one-parameter family of dumbbell-shaped domains. This motivates a
further analysis of the existence of certain near-boundary spikes. Their
location and stability is given in terms of the modified Green's
function. Finally, for the dumbbell-shaped domain, an intricate
bifurcation structure is observed numerically as $D$ is decreased below
some $O(1)$ critical value.

DCDS-B

The existence and stability of localized patterns of criminal activity
are studied for the reaction-diffusion model of urban crime that was
introduced by Short et. al. [Math. Models. Meth. Appl. Sci.,

**18**, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${\mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.
keywords:
Singular perturbations
,
crime
,
reaction-diffusion
,
Hopf Bifurcation.
,
hot-spots
,
nonlocal eigenvalue problem

DCDS-B

Systems of pairwise-interacting particles model a cornucopia of physical systems, from insect swarms and bacterial colonies to nanoparticle self-assembly. We study a continuum model with densities supported on co-dimension one curves for two-species particle interaction in $\mathbb{R}^2$, and apply linear stability analysis of concentric ring steady states to characterize the steady state patterns and instabilities which form. Conditions for linear well-posedness are determined and these results are compared to simulations of the discrete particle dynamics, showing predictive power of the linear theory. Some intriguing steady state patterns are shown through numerical examples.

DCDS-B

We consider a reaction-diffusion system of the form
\[
\left\{
\begin{array}
\ u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\
\tau w_{t}=Dw_{xx}+g(u,w)
\end{array}
\right.
\]
with Neumann boundary conditions on a finite interval. Under certain generic
conditions on the nonlinearities $f,g$ and in the singular limit
$\varepsilon\ll1$ such a system may admit a steady state solution where $u$
has sharp interfaces. It is also known that such interfaces may be
destabilized due to a Hopf bifurcation [Y. Nishiura and M. Mimura. SIAM
J.Appl. Math., 49:481--514, 1989], as $\tau$ is increased beyond a certain
threshold $\tau_{h}$. In this paper, we study what happens for $\tau>\tau
_{h},$ or even $\tau\rightarrow\infty,$ for a solution that consists of either
one or two interfaces. Under the additional assumption $D\gg1,$ using singular
perturbation theory, we determine the existence of another threshold $\tau
_{c}>\tau_{h}$ (where $\tau_{c}$ is allowed to be infinite) such that if
$\tau_{h}<\tau<\tau_{c}$ then the system admits a solution consisting of
periodically oscillating interfaces. On the other hand if $\tau>\tau_{c},$ the
extent of the oscillation eventually exceeds the spatial domain size, even
though very long transient dynamics can preceed this occurence. We make use of
recently developed numerical software (that employs adaptive error control in
space and time) to accurately compute an approximate solution. Excellent
agreement with the analytical theory is observed.

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