## 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-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

An indefinite weight eigenvalue problem characterizing the threshold
condition for extinction of a population based on the single-species
diffusive logistic model in a spatially heterogeneous environment is
analyzed in a bounded two-dimensional domain with no-flux boundary
conditions. In this eigenvalue problem, the spatial heterogeneity of
the environment is reflected in the growth rate function, which is
assumed to be concentrated in $n$ small circular disks, or portions of
small circular disks, that are contained inside the domain. The
constant bulk or background growth rate is assumed to be spatially
uniform. The disks, or patches, represent either strongly favorable or
strongly unfavorable local habitats. For this class of piecewise constant
bang-bang growth rate function, an asymptotic expansion for the
persistence threshold λ

_{1}, representing the positive principal eigenvalue for this indefinite weight eigenvalue problem, is calculated in the limit of small patch radii by using the method of matched asymptotic expansions. By analytically optimizing the coefficient of the leading-order term in the asymptotic expansion of λ_{1}, general qualitative principles regarding the effect of habitat fragmentation are derived. In certain degenerate situations, it is shown that the optimum spatial arrangement of the favorable habit is determined by a higher-order coefficient in the asymptotic expansion of the persistence threshold.
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

## Year of publication

## Related Authors

## Related Keywords

[Back to Top]