arXiv subjects

Henri Berestycki

Publications and source records attributed to Henri Berestycki.

46 records · Page 3Linked to original sources

arXiv2013

Fisher-KPP propagation in the presence of a line: further effects

This paper is a continuation of [2] where a new model of biological invasions in the plane directed by a line was introduced. Here we include new features such as transport and reaction terms on the line. Their interaction with the pure diffusivity in the plane is quantified in terms of enhancement of the propagation speed. We establish conditions that determine whether the spreading speed exceeds the standard Fisher KPP invasion speed. These conditions involve the ratio of the diffusivities on the line and in the field, the transport term and the reactions. We derive the asymptotic behaviour for large diffusions or large transports. We also discuss the biological interpretation of these findings.

math.AP↗

arXiv2013

Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity

We study a reaction-diffusion system of partial differential equations, which can be taken to be a basic model for criminal activity. We show that the assumption of a populations natural tendency towards crime significantly changes the long-time behavior of criminal activity patterns. Under the right assumptions on these natural tendencies we first show that there exists traveling wave solutions connecting zones with no criminal activity and zones with high criminal activity, known as hotspots. This corresponds to an invasion of criminal activity onto all space. Second, we study the problem of preventing such invasions by employing a finite number of resources that reduce the payoff committing a crime in a finite region. We make the concept of wave propagation mathematically rigorous in this situation by proving the existence of entire solutions that approach traveling waves as time approaches negative infinity. Furthermore, we characterize the minimum amount of resources necessary to prevent the invasion in the case when prevention is possible. Finally, we apply our theory to what is commonly known as the gap problem in the excitable media literature, proving existing conjectures in the literature.

math.AP↗

arXiv2013

Self-organization versus top-down planning in the evolution of a city

Interventions of central, top-down planning are serious limitations to the possibility of modelling the dynamics of cities. An example is the city of Paris (France), which during the 19th century experienced large modifications supervised by a central authority, the `Haussmann period'. In this article, we report an empirical analysis of more than 200 years (1789-2010) of the evolution of the street network of Paris. We show that the usual network measures display a smooth behavior and that the most important quantitative signatures of central planning is the spatial reorganization of centrality and the modification of the block shape distribution. Such effects can only be obtained by structural modifications at a large-scale level, with the creation of new roads not constrained by the existing geometry. The evolution of a city thus seems to result from the superimposition of continuous, local growth processes and punctual changes operating at large spatial scales.

physics.soc-ph↗

arXiv2012

The influence of a line with fast diffusion on Fisher-KPP propagation

We propose here a new model to describe biological invasions in the plane when a strong diffusion takes place on a line. We establish the main properties of the system, and also derive the asymptotic speed of spreading in the direction of the line. For low diffusion, the line has no effect, whereas, past a threshold, the line enhances global diffusion in the plane and the propagation is directed by diffusion on the line. It is shown here that the global asymptotic speed of spreading in the plane, in the direction of the line, grows as the square root of the diffusion on the line. The model is much relevant to account for the effects of fast diffusion lines such as roads on spreading of invasive species.

math.AP↗

arXiv2012

On Entire Solutions of an Elliptic System Modeling Phase Separations

We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: Δu=u v^2 in R^n, Δv= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove uniqueness of the one-dimensional profile. In dimension 2, we prove that stable solutions with linear growth must be one-dimensional. Then we construct entire solutions in $\R^2$ with polynomial growth $|x|^d$ for any positive integer $d \geq 1$. For $d\geq 2$, these solutions are not one-dimensional. The construction is also extended to multi-component elliptic systems.

math.AP↗

arXiv2010

Generalized transition waves and their properties

In this paper, we generalize the usual notions of waves, fronts and propagation speeds in a very general setting. These new notions, which cover all classical situations, involve uniform limits, with respect to the geodesic distance, to a family of hypersurfaces which are parametrized by time. We prove the existence of new such waves for some time-dependent reaction-diffusion equations, as well as general intrinsic properties, some monotonicity properties and uniqueness results for almost planar fronts. The classification results, which are obtained under appropriate assumptions, show the robustness of our general definition

math.AP↗

arXiv2010

On least Energy Solutions to A Semilinear Elliptic Equation in A Strip

We consider the following semilinear elliptic equation on a strip: \[ \left\{{array}{l} Δu-u + u^p=0 \ {in} \ \R^{N-1} \times (0, L), u>0, \frac{\partial u}{\partial ν}=0 \ {on} \ \partial (\R^{N-1} \times (0, L)) {array} \right.\] where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 0$ such that for $L \leq L_{*}$, the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for $L >L_{*}$, the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers $L_{*} L_{**}$. A connection with Delaunay surfaces in CMC theory is also made.

math.AP↗

arXiv2010

Disentangling collective trends from local dynamics

A single social phenomenon (such as crime, unemployment or birth rate) can be observed through temporal series corresponding to units at different levels (cities, regions, countries...). Units at a given local level may follow a collective trend imposed by external conditions, but also may display fluctuations of purely local origin. The local behavior is usually computed as the difference between the local data and a global average (e.g. a national average), a view point which can be very misleading. We propose here a method for separating the local dynamics from the global trend in a collection of correlated time series. We take an independent component analysis approach in which we do not assume a small unbiased local contribution in contrast with previously proposed methods. We first test our method on synthetic series generated by correlated random walkers. We then consider crime rate series (in the US and France) and the evolution of obesity rate in the US, which are two important examples of societal measures. For crime rates, the separation between global and local policies is a major subject of debate. For the US, we observe large fluctuations in the transition period of mid-70's during which crime rates increased significantly, whereas since the 80's, the state crime rates are governed by external factors and the importance of local specificities being decreasing. In the case of obesity, our method shows that external factors dominate the evolution of obesity since 2000, and that different states can have different dynamical behavior even if their obesity prevalence is similar.

physics.soc-ph↗

arXiv2009

The explosion problem in a flow

We consider the explosion problem in an incompressible flow introduced in the paper of H. Berestycki, L. Kagan, G. Joulin and G. Sivashinsky. We use a novel $L^p-L^\infty$ estimate for elliptic advection-diffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flow. We also identify the flows for which the explosion threshold tends to infinity as their amplitude grows and obtain an effective description of the explosion threshold in the strong flow asymptotics in a two-dimensional one-cell flow.

math.AP↗

arXiv2009

A population facing climate change: joint influences of Allee effects and environmental boundary geometry

As a result of climate change, many populations have to modify their range to follow the suitable areas - their "climate envelope" - often risking extinction. During this migration process, they may face absolute boundaries to dispersal, because of external environmental factors. Consequently, not only the position, but also the shape of the climate envelope can be modified. We use a reaction-diffusion model to analyse the effects on population persistence of simultaneous changes in the climate envelope position and shape. When the growth term is of logistic type, we show that extinction and persistence are principally conditioned by the species mobility and the speed of climate change, but not by the shape of the climate envelope. However, with a growth term taking an Allee effect into account, we find a high sensitivity to the variations of the shape of the climate envelope. In this case, the species which have a high mobility, although they could more easily follow the migration of the climate envelope, would be at risk of extinction when encountering a local narrowing of the boundary geometry. This effect can be attenuated by a progressive opening of the available space at the exit of the narrowing, even though it transiently leads to a diminished area of the climate envelope.

math.AP↗