Propagation in a kinetic reaction-transport equation: travelling waves and accelerating fronts

In this paper, we study the existence and stability of travelling wave solutions of a kinetic reaction-transport equation. The model describes particles moving according to a velocity-jump process, and proliferating thanks to a reaction term of monostable type. The boundedness of the velocity set appears to be a necessary and sufficient condition for the existence of positive travelling waves. The minimal speed of propagation of waves is obtained from an explicit dispersion relation. We construct the waves using a technique of sub- and supersolutions and prove their \eb{weak} stability in a weighted $L^2$ space. In case of an unbounded velocity set, we prove a superlinear spreading. It appears that the rate of spreading depends on the decay at infinity of the velocity distribution. In the case of a Gaussian distribution, we prove that the front spreads as $t^{3/2}$

Large deviations for velocity-jump processes and non-local Hamilton-Jacobi equations

We establish a large deviation theory for a velocity jump process, where new random velocities are picked at a constant rate from a Gaussian distribution. The Kolmogorov forward equation associated with this process is a linear kinetic transport equation in which the BGK operator accounts for the changes in velocity. We analyse its asymptotic limit after a suitable rescaling compatible with the WKB expansion. This yields a new type of Hamilton Jacobi equation which is non local with respect to velocity variable. We introduce a dedicated notion of viscosity solution for the limit problem, and we prove well-posedness in the viscosity sense. The fundamental solution is explicitly computed, yielding quantitative estimates for the large deviations of the underlying velocity-jump process {\em \`a la Freidlin-Wentzell}. As an application of this theory, we conjecture exact rates of acceleration in some nonlinear kinetic reaction-transport equations

Characteristics of Territories-in-between

Much of physical territory of the Europe does not fit classic ‘urban–rural’ typologies but can best be described as ‘territories-in-between’ (TiB). There is considerable agreement that TiB is pervasive and very significant. However, typologies of territory or spatial development continue to employ only degrees of either urban or rural. Similarly, spatial planning and territorial development policies rarely make use of the notion of in-between areas but tend instead to divide the territory into urban and rural zones. Questions have been raised therefore about the lack of understanding of territories-in-between and their negligence in planning policy. This paper contributes to a better understanding of TiB, by proposing a method for their characterisation and mapping. It asks if there can be a common definition of TiB that reflects consistent and distinctive characteristics across the great variety of spatial development contexts in Europe. It proposes spatial and demographic criteria for their definition, mapping and comparison. The comparison with widely used urban–rural classifications shows that the presented classification of TiB has three advantages: (i) it maps the complexity of the spatial structure of urbanised areas on a regional scale, and thereby helps to overcome the prevalent idea that urbanised regions are characterised by a spatial gradient from urban centre(s) to rural periphery; (ii) it emphasises the network structure of territories-in-between and the underlying connectivity of places with different functions and (iii) it raises awareness that in some parts of Europe a settlement pattern has developed that cannot be understood as either urban or rural

Hyperbolic traveling waves driven by growth

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed $\epsilon^{-1}$ (\epsilon\textgreater{}0), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter $\epsilon$: for small $\epsilon$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $\epsilon$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $\epsilon^{-1}$. The traveling fronts with minimal speed are linearly stable in weighted $L^2$ spaces. We also prove local nonlinear stability of the traveling front with minimal speed when $\epsilon$ is smaller than the transition parameter.Comment: 24 page

Understanding European Regional Diversity - Lessons learned from Case Studies

The content of this report is a deliverable to the FP 7 project RUFUS (Rural future Networks) concerning the case studies made within the project. As a deliverable in a EU framework project it reports extensively on the methods and empirical data collected in the project’s case studies. The work has as an overarching motive to translate research findings into implications that are relevant for policy makers in the EU. The conclusions from the case studies are therefore of two types – the findings made and the implications they might give for policy making within the field of rural development

Mathematical description of bacterial traveling pulses

The Keller-Segel system has been widely proposed as a model for bacterial waves driven by chemotactic processes. Current experiments on {\em E. coli} have shown precise structure of traveling pulses. We present here an alternative mathematical description of traveling pulses at a macroscopic scale. This modeling task is complemented with numerical simulations in accordance with the experimental observations. Our model is derived from an accurate kinetic description of the mesoscopic run-and-tumble process performed by bacteria. This model can account for recent experimental observations with {\em E. coli}. Qualitative agreements include the asymmetry of the pulse and transition in the collective behaviour (clustered motion versus dispersion). In addition we can capture quantitatively the main characteristics of the pulse such as the speed and the relative size of tails. This work opens several experimental and theoretical perspectives. Coefficients at the macroscopic level are derived from considerations at the cellular scale. For instance the stiffness of the signal integration process turns out to have a strong effect on collective motion. Furthermore the bottom-up scaling allows to perform preliminary mathematical analysis and write efficient numerical schemes. This model is intended as a predictive tool for the investigation of bacterial collective motion

Inhibition of Ras activity coordinates cell fusion with cell-cell contact during yeast mating.

In the fission yeast Schizosaccharomyces pombe, pheromone signaling engages a signaling pathway composed of a G protein-coupled receptor, Ras, and a mitogen-activated protein kinase (MAPK) cascade that triggers sexual differentiation and gamete fusion. Cell-cell fusion requires local cell wall digestion, which relies on an initially dynamic actin fusion focus that becomes stabilized upon local enrichment of the signaling cascade on the structure. We constructed a live-reporter of active Ras1 (Ras1-guanosine triphosphate [GTP]) that shows Ras activity at polarity sites peaking on the fusion structure before fusion. Remarkably, constitutive Ras1 activation promoted fusion focus stabilization and fusion attempts irrespective of cell pairing, leading to cell lysis. Ras1 activity was restricted by the guanosine triphosphatase-activating protein Gap1, which was itself recruited to sites of Ras1-GTP and was essential to block untimely fusion attempts. We propose that negative feedback control of Ras activity restrains the MAPK signal and couples fusion with cell-cell engagement

Do we need a European scale of regional planning and design?

Conferència de Vincent Nadin dins el cicle "Projectar el territori" / "Regional Design in Europe" organitzat pel Professor Joaquim Sabate del Màster de Projectació Urbanistica del DUOT-UP