Search

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L $\infty$ norm of these solutions.

In this paper, we start from a very natural system of cross-diffusion equations, which can be seen as a gradient flow for the Wasserstein distance of a certain functional. Unfortunately, the cross-diffusion system is not well-posed, as a consequence of the fact that the underlying functional is not lower semi-continuous. We then consider the relaxation of the functional, and prove existence of a solution in a suitable sense for the gradient flow of (the relaxed functional). This gradient flow has also a cross-diffusion structure, but the mixture between two different regimes, that are determined by the relaxation, makes this study non-trivial., Comment: 38 pages, 2 figures, accepted version with minor revisions

The paper considers a forward-backward system of parabolic PDEs arising in a Mean Field Game (MFG) model where every agent controls the drift of a trajectory subject to Brownian diffusion, trying to escape a given bounded domain $\Omega$ in minimal expected time. The important point is that agents are constrained by a bound on the drift depending on the density of the other agents at their location (the higher the density, the smaller the velocity). Existence for a finite time horizon $T$ is proven via a fixed point argument but, because of the diffusion, the model should be studied in infinite horizon as the total mass inside the domain decreases in time, but never reaches zero in finite time. Hence, estimates are needed to pass the solution to the limit as $T\to\infty$, and the asymptotic behavior of the solution which is obtained in this way is also studied. This passes through a combination of classical parabolic arguments together with specific computations for MFGs. Both the Fokker--Planck equation ruling the evolution of the density of agents and the Hamilton--Jacobi--Bellman equation on the value function display Dirichlet boundary conditions as a consequence of the fact that agents stop as soon as they reach $\partial\Omega$. The initial datum for the density is given (and its regularity is discussed so as to have the sharpest result), and the long-time limit of the value function is characterized as the solution of a stationary problem.

We introduce a model designed to account for the influence of a line with fast diffusion-such as a road or another transport network-on the dynamics of a population in an ecological niche. This model consists of a system of coupled reaction-diffusion equations set on domains with different dimensions (line / plane). We first show that the presence of the line is always deleterious and can even lead the population to extinction. Next, we consider the case where the niche is subject to a displacement, representing the effect of a climate change or of seasonal variation of resources. We find that in such case the presence of the line with fast diffusion can help the population to persist. We also study several qualitative properties of this system. The analysis is based on a notion of generalized principal eigenvalue developed by the authors in [5].

We study some anisotropic heterogeneous nonlinear integral equations arising in epidemiology. We focus on the case where the heterogeneities are spatially periodic. In the first part of the paper, we show that the equations we consider exhibit a "threshold phenomenon". In the second part, we study the existence and non-existence of "traveling waves", and we provide a formula for the admissible speeds. In a third part, we apply our results to a spatial heterogeneous SIR model.

International audience; We investigate the large time behavior of solutions of reaction-diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous equation set in a domain with periodic holes, and specifically in the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.

We study the phenomenon of invasion for heterogeneous reaction-diffusion equations in periodic domains with monostable and combustion reaction terms. We give an answer to a question raised by Berestycki, Hamel and Nadirashvili concerning the connection between the speed of invasion and the critical speed of fronts. To do so, we extend the classical Freidlin–Gartner formula to such equations and we derive some bounds on the speed of invasion using estimates on the heat kernel. We also give geometric conditions on the domain that ensure that the spreading occurs at the critical speed of fronts.

This paper develops the notion and properties of the generalized principal eigenvalue for an elliptic system coupling an equation in a plane with one on a line in this plane, together with boundary conditions that express exchanges taking place between the plane and the line. This study is motivated by the reaction–diffusion model introduced by Berestycki, Roquejoffre and Rossi [The influence of a line with fast diffusion on Fisher–KPP propagation, J. Math. Biol. 66(4–5) (2013) 743–766] to describe the effect on biological invasions of networks with fast diffusion imbedded in a field. Here we study the eigenvalue associated with heterogeneous generalizations of this model. In a forthcoming work [Influence of a line with fast diffusion on an ecological niche, preprint (2018)] we show that persistence or extinction of the associated nonlinear evolution equation is fully accounted for by this generalized eigenvalue. A key element in the proofs is a new Harnack inequality that we establish for these systems and which is of independent interest.

Lower critical solubility temperature (LCST) for 3 polyolefins, polybutene-1 (PB1), polypentene-1 (PP1) and poly(4-methylpentene-1) (P4MP1), and the x interaction parameter in concentrated solutions for PB1 and the 33% ethylene ethylene-propylene copolymer have been measured in linear, branched, cyclic alkanes and some other solvents. Effects on x of the equation of state term, of correlations of molecular orientations (CMO) and of the solvent steric hindrance were investigated. The solvent density ds is found to be a good empirical parameter to characterize the equation of state term and to correlate the LCST. The parameter d s d p (where dp is the polymer density) affords an excellent correlation for the LCST of all polyolefins in normal and branched alkanes (polyethylene (PE) excepted). In dilute solution (at the LCST) the effect of CMO and solvent steric hindrance could not be distinguished from equation of state effects. However, values of x, found to be higher in branched than in linear alkanes in solutions of the linear polymer (PE) but not with the branched PB1 and the copolymer, are indicative of the importance in concentrated solutions of CMO even at high temperatures (100°–135°C). Furthermore, the lowering of x from linear PE to the branched PE and to the ethylene-propylene copolymers, following the expected diminution of CMO in the corresponding melts, is another indication of the persistance of CMO at high temperature. Solvent steric hindrance is seen to lower x (measured here by gas-liquid chromatography).