arXiv subjects

Grégoire Nadin

Publications and source records attributed to Grégoire Nadin.

33 records · Page 2Linked to original sources

arXiv2017

Competition in periodic media: II -- Segregative limit of pulsating fronts and "Unity is not Strength"-type result

This paper is concerned with the limit, as the interspecific competition rate goes to infinity, of pulsating front solutions in space-periodic media for a bistable two-species competition--diffusion Lotka--Volterra system. We distinguish two important cases: null asymptotic speed and non-null as-ymptotic speed. In the former case, we show the existence of a segregated stationary equilibrium. In the latter case, we are able to uniquely characterize the segregated pulsating front, and thus full convergence is proved. The segregated pulsating front solves an interesting free boundary problem. We also investigate the sign of the speed as a function of the parameters of the competitive system. We are able to determine it in full generality, with explicit conditions depending on the various parameters of the problem. In particular, if one species is sufficiently more motile or competitive than the other, then it is the invader. This is an extension of our previous work in space-homogeneous media.

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arXiv2017

Hindrances to bistable front propagation: application to Wolbachia invasion

We study the biological situation when an invading population propagates and replaces an existing population with different characteristics. For instance, this may occur in the presence of a vertically transmitted infection causing a cytoplasmic effect similar to the Allee effect (e.g. Wolbachia in Aedes mosquitoes): the invading dynamics we model is bistable. After quantification of the propagules, a second question of major interest is the invasive power. How far can such an invading front go, and what can stop it? We rigorously show that a heterogeneous environment inducing a strong enough population gradient can stop an invading front, which will converge in this case to a stable front. We characterize the critical population jump, and also prove the existence of unstable fronts above the stable (blocking) fronts. Being above the maximal unstable front enables an invading front to clear the obstacle and propagate further. We are particularly interested in the case of artificial Wolbachia infection, used as a tool to fight arboviruses.

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arXiv2016

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 $ε^{-1}$ ($ε\textgreater{}0$), 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 $ε$: for small $ε$ the behaviour is essentially the same as for the diffusive Fisher-KPP equation. However, for large $ε$ the traveling front with minimal speed is discontinuous and travels at the maximal speed $ε^{-1}$. 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 $ε$ is smaller than the transition parameter.

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arXiv2016

Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalueproblem in a fixed box $Ω$ and we consider the general case of Robin boundary conditions on $\partialΩ$. It is well known that it suffices to consider {\it bang-bang} weights taking two values of different signs, that can be parametrized by the characteristic function of the subset $E$ of $Ω$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to $E$, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.

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arXiv2016

Traveling fronts in space-time periodic media

This paper is concerned with the existence of pulsating traveling fronts for the equation: $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = f (t, x, u)$, (1) where the diffusion matrix $A$, the advection term $q$ and the reaction term $f$ are periodic in $t$ and $x$. We prove that there exist some speeds $c^*$ and $c^{**}$ such that there exists a pulsating traveling front of speed $c$ for all $c\ge c^{**}$ and that there exists no such front of speed $c<c^*$. We also give some spreading properties for front-like initial data. In the case of a KPP-type reaction term, we prove that $c^* = c^{**}$ and we characterize this speed with the help of a family of eigenvalues associated with the equation. If $f$ is concave with respect to $u$, we prove some Lipschitz continuity for the profile of the pulsating traveling front. Cet articl{é} etudie l'existence de fronts pulsatoires pour l'équation : $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = f (t, x, u)$, (2) où la matrice de diffusion A, le terme d'advection $q$ et le terme de r{é}action $f$ sont p{é}riodiques en $t$ et en $x$. Nous prouvons l'existence de deux vitesses $c^*$ et $c^{**}$ telles qu'il existe un front pulsatoire de vitesse $c$ pour tout $c \ge c^{**}$ et qu'il n'existe pas de tel front de vitesse $c<c^*$. Nous donnons egalement des propri{é}t{é}s de spreading pour des donn{é}es initiales ressemblant a des fronts. Dans le cas d'un terme de r{é}action de type KPP, nous prouvons que $c^* = c^{**}$ et nous caract{é}risons cette vitess{è} a l'aide d'une famille de valeurs propres associ{é}{è} a l'équation. Si $f$ est concave en $u$, nous montrons que le profil du front pulsatoire construit est lipschitzien.

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arXiv2016

Generalized transition fronts for one-dimensional almost periodic Fisher-KPP equations

This paper investigates the existence of generalized transition fronts for Fisher-KPP equations in one-dimensional, almost periodic media. Assuming that the linearized elliptic operator near the unstable steady state admits an almost periodic eigenfunction, we show that such fronts exist if and only if their average speed is above an explicit threshold. This hypothesis is satisfied in particular when the reaction term does not depend on x or (in some cases) is small enough. Moreover, except for the threshold case, the fronts we construct and their speeds are almost periodic, in a sense. When our hypothesis is no longer satisfied, such generalized transition fronts still exist for an interval of average speeds, with explicit bounds. Our proof relies on the construction of sub and super solutions based on an accurate analysis of the properties of the generalized principal eigenvalues.

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arXiv2016

Some dependence results between the spreading speed and the coefficients of the space--time periodic Fisher--KPP equation

We investigate in this paper the dependence relation between the space-time periodic coefficients $A, q$ and $μ$ of the reaction-diffusion equation $\partial_t u - \nabla \cdot (A(t, x)\nabla u) + q(t, x) \cdot \nabla u = μ(t, x) u(1 - u)$, and the spreading speed of the solutions of the Cauchy problem associated with this equation and compactly supported initial data. We prove in particular that (1) taking the spatial or temporal average of $μ$ decreases the minimal speed, (2) if the coefficients do not depend on $t$ and $q\not\equiv 0$, then increasing the amplitude of the diffusion matrix $A$ increases the minimal speed, (3) if $A = I_N$, $μ$ is a constant, then the introduction of a space periodic drift term $q = \nabla Q$ increases the minimal speed. To prove these results, we use a variational characterization of the spreading speed that involves a family of periodic principal eigenvalues associated with the linearization of the equation near 0. We are thus back to the investigation of the dependence relation between this family of eigenvalues and the coefficients.

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arXiv2016

The Effect of the Schwarz Rearrangement on the Periodic Principal Eigenvalue of a Nonsymmetric Operator

This paper is concerned with the periodic principal eigenvalue $k_λ(μ)$ associated with the operator $- {d^2\over dx^2} - 2λ{d\over dx} - μ(x) - λ^2$ , (1) where $λ\in \mathbb{R}$ and $μ$ is continuous and periodic in $x\in\mathbb{R}$. Our main result is that $k_λ(μ^*) \le k_λ(μ)$, where $μ^*$ is the Schwarz rearrangement of the function $μ$. From a population dynamics point of view, using reaction-diffusion modeling, this result means that the fragmentation of the habitat of an invading population slows down the invasion. We prove that this property does not hold in higher dimension, if $μ^*$ is the Steiner symmetrization of $μ$. For heterogeneous diffusion and advection, we prove that increasing the period of the coefficients decreases $k_λ$ and we compute the limit of $k_λ$ when the period of the coefficients goes to 0. Lastly, we prove that, in dimension 1, rearranging the diffusion term decreases $k_λ$. These results rely on some new formula for the periodic principal eigenvalue of a nonsymmetric operator.

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arXiv2016

How does the spreading speed associated with the Fisher-KPP equation depend on random stationary diffusion and reaction terms?

We consider one-dimensional reaction-diffusion equations of Fisher-KPP type with random stationary ergodic coefficients. A classical result of Freidlin and Gartner [16] yields that the solutions of the initial value problems associated with compactly supported initial data admit a linear spreading speed almost surely. We use in this paper a new characterization of this spreading speed recently proved in [8] in order to investigate the dependence of this speed with respect to the heterogeneity of the diffusion and reaction terms. We prove in particular that adding a reaction term with null average or rescaling the coefficients by the change of variables x $\rightarrow$ x/L, with L \textgreater{} 1, speeds up the propagation. From a modelling point of view, these results mean that adding some heterogeneity in the medium gives a higher invasion speed, while fragmentation of the medium slows down the invasion.

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arXiv2016

Transition waves for Fisher-KPP equations with general time-heterogeneous and space-periodic coefficients

This paper is devoted to existence and non-existence results for generalized tran-sition waves solutions of space-time heterogeneous Fisher-KPP equations. When the coefficients of the equation are periodic in space but otherwise depend in a fairly gen-eral fashion on time, we prove that such waves exist as soon as their speed is sufficiently large in a sense. When this speed is too small, transition waves do not exist anymore, this result holds without assuming periodicity in space. These necessary and sufficient conditions are proved to be optimal when the coefficients are periodic both in space and time. Our method is quite robust and extends to general non-periodic space-time heterogeneous coefficients, showing that transition waves solutions of the nonlinear equation exist as soon as one can construct appropriate solutions of a given linearized equation.

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arXiv2016

Spreading speeds for one-dimensional monostable reaction-diffusion equations

We establish in this article spreading properties for the solutions of equations of the type $\partial$ t u -- a(x)$\partial$ xx u -- q(x)$\partial$ x u = f (x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable KPP type between two steady states 0 and 1 and the initial datum is compactly sup-ported. Using homogenization techniques, we construct two speeds w $\le$ w such that lim t$\rightarrow$+$\infty$ sup 0$\le$x$\le$wt |u(t, x)--1| = 0 for all w $\in$ (0, w) and lim t$\rightarrow$+$\infty$ sup x$\ge$wt |u(t, x)| = 0 for all w \textgreater{} w. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particu-lar, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where w = w).

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arXiv2015

Travelling waves for diffusive and strongly competitive systems: relative motility and invasion speed

Our interest here is to find the invader in a two species, diffusive and competitive Lotka-Volterra system in the particular case of travelling wave solutions. We investigate the role of diffusion in homogeneous domains. We might expect a priori two different cases: strong interspecific competition and weak interspecific competition. In this paper, we study the first one and obtain a clear conclusion: the invading species is, up to a fixed multiplicative constant, the more diffusive one.

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arXiv2014

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 $t^{3/2}$.

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arXiv2014

What is the optimal shape of a fin for one dimensional heat conduction?

This article is concerned with the shape of small devices used to control the heat flowing between a solid and a fluid phase, usually called \textsl{fin}. The temperature along a fin in stationary regime is modeled by a one-dimensional Sturm-Liouville equation whose coefficients strongly depend on its geometrical features. We are interested in the following issue: is there any optimal shape maximizing the heat flux at the inlet of the fin? Two relevant constraints are examined, by imposing either its volume or its surface, and analytical nonexistence results are proved for both problems. Furthermore, using specific perturbations, we explicitly compute the optimal values and construct maximizing sequences. We show in particular that the optimal heat flux at the inlet is infinite in the first case and finite in the second one. Finally, we provide several extensions of these results for more general models of heat conduction, as well as several numerical illustrations.

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arXiv2011

Propagation phenomena for time heterogeneous KPP reaction-diffusion equations

We investigate in this paper propagation phenomena for the heterogeneous reaction-diffusion equation $\partial_t u -Δu = f(t,u)$, $x\in R^N$, $t\in\R$, where f=f(t,u) is a KPP monostable nonlinearity which depends in a general way on t. A typical f which satisfies our hypotheses is f(t,u)=m(t) u(1-u), with m bounded and having positive infimum. We first prove the existence of generalized transition waves (recently defined by Berestycki and Hamel, Shen) for a given class of speeds. As an application of this result, we obtain the existence of random transition waves when f is a random stationary ergodic function with respect to t. Lastly, we prove some spreading properties for the solution of the Cauchy problem.

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