arXiv subjects

Lionel Roques

Publications and source records attributed to Lionel Roques.

28 records · Page 2Linked to original sources

arXiv2011

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.

math.AP↗

arXiv2011

Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $μ_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,ε).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases N=2 and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

math.AP↗

arXiv2010

On population resilience to external perturbations

We study a spatially explicit harvesting model in periodic or bounded environments. The model is governed by a parabolic equation with a spatially dependent nonlinearity of Kolmogorov--Petrovsky--Piskunov type, and a negative external forcing term $-δ$. Using sub- and supersolution methods and the characterization of the first eigenvalue of some linear elliptic operators, we obtain existence and nonexistence results as well as results on the number of stationary solutions. We also characterize the asymptotic behavior of the evolution equation as a function of the forcing term amplitude. In particular, we define two critical values $δ^*$ and $δ_2$ such that, if $δ$ is smaller than $δ^*$, the population density converges to a "significant" state, which is everywhere above a certain small threshold, whereas if $δ$ is larger than $δ_2$, the population density converges to a "remnant" state, everywhere below this small threshold. Our results are shown to be useful for studying the relationships between environmental fragmentation and maximum sustainable yield from populations. We present numerical results in the case of stochastic environments.

math.AP↗

arXiv2010

Models of population dynamics under the influence of external perturbations: mathematical results

In this Note, we describe the stationary equilibria and the asymptotic behaviour of an heterogeneous logistic reaction-diffusion equation under the influence of autonomous or time-periodic forcing terms. We show that the study of the asymptotic behaviour in the time-periodic forcing case can be reduced to the autonomous one, the last one being described in function of the "size" of the external perturbation. Our results can be interpreted in terms of maximal sustainable yields from populations. We briefly discuss this last aspect through a numerical computation.

math.AP↗

arXiv2010

On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation

This paper is devoted to the analysis of some uniqueness properties of a classical reaction-diffusion equation of Fisher-KPP type, coming from population dynamics in heterogeneous environments. We work in a one-dimensional interval $(a,b)$ and we assume a nonlinear term of the form $u \, (μ(x)-γu)$ where $μ$ belongs to a fixed subset of $C^{0}([a,b])$. We prove that the knowledge of $u$ at $t=0$ and of $u$, $u_x$ at a single point $x_0$ and for small times $t\in (0,\varepsilon)$ is sufficient to completely determine the couple $(u(t,x),μ(x))$ provided $γ$ is known. Additionally, if $u_{xx}(t,x_0)$ is also measured for $t\in (0,\varepsilon)$, the triplet $(u(t,x),μ(x),γ)$ is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of $u$ and $u_x$ at a single point $x_0$ (and for $t\in (0,\varepsilon)$) are sufficient to obtain a good approximation of the coefficient $μ(x).$ These numerical simulations also show that the measurement of the derivative $u_x$ is essential in order to accurately determine $μ(x)$.

math.AP↗

arXiv2009

Homogenization and influence of fragmentation in a biological invasion model

In this paper, some properties of the minimal speeds of pulsating Fisher-KPP fronts in periodic environments are established. The limit of the speeds at the homogenization limit is proved rigorously. Near this limit, generically, the fronts move faster when the spatial period is enlarged, but the speeds vary only at the second order. The dependence of the speeds on habitat fragmentation is also analyzed in the case of the patch model.

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↗

arXiv2009

Does reaction-diffusion support the duality of fragmentation effect?

There is a gap between single-species model predictions, and empirical studies, regarding the effect of habitat fragmentation per se, i.e., a process involving the breaking apart of habitat without loss of habitat. Empirical works indicate that fragmentation can have positive as well as negative effects, whereas, traditionally, single-species models predict a negative effect of fragmentation. Within the class of reaction-diffusion models, studies almost unanimously predict such a detrimental effect. In this paper, considering a single-species reaction-diffusion model with a removal -- or similarly harvesting -- term, in two dimensions, we find both positive and negative effects of fragmentation of the reserves, i.e. the protected regions where no removal occurs. Fragmented reserves lead to higher population sizes for time-constant removal terms. On the other hand, when the removal term is proportional to the population density, higher population sizes are obtained on aggregated reserves, but maximum yields are attained on fragmented configurations, and for intermediate harvesting intensities.

math.AP↗

arXiv2009

Biological invasions: deriving the regions at risk from partial measurements

We consider the problem of forecasting the regions at higher risk for newly introduced invasive species. Favourable and unfavourable regions may indeed not be known a priori, especially for exotic species whose hosts in native range and newly-colonised areas can be different. Assuming that the species is modelled by a logistic-like reaction-diffusion equation, we prove that the spatial arrangement of the favourable and unfavourable regions can theoretically be determined using only partial measurements of the population density: 1) a local "spatio-temporal" measurement, during a short time period and, 2) a "spatial" measurement in the whole region susceptible to colonisation. We then present a stochastic algorithm which is proved analytically, and then on several numerical examples, to be effective in deriving these regions.

math.AP↗

arXiv2009

Fast propagation for KPP equations with slowly decaying initial conditions

This paper is devoted to the analysis of the large-time behavior of solutions of one-dimensional Fisher-KPP reaction-diffusion equations. The initial conditions are assumed to be globally front-like and to decay at infinity towards the unstable steady state more slowly than any exponentially decaying function. We prove that all level sets of the solutions move infinitely fast as time goes to infinity. The locations of the level sets are expressed in terms of the decay of the initial condition. Furthermore, the spatial profiles of the solutions become asymptotically uniformly flat at large time. This paper contains the first systematic study of the large-time behavior of solutions of KPP equations with slowly decaying initial conditions. Our results are in sharp contrast with the well-studied case of exponentially bounded initial conditions.

math.AP↗