arXiv subjects

Thomas Giletti

Publications and source records attributed to Thomas Giletti.

At least 19 recordsLinked to original sources

arXiv2026

Spreading dynamics for diffusive competition systems in shifting environments

We study the spreading dynamics for two-species diffusive competition systems in shifting environments caused by climate changes. Our main goal is to derive conditions for extinction and persistence of each species in the case of a strong-weak competition. Depending on the pace of the climate change, but also on whether the strong competitor is faster or slower, we will uncover dramatically different outcomes in the asymptotic behavior of solutions. For instance, it may be that a weak and fast competitor survives while a strong and slow one does not. Furthermore, we find some parameter regime where the outcome depends not only on the climate change speed, but also on both species' resilience to it, and even on the initial populations' distributions. Our results show how strikingly complex the effect of climate change may be on population dynamics as soon as several species are considered.

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arXiv2026

Traveling waves of a reaction-diffusion system with partially coupled diffusion

This work is devoted to the study of traveling wave solutions of reaction-diffusion systems, where the diffusion rate of~$v$, the second component, depends on~$u$, the first component. Such systems arise in prey-predator models, where the predator~$v$ is actively hunting its prey~$u$, or in epidemiological models where the disease induces erratic behavior, for example rabies. This results in a quasilinear coupling in the highest-order term of the equation for~$v$. From the theoretical point of view, we fully classify traveling wave solutions when the first component does not diffuse, in which case the problem can be reduced to a nonlinear scalar equation. The case where both components diffuse is investigated numerically.

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arXiv2025

Stability of propagating terraces in spatially periodic multistable equations in $\mathbb{R}^N$

In this paper, we study the large time behaviour of solutions of multistable reaction-diffusion equations in $\mathbb{R}^N$, with a spatially periodic heterogeneity. By multistable, we mean that the problem admits a finite -- but arbitrarily large -- number of stable, periodic steady states. In contrast with the more classical monostable and bistable frameworks, which exhibit the emergence of a single travelling front in the long run, in the present case the large time dynamics is governed by a family of stacked travelling fronts, involving intermediate steady states, called propagating terrace. Their existence in the multidimensional case has been established in our previous work [13]. The first result of the present paper is their uniqueness. Next, we show that the speeds of the propagating terraces in different directions dictate the spreading speeds of solutions of the Cauchy problem, for both planar-like and compactly supported initial data. The latter case turns out to be much more intricate than the former, due to the fact that the propagating terraces in distinct directions may involve different sets of intermediate steady states. Another source of difficulty is that the Wulff shape of the speeds of travelling fronts can be non-smooth, as we show in the bistable case using a result of [4].

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arXiv2024

The Bramson correction in the Fisher-KPP equation: from delay to advance

We consider the solution to the scalar Fisher-KPP equation with front-like initial data, focusing on the location of its level sets at large times, particularly their deviation from points moving at the known spreading speed. We consider an intermediate case for the tail of the initial data, where the decay rate approaches, up to a polynomial term, that of the traveling wave with minimal speed. This approach enables us to capture deviations of the form $-r \ln t$ with $r \< \frac{3}{2}$, which corresponds to a logarithmic delay when $0 \< r \< \frac{3}{2}$ and a logarithmic advance when $r \< 0$. The critical case $r=\frac 32$ is also studied, revealing an extra $\mathcal O(\ln \ln t)$ term. Our arguments involve the construction of new sub- and super-solutions based on preliminary formal computations on the equation with a moving Dirichlet condition. Finally, convergence to the traveling wave with minimal speed is addressed.

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arXiv2024

Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size

This paper is concerned with spreading properties of space-time heterogeneous Fisher--KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or intermediate growth, a central patch with fast growth and a right half-line with slow or intermediate growth. The central patch moves at various speeds. The behavior of the front changes drastically depending on the speed of the central patch. Among other things, intriguing phenomena such as nonlocal pulling and locking may occur, which would make the behavior of the front further complicated. The problem we discuss here is closely related to questions in biomathematical modelling. By considering several special cases, we illustrate the remarkable diversity of possible behaviors. In particular, when the central patch has constant size and constant speed, we provide a complete set of explicit formulas for the spreading speed.

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arXiv2022

Forced waves of a three species predator-prey system with a pair of weak-strong competing preys in a shifting environment

In this paper, we investigate so-called forced wave solutions of a three components reaction-diffusion system from population dynamics. Our system involves three species that are respectively two competing preys and one predator; moreover, the competition between both preys is strong, i.e. in the absence of the predator, one prey is driven to extinction and the other survives. Furthermore, our problem includes a spatio-temporal heterogeneity in a moving variable that typically stands as a model for climate shift. In this context, forced waves are special stationary solutions which are expected to describe the large-time behavior of solutions, and in particular to provide criteria on the climate shift speed to allow survival of either of the three species. We will consider several types of forced waves to deal with various situations depending on which species are indigenous and which species are aboriginal.

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arXiv2022

Terrace solutions for non-Lipschitz multistable nonlinearities

Traveling wave solutions of reaction-diffusion equations are well-studied for Lipschitz continuous monostable and bistable reaction functions. These special solutions play a key role in mathematical biology and in particular in the study of ecological invasions. However, if there are more than two stable steady states, the invasion phenomenon may become more intricate and involve intermediate steps, which leads one to consider not a single but a collection of traveling waves with ordered speeds. In this paper we show that, if the reaction function is discontinuous at the stable steady states, then such a collection of traveling waves exists and even provides a special solution which we call a terrace solution. More precisely, we will address both the existence and uniqueness of the terrace solution.

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arXiv2022

Convergence to a terrace solution in multistable reaction-diffusion equations with discontinuities

In this paper we address the large-time behavior of solutions of bistable and multistable reaction-diffusion equations with discontinuities around the stable steady states. We show that the solution always converges to a special solution, which may either be a traveling wave in the bistable case, or more generally a terrace (i.e. a collection of stacked traveling waves with ordered speeds) in the multistable case.

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arXiv2022

A PDE-ODE Coupled Spatio-Temporal Mathematical Model for Fire Blight During Bloom

Fire blight is a bacterial plant disease that affects apple and pear trees. We present a mathematical model for its spread in an orchard during bloom. This is a PDE-ODE coupled system, consisting of two semilinear PDEs for the pathogen, coupled to a system of three ODEs for the stationary hosts. Exploratory numerical simulations suggest the existence of travelling waves, which we subsequently prove, under some conditions on parameters, using the method of upper and lower bounds and Schauder's fixed point theorem. Our results are likely not optimal in the sense that our constraints on parameters, which can be interpreted biologically, are sufficient for the existence of travelling waves, but probably not necessary. Possible implications for fire blight biology and management are discussed.

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arXiv2021

Monostable pulled fronts and logarithmic drifts

In this work we investigate the issue of logarithmic drifts in the position of the level sets of solutions of monostable reaction-diusion equations, with respect to the traveling front with minimal speed. On the one hand, it is a celebrated result of Bramson that such a logarithmic drift occurs when the reaction is of the KPP (or sublinear) type. On the other hand, it is also known that this drift phenomenon disappears when the minimal front speed is nonlinearly determined. However, some monostable reaction-diusion equations fall in neither of those cases and our aim is to fill that gap. We prove that a logarithmic drift always occurs when the speed is linearly determined, but surprisingly we find that the factor in front of the logarithmic term may be different from the KPP case.

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arXiv2021

Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal

We are concerned with the persistence of both predator and prey in a diffusive predator-prey system with a climate change effect, which is modeled by a spatial-temporal heterogeneity depending on a moving variable. Moreover, we consider both the cases of nonlocal and local dispersal. In both these situations, we first prove the existence of forced waves, which are positive stationary solutions in the moving frames of the climate change, of either front or pulse type. Then we address the persistence or extinction of the prey and the predator separately in various moving frames, and achieve a complete picture in the local diffusion case. We show that the survival of the species depends crucially on how the climate change speed compares with the minimal speed of some pulse type forced waves.

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arXiv2021

On the modelling of spatially heterogeneous nonlocal diffusion: deciding factors and preferential position of individuals

We develop general heterogeneous nonlocal diffusion models and investigate their connection to local diffusion models by taking a singular limit of focusing kernels. We reveal the link between the two groups of diffusion equations which include both spatial heterogeneity and anisotropy. In particular, we introduce the notion of deciding factors which single out a nonlocal diffusion model and typically consist of the total jump rate and the average jump length. In this framework, we also discuss the dependence of the profile of the steady state solutions on these deciding factors, thus shedding light on the preferential position of individuals.

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arXiv2021

Asymptotic spreading for Fisher-KPP reaction-diffusion equations with heterogeneous shifting diffusivity

We determine the asymptotic spreading speed of the solutions of a Fisher-KPP reaction-diffusion equation, starting from compactly supported initial data, when the diffusion coefficient is a fixed bounded monotone profile that is shifted at a given forcing speed and satisfies a general uniform ellipticity condition. Depending on the monotony of the profile, we are able to characterize this spreading speed as a function of the forcing speed and the two linear spreading speeds associated to the asymptotic problems. Most notably, when the profile of the coefficient diffusion is increasing we show that there is an intermediate range for the forcing speed where spreading actually occurs at a speed which is larger than the linear speed associated with the homogeneous state around the position of the front. We complement our study with the construction of strictly monotone traveling front solutions with strong exponential decay near the unstable state when the profile of the coefficient diffusion is decreasing and in the regime where the forcing speed is precisely the selected spreading speed.

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arXiv2020

Persistence of preys in a diffusive three species predator-prey system with a pair of strong-weak competing preys

We investigate the traveling wave solutions of a three-species system involving a single predator and a pair of strong-weak competing preys. Our results show how the predation may affect this dynamics. More precisely, we describe several situations where the environment is initially inhabited by the predator and by either one of the two preys. When the weak competing prey is an aboriginal species, we show that there exist traveling waves where the strong prey invades the environment and either replaces its weak counterpart, or more surprisingly the three species eventually co-exist. Furthermore, depending on the parameters, we can also construct traveling waves where the weaker prey actually invades the environment initially inhabited by its strong competitor and the predator. Finally, our results on the existence of traveling waves are sharp, in the sense that we find the minimal wave speed in all those situations.

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arXiv2020

Asymptotic spreading speeds for a predator-prey system with two predators and one prey

This paper investigates the large time behaviour of a three species reaction-diffusion system, modelling the spatial invasion of two predators feeding on a single prey species. In addition to the competition for food, the two predators exhibit competitive interactions and under some parameter conditions ($\mu>0$), they can also be considered as two mutants. When mutations occur in the predator populations, the spatial spread of invasion takes place at a definite speed, identical for both mutants. When the two predators are not coupled through mutation, the spreading behaviour exhibits a more complex propagating pattern, including multiple layers with different speeds. In addition, some parameter conditions reveal situations where a nonlocal pulling phenomenon occurs and in particular where the spreading speed is not linearly determined.

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arXiv2020

Propagation for KPP bulk-surface systems in a general cylindrical domain

In this paper, we investigate propagation phenomena for KPP bulk-surface systems in a cylindrical domain with general section and heterogeneous coefficients. As for the scalar KPP equation, we show that the asymptotic spreading speed of solutions can be computed in terms of the principal eigenvalues of a family of self-adjoint elliptic operators. Using this characterization, we analyze the dependence of the spreading speed on various parameters, including diffusion rates and the size and shape of the section of the domain. In particular, we provide new theoretical results on several asymptotic regimes like small and high diffusion rates and sections with small and large sizes. These results generalize earlier ones which were available in the radial homogeneous case. Finally, we numerically investigate the issue of shape optimization of the spreading speed. By computing its shape derivative, we observe, in the case of homogeneous coefficients, that a disk either maximizes or minimizes the speed, depending on the parameters of the problem, both with or without constraints. We also show the results of numerical shape optimization with non homogeneous coefficients, when the disk is no longer an optimizer.

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arXiv2020

Admissible speeds in spatially periodic bistable reaction-diffusion equations

Spatially periodic reaction-diffusion equations typically admit pulsating waves which describe the transition from one steady state to another. Due to the heterogeneity, in general such an equation is not invariant by rotation and therefore the speed of the pulsating wave may a priori depend on its direction. However, little is actually known in the literature about whether it truly does: surprisingly, it is even known in the one-dimensional monostable Fisher-KPP case that the speed is the same in the opposite directions despite the lack of symmetry. Here we investigate this issue in the bistable case and show that the set of admissible speeds is actually rather large, which means that the shape of propagation may indeed be asymmetrical. More precisely, we show in any spatial dimension that one can choose an arbitrary large number of directions , and find a spatially periodic bistable type equation to achieve any combination of speeds in those directions, provided those speeds have the same sign. In particular, in spatial dimension 1 and unlike the Fisher-KPP case, any pair of (either nonnegative or nonpositive) rightward and leftward wave speeds is admissible. We also show that these variations in the speeds of bistable pulsating waves lead to strongly asymmetrical situations in the multistable equations.

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arXiv2020

Spreading and vanishing for a monostable reaction-diffusion equation with forced speed

Invasion phenomena for heterogeneous reaction-diffusion equations are contemporary and challenging questions in applied mathematics. In this paper we are interested in the question of spreading for a reaction-diffusion equation when the subdomain where the reaction term is positive is shifting/contracting at a given speed $c$. This problem arises in particular in the modelling of the impact of climate change on population dynamics. By placing ourselves in the appropriate moving frame, this leads us to consider a reaction-diffusion-advection equation with a heterogeneous in space reaction term, in dimension $N\geq1$. We investigate the behaviour of the solution $u$ depending on the value of the advection constant~$c$, which typically stands for the velocity of climate change. We find that, when the initial datum is compactly supported, there exists precisely three ranges for $c$ leading to drastically different situations. In the lower speed range the solution always spreads, while in the upper range it always vanishes. More surprisingly, we find that that both spreading and vanishing may occur in an intermediate speed range. The threshold between those two outcomes is always sharp, both with respect to $c$ and to the initial condition. We also briefly consider the case of an exponentially decreasing initial condition, where we relate the decreasing rate of the initial condition with the range of values of~$c$ such that spreading occurs.

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