Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How the patchiness of landscapes affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behavior to local habitat quality and show preferences for some habitat types over others. In this thesis, we focus on how landscape composition and the movement behaviour of individuals at an interface between patches of different quality affect the steady state of a single species. We consider a model of reaction-diffusion equations for the temporal evolution of the density of the population in space. Individual movement is described by a diffusion process, e.g., an uncorrelated random walk. Population net growth is encapsulated in the growth function that considers birth and death of individuals, including nonlinear effects that arise from competition and/or facilitation within the species. We consider the simplest case of two adjacent one-dimensional patches, e.g., two intervals on the real line that share one boundary point. Conditions are homogeneous within a patch but differ between patches. The movement behaviour of individuals between the two patches is incorporated into matching conditions of population flux and density at the interface between patches, i.e., the boundary point that the intervals share. These matching conditions turn out to be continuous in the flux but discontinuous in the density. Several authors have studied similar models recently. Most of these studies consider monostable dynamics on both patches, i.e., logistic growth. Under logistic growth, the net population growth rate is a strictly decreasing function of population density. Logistic population dynamics are very simple: the population extinction state is unstable and a positive steady state is globally asymptotically stable. In this work, we also include bistable dynamics, i.e., an Allee effect. Biologically, an Allee effect occurs when individuals cooperate at some level so that the net population growth rate is increasing with population density for at least some low or intermediate densities. Models with Allee growth typically exhibit bistability: there are two locally stable steady states, one at low density (possibly zero) and one at high density. This bistability makes mathematical analysis more challenging, but leads to more interesting results in return. Mathematically, most existing work on related models is based on linear stability analysis of the extinction state. We focus on the nonlinear models and specifically on positive steady states. We establish the existence, uniqueness and - in some cases - global asymptotic stability of a positive steady state. We classify the shape of these states depending on movement behaviour. We clarify the role of movement in this context. In particular, we investigate the following prior observation: a randomly diffusing population at steady state in a continuously varying habitat can exceed its carrying capacity. Our results clarify when and under which conditions this effect can arise in our two-patch landscape. The analysis of the model with an Allee effect on one of the two patches yields a rich and interesting structure of steady states. Under certain parameter conditions, some of these states are amenable to explicit stability calculations. These yield insights into the possible bifurcations that can occur in our system. Finally, we indicate how the model and analysis here can be extended to systems of reaction-diffusion equations on graphs that represent natural habitats with different geometries, for example watersheds.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第23675号 / 理博第4765号 / 新制||理||1683(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 坂上 貴之, 教授 泉 正己, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM

Many biological populations reside in increasingly fragmented landscapes, which arise from human activities and natural causes. Landscape characteristics may change abruptly in space and create sharp transitions (interfaces) in landscape quality. How patchy landscape affects ecosystem diversity and stability depends, among other things, on how individuals move through the landscape. Individuals adjust their movement behaviour to local habitat quality and show preferences for some habitat types over others. In this dissertation, we focus on how landscape composition and the movement behaviour at an interface between habitat patches of different quality affects the steady states of a single species and a predator-prey system. First, we consider a model for population dynamics in a habitat consisting of two homogeneous one-dimensional patches in a coupled ecological reaction-diffusion equation. Several recent publications by other authors explored how individual movement behaviour affects population-level dynamics in a framework of reaction-diffusion systems that are coupled through discontinuous boundary conditions. The movement between patches is incorporated into the interface conditions. While most of those works are based on linear analysis, we study positive steady states of the nonlinear equations. We establish the existence, uniqueness and global asymptotic stability of the steady state, and we classify their qualitative shape depending on movement behaviour. We clarify the role of nonrandom movement in this context, and we apply our analysis to a previous result where it was shown that a randomly diffusing population in a continuously varying habitat can exceed the carrying capacity at steady state. In particular, we apply our results to study the question of why and under which conditions the total population abundance at steady state may exceed the total carrying capacity of the landscape. Secondly, we model population dynamics with a predator-prey system in a coupled ecological reaction-diffusion equation in a heterogeneous landscape to study Turing patterns that emerge from diffusion-driven instability (DDI). We derive the DDI conditions, which consist of necessary and sufficient conditions for initiation of spatial patterns in a one-dimensional homogeneous landscape. We use a finite difference scheme method to numerically explore the general conditions using the May model, and we present numerical simulations to illustrate our results. Then we extend our studies on Turing-pattern formation by considering a predator-prey system on an infinite patchy periodic landscape. The movement between patches is incorporated into the interface conditions that link the reaction-diffusion equations between patches. We use a homogenization technique to obtain an analytically tractable approximate model and determine Turing-pattern formation conditions. We use numerical simulations to present our results from this approximation method for this model. With this tool, we then explore how differential movement and habitat preference of both species in this model (prey and predator) affect DDI.

This dissertation presents a detailed account of the study undertaken on the subject of heat and mass transfer phenomena in porous media. The current work specifically targets the general reaction-diffusion systems arising in separation processes using porous sorbent particles. These particles are comprised of pore channels spanning length scales over almost three orders of magnitude while involving a variety of physical processes such as mass diffusion, heat transfer and surface adsorption-desorption. A novel methodology is proposed in this work that combines models that account for the multi-scale and multi-physics phenomena involved. Pore-resolving DNS calculations using an immersed boundary method (IBM) framework are used to simulate the macro-scale physics while the phenomena at smaller scales are modeled using a sub-pore modeling technique. The IBM scheme developed as part of this work is applicable to complex geometries on curvilinear grids, while also being very efficient, consuming less than 1% of the total simulation time per time-step. A new method of implementing the conjugate heat transfer (CHT) boundary condition is proposed which is a direct extension of the method used for other boundary conditions and does not involve any complex interpolations like previous CHT implementations using IBM. Detailed code verification and validation studies are carried out to demonstrate the accuracy of the developed method. The developed IBM scheme is used in conjunction with a stochastic reconstruction procedure based on simulated annealing. The developed framework is tested in a two-dimensional channel with two types of porous sections - one created using a random assembly of square blocks and another using the stochastic reconstruction procedure. Numerous simulations are performed to demonstrate the capability of the developed framework. The computed pressure drops across the porous section are compared with predictions from the Darcy-Forchheimer equation for media composed of different structure sizes. The developed methodology is also applied to CO2 diffusion studies in porous spherical particles of varying porosities. For the pore channels that are unresolved by the IBM framework, a sub-pore modeling methodology developed as part of this work which solves a one-dimensional unsteady diffusion equation in a hierarchy of scales represented by a fractal-type geometry. The model includes surface adsorption-desorption, and heat generation and absorption. It is established that the current framework is useful and necessary for reaction-diffusion problems in which the adsorption time scales are very small (diffusion-limited) or comparable to the diffusion time scales. Lastly, parametric studies are conducted for a set of diffusion-limited problems to showcase the powerful capability of the developed methodology. / Ph. D.

This thesis deals with a newly-developed model for the spatiotemporal calcium dynamics within presynaptic terminals. The model is based on single-protein kinetics and has been used to successfully describe different neuron types such as pyramidal neurons in the rat neocortex and the Calyx of Held of neurons from the rat brainstem. A limited number of parameters had to be adjusted to fluorescence measurements of the calcium concentration. These values can be interpreted as a prediction of the model, and in particular the protein densities can be compared to independent experiments. The contribution of single proteins to the total calcium dynamics has been analysed in detail for voltage-dependent calcium channel, plasma-membrane calcium ATPase, sodium-calcium exchanger, and endogenous as well as exogenous buffer proteins. The model can be used to reconstruct the unperturbed calcium dynamics from measurements using fluorescence indicators. The calcium response to different stimuli has been investigated in view of its relevance for synaptic plasticity. This work provides a first step towards a description of the complete synaptic transmission using single-protein data.

Dans différents domaines de recherche, la modélisation est devenue un outil efficace pour étudier et prédire l’évolution possible d’un système, en particulier en épidémiologie. En raison de la mondialisation et de la mutation génétique de certaines maladies ou vecteurs de transmission, plusieurs épidémies sont apparues dans des régions non encore concernées ces dernières années. Dans cette thèse, un modèle décrivant la transmission de l’épidémie de chikungunya à la population humaine est étudié. Ce modèle prend en compte la mobilité spatiale des humains, ce qui est nouveau. En effet, c’est un facteur intéressant qui a influencé la réapparition de plusieurs maladies épidémiques. Le déplacement des moustiques est omis puisqu’il est limité à quelques mètres. Le modèle complet (modèle EDOs-EDPs) est alors composé d’un système à réaction-diffusion (prenant la forme d’équations différentielles partielles (EDPs) paraboliques semi-linéaires) couplé à des équations différentielles ordinaires (EDOs). Nous démontrons pour ce modèle, d’abord l’existence et l’unicité de la solution globale, sa positivité et sa bornitude, puis nous donnons quelques simulations numériques. Dans ce modèle, certains paramètres ne sont pas directement accessibles à partir des expériences et doivent être estimés numériquement. Cependant, avant de rechercher leurs valeurs, il est essentiel de vérifier l’identifiabilité des paramètres pour déterminer si l’ensemble des paramètres inconnus peut être déterminé de manière unique à partir des données. Cette étude permettra de s’assurer que les procédures numériques peuvent être couronnées de succès. Si l’identifiabilité n’est pas assurée, certaines données supplémentaires doivent être ajoutées. En fait, une première étude d’identifiabilité a été effectuée pour le modèle EDOs en considérant que le nombre d’œufs peut être facilement compté. Toutefois, après avoir discuté avec les chercheurs épidémiologistes, il apparaît que c’est le nombre de larves qui peut être estimé semaines par semaines. Ainsi, nous ferons une étude d’identifiabilité pour le nouveau modèle EDOs-EDPs avec cette hypothèse. Grâce à l’intégration de l’une des équations du modèle, on obtient des équations plus faciles reliant les entrées, les sorties et les paramètres, ce qui simplifie vraiment l’étude d’identifiabilité. A partir de l’étude d’identifiabilité, une méthode et une procédure numérique sont proposés pour estimer les paramètres sans en avoir connaissance. / In different fields of research, modeling has become an effective tool for studying and predicting the possible evolution of a system, particularly in epidemiology. Due to the globalization and the genetic mutation of certain diseases or transmission vectors, several epidemics have appeared in regions not yet concerned in the last years. In this thesis, a model describing the transmission of the chikungunya epidemic to the human population is studied. As a novelty, this model incorporates the spatial mobility of humans. Indeed, it is an interesting factor that has influenced the re-emergence of several epidemic diseases. The displacement of mosquitoes is omitted since it is limited to a few meters. The complete model (ODEs-PDEs model) is then composed of a reaction-diffusion system (taken the form of semi-linear parabolic partial differential equations (PDEs)) coupled with ordinary differential equations (ODEs). We prove the existence, uniqueness, positivity and boundedness of a global solution of this model at first and then give some numerical simulations. In such a model, some parameters are not directly accessible from experiments and have to be estimated numerically. However, before searching for their values, it is essential to verify the identifiability of parameters in order to assess whether the set of unknown parameters can be uniquely determined from the data. This study will insure that numerical procedures can be successful. If the identifiability is not ensured, some supplementary data have to be added. In fact, a first identifiability study had been done for the ODEs model by considering that the number of eggs can be easily counted. However, after discussing with epidemiologist searchers, it appears that it is the number of larvae which can be estimated weeks by weeks. Thus, we will do an identifiability study for the novel ODEs-PDEs model with this assumption. Thanks to an integration of one of the model equations, some easier equations linking the inputs, outputs and parameters are obtained which really simplify the study of identifiability. From the identifiability study, a method and numerical procedure are proposed for estimating the parameters without any knowledge of them.

This thesis deals with a newly-developed model for the spatiotemporal calcium dynamics within presynaptic terminals. The model is based on single-protein kinetics and has been used to successfully describe different neuron types such as pyramidal neurons in the rat neocortex and the Calyx of Held of neurons from the rat brainstem. A limited number of parameters had to be adjusted to fluorescence measurements of the calcium concentration. These values can be interpreted as a prediction of the model, and in particular the protein densities can be compared to independent experiments. The contribution of single proteins to the total calcium dynamics has been analysed in detail for voltage-dependent calcium channel, plasma-membrane calcium ATPase, sodium-calcium exchanger, and endogenous as well as exogenous buffer proteins. The model can be used to reconstruct the unperturbed calcium dynamics from measurements using fluorescence indicators. The calcium response to different stimuli has been investigated in view of its relevance for synaptic plasticity. This work provides a first step towards a description of the complete synaptic transmission using single-protein data.

La présente thèse est divisée en deux parties. La première partie concerne l'analyse mathématique et la contrôlabilité exacte à zéro pour une catégorie de systèmes structurés décrivant la dynamique d'une population d'insectes. La seconde partie est consacrée à l'étude de la stabilité de la conductivité d'un système de réaction diffusion modélisant l'activité électrique du coeur.Dans le chapitre 2, on considère que la population d'adultes se diffuse dans la vignoble,la fonction de la croissance des individus à chaque stade dépend des variations climatiques et de la variété des raisins. En utilisant la méthode de point fixe, on obtient l'existence et l'unicité des solutions du modèle. On démontre ensuite l'existence d'un attracteur global pour le système dynamique. Enfin, on utilise la théorie des opérateurs compacts et le théorème de point fixe de Krasnoselskii pour prouver l'existence des états stationnaires.Dans le chapitre 3, on traite le problème de contrôlabilité exacte du modèle de Lobesia Botrana, lorsque la fonction de croissance est égale à 1. On suppose que les quatre sous-catégories de ce système sont dans une phase statique. On obtient que la population d'oeufs peut être contrôlée à zéro. Ce résultat est basé sur des estimations à priori combinées avec un théorème de point fixe.Lorsque les papillons adultes se dispersent spatialement, on introduit un contrôle sur la population d'oeufs, de larves et de femelles dans une petite région du vignoble. On montre alors la contrôlabilité exacte à zéro pour les femelles.Dans la deuxième partie de cette thèse, on analyse la stabilité des coefficients de diffusion d'un système parabolique qui modélise l'activité électrique du coeur. On établit une estimation de Carleman pour le système de réaction-diffusion. En combinant cette estimation avec des estimations d'énergie avec poids on obtient le résultat de stabilité. / This thesis is divided into two parts.One is mainly devoted to make a qualitative analysis and exact null controlfor a class of structured population dynamical systems, and the other concernsstability of conductivities in an inverse problem of a reaction-diffusion systemarising in electrocardiology.In the first part, we study the dynamics ofEuropean grape moth, which has caused serious damages on thevineyards in Europe,North Africa, and even some Asian countries.To model this grapevine insect, physiologically structured multistage population systems are proposed.These systemshave nonlocal boundary conditions arising in nonlocal transition processes in ecosystem.We consider the questions of spatial spread of the populationunder physiological age and stage structures,and show global dynamical properties for the model.Furthermore, we investigate the control problem for this Lobesia botrana modelwhen the growth function is equal to $1$.For the case that four subclasses of this system are all in static station,we conclude that the population of eggs can be controlled to zero at acertain moment by acting on eggs.While the adult moths can disperse,we describe a control by a removal of egg and larvapopulation, and also on female moths in a small region of the vineyard.Then the null controllability for female mothsin a nonempty open sub-domain at a given time is obtained.In the second part, a reaction-diffusion system approximating a parabolic-elliptic systemwas proposed tomodel electrical activity in the heart. We are interested inthe stability analysis of an inverse problem for this model.Then we use the method of Carleman estimates and certain weight energyestimatesfor the identification of diffusion coefficients for the parabolicsystem to draw the conclusion.

L'objet de cette thèse est d'étudier le comportement en temps long de solutions d'équations d'évolution non locales ainsi que la limite singulière d'équations et de systèmes d'équations aux dérivées partielles, où intervient un petit paramètre epsilon. Au Chapitre 1, nous considérons une équation de réaction-diffusion non locale avec conservation au cours du temps de l'intégrale en espace de la solution; cette équation a été initialement proposée par Rubinstein et Sternberg pour modéliser la séparation de phase dans un mélange binaire. Le problème de Neumann associé possède une fonctionnelle de Lyapunov, c'est-à-dire une fonctionnelle qui décroit selon les orbites. Après avoir prouvé que la solution est confinée dans une région invariante, nous étudions son comportement en temps long. Nous nous appuyons sur une inégalité de Lojasiewicz pour montrer qu'elle converge vers une solution stationnaire quand t tend vers l'infini. Nous évaluons également le taux de la convergence et calculons précisément la solution stationnaire limite en dimension un d'espace. Le Chapitre 2 est consacré à l'étude de l'équation différentielle non locale que l'on obtient en négligeant le terme de diffusion dans l'équation d'Allen-Cahn non locale étudiée au Chapitre 1. Sans le terme de diffusion, la solution ne peut pas être plus régulière que la fonction initiale. C'est la raison pour laquelle on ne peut pas appliquer la méthode du Chapitre 1 pour l'étude du comportement en temps long de la solution. Nous présentons une nouvelle méthode basée sur la théorie des réarrangements et sur l'étude du profil de la solution. Nous montrons que la solution est stable pour les temps grands et présentons une caractérisation détaillée de sa limite asymptotique quand t tend vers l'infini. Plus précisément, la fonction limite est une fonction en escalier, qui prend au plus deux valeurs, qui coïncident avec les points stables d'une équation différentielle associée. Nous montrons aussi par un contre-exemple non trivial que, quand une hypothèse sur la fonction initiale n'est pas satisfaite, la fonction limite peut prendre trois valeurs, qui correspondent aux points instable et stables de l'équation différentielle associée. Nous étudions au Chapitre 3 une équation différentielle ordinaire non locale qui a éte proposée par M. Nagayama. Une difficulté essentielle est que le dénominateur dans le terme de réaction non local peut s'annuler. Nous appliquons un théorème de point fixe lié a une application contractante pour démontrer que le problème à valeur initiale correspondant possède une solution unique qui reste connée dans un ensemble invariant. Ce problème possède une fonctionnelle de Lyapunov, qui est un ingrédient essentiel pour démontrer que la solution converge vers une solution stationnaire constante par morceaux quand t tend vers l'infini. Au Chapitre 4, nous considérons un modèle d'interface diffuse pour la croissance de tumeurs, où intervient une équation d'ordre quatre de type Cahn Hilliard. Après avoir introduit un modèle de champ de phase associé, on étudie formellement la limite singulière de la solution quand le coefficient du terme de réaction tend vers l'infini. Plus précisément, nous montrons que la solution converge vers la solution d'un problème à frontière libre. AMS subject classifications. 35K57, 35K50, 35K20, 35R35, 35R37, 35B40, 35B25. / The aim of this thesis is to study the large time behavior of solutions of nonlocal evolution equations and to also study the singular limit of equations and systems of parabolic partial differential equations involving a small parameter epsilon. In Chapter 1, we consider a nonlocal reaction-diffusion equation with mass conservation, which was originally proposed by Rubinstein and Sternberg as a model for phase separation in a binary mixture. The corresponding Neumann problem possesses a Lyapunov functional, namely a functional which decreases in time along solution orbits. After having proved that the solution is conned in an invariant region, we study its large time behavior and apply a Lojasiewicz inequality to show that it converges to a stationary solution as t tends to infinity. We also evaluate the rate of convergence and precisely compute the limiting stationary solution in one space dimension. Chapter 2 is devoted to the study of a nonlocal evolution equation which one obtains by neglecting the diffusion term in the nonlocal Allen-Cahn equation studied in Chapter 1. Without the diffusion term, the solution can not be expected to be more regular than the initial function. Moreover, because of the absence of the diusion term, the method of Chapter 1 can not be applied to study the large time behavior of the solution. We present a new method based up on rearrangement theory and the study of the solution profile. We show that the solution stabilizes for large times and give a detailed characterization of its asymptotic limit as t tends to infinity. More precisely, it turns out that the limiting function is a step function, which takes at most two values, which are stable points of a corresponding ordinary dierential equation. We also show by means of a nontrivial counterexample that, when a certain hypothesis on the initial function does not hold, the limiting function may take three values. One of them is the unstable point and the two others are the stable points of the ordinary dierential equation. We study in Chapter 3 a nonlocal ordinary dierential equation which has been proposed by M. Nagayama. The nonlocal term involves a denominator which may vanish. We apply a contraction fixed point theorem to prove the existence of a unique solution which stays confined in an invariant region. We also show that the corresponding initial value problem possesses a Lyapunov functional and prove that the solution stabilizes for large times to a step function, which takes at most two values. In Chapter 4, we consider a diffuse-interface tumor-growth model which involves a fourth order Cahn-Hilliard type equation. Introducing a related phase-field model, we formally study the singular limit of the solution as the reaction coecient tends to infinity. More precisely, we show that the solution converges to the solution of a moving boundary problem. AMS subject classifications. 35K57, 35K50, 35K20, 35R35, 35R37, 35B40, 35B25.

Dans cette thèse, nous nous sommes intéressés aux équations différentielles fractionnaires. Nous avons commencé par l'étude d'une équation différentielle fractionnaire en temps. Ensuite, nous avons étudié trois systèmes fractionnaires non linéaires ; le premier avec un Laplacien fractionnaire et les autres avec une dérivée fractionnaire en temps définie au sens de Caputo. Dans le premier chapitre, nous avons établi les propriétés qualitatives de la solution d'une équation différentielle fractionnaire en temps qui modélise l'évolution d'une certaine espèce. Plus précisément, l'existence et l'unicité de la solution globale sont démontrées pour certaines valeurs de la condition initiale. Dans ce cas, nous avons obtenu le comportement asymptotique de la solution en t^α. Sous une autre condition sur la donnée initiale, la solution explose en temps fini. Le profil de la solution et l'estimation du temps d'explosion sont établis et une confirmation numérique de ces résultats est présentée. Les chapitres 4, 5 et 6 sont consacrés à l'étude théorique de trois systèmes fractionnaires : un système de la diffusion anormale qui décrit la propagation d'une épidémie infectieuse de type SIR dans une population confinée, le Brusselator avec une dérivée fractionnaire en temps et un système fractionnaire en temps avec une loi de balance. Pour chaque système, on présente l'existence globale et le comportement asymptotique des solutions. L'existence et l'unicité de la solution locale pour les trois systèmes sont obtenues par le théorème de point fixe de Banach. Cependant, le comportement asymptotique est établi par des techniques différentes : le comportement asymptotique de la solution du premier système est démontré en se basant sur les estimations du semi-groupe et le théorème d'injection de Sobolev. Concernant le Brusselator fractionnaire, la technique utilisée s'appuie sur un argument de feedback. Finalement, un résultat de régularité maximale est utilisé pour l'étude du dernier système. / In this thesis, we are interested in fractional differential equations. We begin by studying a time fractional differential equation. Then we study three fractional nonlinear systems ; the first system contains a fractional Laplacian, while the others contain a time fractional derivative in the sense of Caputo. In the second chapter, we establish the qualitative properties of the solution of a time fractional equation which describes the evolution of certain species. The existence and uniqueness of the global solution are proved for certain values of the initial condition. In this case, the asymptotic behavior of the solution is dominated by t^α. Under another condition, the solution blows-up in a finite time. The solution profile and the blow-up time estimate are established and a numerical confirmation of these results is presented. The chapters 4, 5 and 6 are dedicated to the study of three fractional systems : an anomalous diffusion system which describes the propagation of an infectious disease in a confined population with a SIR type, the time fractional Brusselator and a time fractional reaction-diffusion system with a balance law. The study includes the global existence and the asymptotic behavior. The existence and uniqueness of the local solution for the three systems are obtained by the Banach fixed point theorem. However, the asymptotic behavior is investigated by different techniques. For the first system our results are proved using semi-group estimates and the Sobolev embedding theorem. Concerned the time fractional Brusselator, the used technique is based on an argument of feedback. Finally, a maximal regularity result is used for the last system.