We study anomalous diffusion on fractals with a static external field applied. We utilise the master equation to calculate particle distributions and from that important quantities as for example the mean square displacement <r^2(t)>. Applying different bias amplitudes on several regular Sierpinski carpets we obtain maximal drift velocities for weak field strengths. According to <r^2(t)>~t^(2/d_w), we determine random walk dimensions of d_w<2 for applied external fields. These d_w corresponds to superdiffusion, although diffusion is hindered by the structure of the carpet, containing dangling ends. This seems to result from two competing effects arising within an external field. Though the particles prefer to move along the biased direction, some particles get trapped by dangling ends. To escape from there they have to move against the field direction. Due to the by the bias accelerated particles and the trapped ones the probability distribution gets wider and thus d_w<2. / In dieser Arbeit untersuchen wir anomale Diffusion auf Fraktalen unter Einwirkung eines statisches äußeres Feldes. Wir benutzen die Mastergleichung, um die Wahrscheinlichkeitsverteilung der Teilchen zu berechnen, um daraus wichtige Größen wie das mittlere Abstandsquadrat <r^2(t)> zu bestimmen. Wir wenden unterschiedliche Feldstärken bei verschiedenen regelmäßigen Sierpinski-Teppichen an und erhalten maximale Driftgeschwindigkeiten für schwache Feldstärken. Über <r^2(t)>~t^{2/d_w} bestimmen wir die Random-Walk-Dimension d_w als d_w<2. Dieser Wert für d_w entspricht der Superdiffusion, obwohl der Diffusionsprozess durch Strukturen des Teppichs, wie Sackgassen, behindert wird. Es schient, dass dies das Ergebnis zweier konkurrierender Effekte ist, die durch das Anlegen eines äußeren Feldes entstehen. Einerseits bewegen sich die Teilchen bevorzugt entlang der Feldrichtung. Andererseits gelangen einige Teilchen in Sackgassen. Um die Sackgassen, die in Feldrichtung liegen, zu verlassen, müssen sich die Teilchen entgegen der Feldrichtung bewegen. Somit sind die Teilchen eine gewisse Zeit in der Sackgasse gefangen. Infolge der durch das äußere Feld beschleunigten und der gefangenen Teilchen, verbreitert sich die Wahrscheinlichkeitsverteilung der Teilchen und somit ist d_w<2.

The increasing complexity of heterogeneous cellular networks (HetNets) due to the irregular deployment of small cells demands significant rethinking in the way cellular networks are perceived, modeled and analyzed. In addition to threatening the relevance of classical models, this new network paradigm also raises questions regarding the feasibility of state-of-the-art simulation-based approach for system design. This dissertation proposes a fundamentally new approach based on random spatial models that is not only tractable but also captures current deployment trends fairly accurately. First, this dissertation presents a general baseline model for HetNets consisting of K different types of base stations (BSs) that may differ in terms of transmit power, deployment density and target rate. Modeling the locations of each class of BSs as an independent Poisson Point Process (PPP) allows the derivation of surprisingly simple expressions for coverage probability and average rate. One interpretation of these results is that adding more BSs or tiers does not necessarily change the coverage probability, which indicates that fears of "interference overload" in HetNets are probably overblown. Second, a flexible notion of BS load is incorporated by introducing a new idea of conditionally thinning the interference field. For this generalized model, the coverage probability is shown to increase when lightly loaded small cells are added to the existing macrocellular networks. This is due to the fact that owing to the smaller loads, small cells typically transmit less often than macrocells, thus contributing less to the interference power. The same idea of conditional thinning is also shown to be useful in modeling the non-uniform user distributions, especially when the users lie closer to the BSs. Third, the baseline model is extended to study multi-antenna HetNets, where BSs across tiers may additionally differ in terms of the number of transmit antennas, number of users served and the multi-antenna transmission strategy. Using novel tools from stochastic orders, a tractable framework is developed to compare the performance of various multi-antenna transmission strategies for a fairly general spatial model, where the BSs may follow any general stationary distribution. The analysis shows that for a given total number of transmit antennas in the network, it is preferable to spread them across many single-antenna BSs vs. fewer multi-antenna BSs. Fourth, accounting for the load on the serving BS, downlink rate distribution is derived for a generalized cell selection model, where shadowing, following any general distribution, impacts cell selection while fading does not. This generalizes the baseline model and all its extensions, which either ignore the impact of channel randomness on cell selection or lumps all the sources of randomness into a single random variable. As an application of these results, it is shown that in certain regimes, shadowing naturally balances load across various tiers and hence reduces the need for artificial cell selection bias. Fifth and last, a slightly futuristic scenario of self-powered HetNets is considered, where each BS is powered solely by a self-contained energy harvesting module that may differ across tiers in terms of the energy harvesting rate and energy storage capacity. Since a BS may not always have sufficient energy, it may not always be available to serve users. This leads to a notion of availability region, which characterizes the fraction of time each type of BS can be made available under variety of strategies. One interpretation of this result is that the self-powered BSs do not suffer performance degradation due to the unreliability associated with energy harvesting if the availability vector corresponding to the optimal system performance lies in the availability region. / text

This thesis aims to increase our understanding of mechanisms that influence larval dispersal in marine benthic invertebrates, particularly in the absence of strong oceanographic features (e.g. estuarine plumes, upwelling events, or markedly different water masses). Laboratory experiments identified behavioural mechanisms that regulate the vertical distribution of larvae in response to thermal stratification, and field studies in St. George’s Bay, Nova Scotia (NS), Canada, examined the relationship between larval abundance and physical variables (temperature, salinity, fluorescence, etc) and identified mechanisms that regulate larval distributions in situ. In the laboratory, I demonstrated that thermal stratification affects the vertical distribution of larvae by acting as a barrier to migration, or through temperature-dependent vertical swimming velocities. I also developed a random walk based model which highlighted that the key to successfully simulating larval response to temperature was 1) determining the temperature-dependent distribution of vertical swimming velocities and 2) the temporal autocorrelation in these velocities. In the field, the most striking pattern was that the larval distributions for species with similar swimming abilities were significantly correlated to one another at all scales (0.5 to 40 km). This suggests a common mechanism, related to larval swimming ability, which greatly influences the horizontal larval distribution. I found that the spatial scale of variability in larval distributions (~ 3 km) matches that in both the environmental variables and of coherent structures in current velocities (i.e. the tidal excursion). Results from an aggregation-diffusion model suggest that horizontal larval swimming could not be responsible for the observed level of aggregation in the larval horizontal distributions. I suggest that these horizontal patterns are the result of 1) an aggregative process (i.e. larvae swimming against a vertical current and maintaining their vertical position) and 2) a diffusive process which scales the aggregations to the scale of the coherent structures in current velocity (i.e. tidal excursion). In conclusion, this thesis increases our understanding of larval behaviour and its effects on larval dispersal. The results will be particularly useful to those who are interested in mechanisms regulate population connectivity, particularly those using bio-physical models to model dispersal trajectories.

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

Mémoire numérisé par la Division de la gestion de documents et des archives de l'Université de Montréal

In this dissertation, we study integrodifference equations in patchy landscapes. Specifically, we provide a framework for linking individual dispersal behavior with population-level dynamics in patchy landscapes by integrating recent advances in modeling dispersal into an integrodifference equation. First, we formulate a random-walk model in a patchy landscape with patch-dependent diffusion, settling, and mortality rates. We incorporate mechanisms for individual behavior at an interface which, in general, results in the probability-density function of the random walker being discontinuous at an interface. We show that the dispersal kernel can be characterized as the Green's function of a second-order differential operator and illustrate the kind of (discontinuous) dispersal kernels that arise from our approach. We examine the dependence of obtained kernels on model parameters. Secondly, we analyze integrodifference equations in patchy landscapes equipped with discontinuous kernels. We obtain explicit formulae for the critical-domain-size problem, as well as, explicit formulae for the analogous critical size of good patches on an infinite, periodic, patchy landscape. We examine the dependence of obtained formulae on individual behavior at an interface. Through numerical simulations, we observe that, if the population can persist on an infinite, periodic, patchy landscape, its spatial profile can evolve into a discontinuous traveling periodic wave. We derive a dispersion relation for the speed of the wave and illustrate how interface behavior affects invasion speeds. Lastly, we develop a strategic model for the spread of the emerald ash borer and its interaction with host trees. A thorough literature search provides point estimates and interval ranges for model parameters. Numerical simulations show that the spatial profile of an emerald ash borer invasion evolves into a pulse-like solution that moves with constant speed. We employ Latin hypercube sampling to obtain a plausible collection of parameter values and use a sensitivity analysis technique, partial rank correlation coefficients, to identify model parameters that have the greatest influence on obtained speeds. We illustrate the applicability of our framework by exploring the effectiveness of barrier zones on slowing the spread of the emerald ash borer invasion.

Cette thèse contient quatre travaux sur le modèle d'Agrégation Limitée par Diffusion Interne (iDLA), qui est un modèle de croissance pour la construction récursive d'ensembles aléatoires. Le premier travail concerne la dimension 1 et étudie le cas où les marches aléatoires formant l'agrégat évoluent dans un milieu aléatoire. L'agrégat normalisé converge alors non pas vers une forme limite déterministe comme dans le cas de marches aléatoires simples mais converge en loi vers un segment contenant l'origine dont les extrémités suivent la loi de l'Arcsinus. Dans le deuxième travail, on considère le cas où l'agrégat est formé par des marches aléatoires simples en dimension d > 1. On donne alors des résultats de convergence et de fluctuations sur la fonction odomètre introduite par Levine et Peres, qui compte en chaque point le nombre de passages des marches ayant formé l'agrégat. Dans le troisième travail, on s'intéresse au cas où l'agrégat est formé par des marches aléatoires multidimensionnelles qui ne sont pas centrées. On montre que sous une normalisation appropriée, l'agrégat converge vers une forme limite qui s'identifie à une vraie boule de chaleur. Nous répondons ainsi à une question ouverte en analyse concernant l'existence d'une telle boule bornée. Le quatrième travail concerne le cas particulier où une borne intérieure est connue pour l'agrégat. On donne alors des conditions suffisantes sur le graphe ainsi que sur la nature de cette borne pour qu'elle implique une borne extérieure. Ce résultat est appliqué au cas de marches évoluant sur un amas de percolation par arêtes surcritique, complétant ainsi un résultat de Shellef. / This thesis contains four works on the Internal Diffusion Limited Aggregation model (iDLA), which is a growth model that recursively builds random sets. The first work is set in dimension 1 and studies the case where the random walks that build the aggregate evolve in a random environment. The normalised aggregate then does not converges towards a deterministic limiting shape as it is the case for simple random walks, but converges in law towards a segment that contains the origin and which extremal points follow the Arcsine law. In the second work, we consider the case where the aggregate is built by simple random walks in dimension d > 1. We give convergence and fluctuation results on the odometer function introduced by Levine and Peres, which counts at each point the number of visits of walkers throughout the construction of the aggregate. In the third work, we examine the case where the aggregate is built using multidimensional drifted random walks. We show that under a suitable normalisation, the aggregate converges towards a limiting shape which is identified as a true heat ball. We thus give an answer to an open question in analysis concerning the existence of such a bounded shape. The last work deals with the special case where an interior bound is known for the aggregate. We give a set of conditions on the graph and on the nature of this interior bound that are sufficient to imply an outer bound. This result is applied to the case of random walks on the supercritical bond percolation cluster, thus completing a result by Shellef.

Cette thèse est consacrée à la modélisation de la formation spontanée de chemins préférentiels par des marcheurs déposant des traces attractives sur leurs trajectoires. Plus précisément, par une démarche pluridisciplinaire couplant modélisation et expérimentation, elle vise à dégager un ensemble de règles minimales individuelles permettant l'apparition d'un tel phénomène. Dans ce but, nous avons étudié sous différents angles les modèles minimaux que sont les marches aléatoires renforcées (MAR).Ce travail comporte deux parties principales. La première démontre de nouveaux résultats dans le domaine des probabilités et statistiques. Nous avons généralisé le travail publié par M. Benaïm et O. Raimond en 2010 afin d'étudier l'asymptotique d'une classe de MAR auxquelles les demi-tours sont interdits. Nous avons également développé une procédure statistique permettant, sous certaines conditions adéquates de régularité, d'estimer les paramètres de MAR paramétrées et d'évaluer des marges d'erreur.Dans la seconde partie, sont décrits les résultats et analyses d'une étude comportementale et expérimentale de la fourmi Linepithema humile. Une partie de notre réflexion est centrée sur le rôle et la valeur des paramètres du modèle proposé par J.-L. Deneubourg et al. en 1990. Nous nous sommes aussi demandés dans quelle mesure une MAR peut reproduire les déplacements d'une fourmi dans un réseau. Dans ces objectifs, nous avons mené des expériences confrontant des fourmis à des réseaux à une ou plusieurs bifurcations. Nous avons appliqué aux données expérimentales les outils statistiques développés dans cette thèse. Nous avons aussi effectué une étude comparative entre les simulations de plusieurs modèles et les expériences. / This thesis is devoted to the modelisation of the spontaneous formation of preferential paths by walkers that deposit attractive trails on their trajectories. More precisely, through a multidisciplinary approach, which combines modelisation and experimentation, this thesis aims to bring out a set of minimal individual rules that allow the apparition of this phenomena. In this purpose, we study in several ways the minimal models, which are the Reinforced Random Walks (RRW).This work contains two main parts. The first one proves some new results in the field of probability and statistics. We have generalized the work published by M. Benaïm and O. Raimond in 2010 in order to study the asymptotics of a class of RRW, to which U-turns are forbidden. We developped also a statistical procedure that allows under some appropriate regularity hypotheses to estimate the parameters of parametized RRW and to evaluate margins of error.In the second part, we describe the results and the analyses of a experimental and behavioral study of the Linepithema humile ants. One part of our reflection is centered on the role and the value of the parameters of the model defined by J.-L. Deneubourg et al. in 1990. We investigated also the extent to which RRW could reproduce the moving of an ant in a network. To these purposes, we performed experiments that confront ants to a network of one or several forks. We applied to experimental data the statistical tools developed in this thesis and we performed a comparative study between experiments and simulations of several models.

In this work, we analyze search dynamics and the statistical properties of an organism in search of a target of interest. In general terms, there are many interesting aspects of studies of this nature. For example, in the biological context, organisms in Nature constantly interact one with another, both of the same as well as of different species. The general objectives of random searches are diverse, ranging from searches for food, reproductive partners, etc. of living organisms to socio-economically relevant processes, such as searches for missing children, fugitive terrorists, or searches for petroleum. In our specific model, we consider the searcher and the target moving randomly in a one dimensional lattice of size with periodic boundary conditions. The type of diffusion in the system is determined by the choice of the probability distribution function for the steps sizes for the individual walkers. We assume a power law distribution, characteristic of Levy processes, . Considering an initial energy for the searcher, an energetic expenditure for the walk and an energetic gain g for each target found, we discuss relevant physical quantities, such as energy fluctuations, the fraction of survival searchers and the cumulative energy for N time steps, as a function of the parameters, e.g., the lattice size . We find that searches with ballistic diffusion are more efficient than Brownian ones, allowing the survival of the searcher in situations of ultra-low target density. This extreme behavior guarantees the differential survival of such searchers. We also find strong evidence of a continuous phase transition, in which one phase has survival and the other phase has extinction. We calculate the critical densities which depend on the parameters of diffusion adopted by the organisms. We also obtain the critical exponents for the transition. Our results suggest a universality of the critical exponents, which independent of the type of diffusion of the organisms. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Neste trabalho, analisamos a dinâmica de busca e propriedades estatísticas de um organismo buscador ( searcher ) à procura de um alvo de interesse ( target ). De forma geral, muitos são os aspectos de interesse nesse tipo de estudo. Por exemplo, se pensarmos no contexto biológico, temos que na natureza constantemente organismos interagem uns com os outros, tanto dentro da mesma como entre diferentes espécies. Os objetivos gerais da busca aleatória são os mais variados, indo desde busca de alimentos, parceiro para reprodução etc, em seres vivos, até processos de interesse socio-econômicos, como busca por crianças desaparecidas, terroristas fugitivos ou então busca por petróleo. Em nosso modelo específico, consideramos o buscador e o alvo caminhando aleatoriamente numa rede unidimensional de tamanho e com condições periódicas de contorno. O tipo de difusão no sistema é determinado pela escolha da função de distribuição de probabilidade para os passos individuais dos indivíduos. Assumimos uma distribuição tipo lei de potência, característica de processos de Lévy . Considerando uma energia inicial do buscador , um gasto energético de caminhada e um ganho de energia g cada vez que o buscador encontra o alvo, discutimos algumas quantidades físicas relevantes, como flutuação energética, fração de buscadores sobreviventes e energia acumulada para N passos realizados - tempo de busca - como função de diferentes parâmetros, por exemplo, o comprimento de rede . Constatamos que o processo de busca com difusão balística é mais eficiente do que a Browniana, ocasionando a sobrevivência do organismo buscador em situações de densidade de alvos muito baixas. Este comportamento extremo garante a relativa sobrevivência do buscador. Também verificamos fortes evidências de uma transição contínua, para a qual numa dada fase temos sobrevivênvia e em outra temos extinção. Calculamos as densidades críticas que dependem dos parâmetros de difusão adotados pelos organismos. Também obtemos os expoentes críticos relacionados a tal transição. Nossos resultados sugerem uma universalidade dos expoentes críticos, que independente do tipo de difusão seguida pelos organismos.

Nowadays there has been a growing interest in anomalous diffusion: the super difusive and sub-difusive processes. The problem about normal diffusion already well established whereas many problems still exist in anomalous diffusion. Several mathematical models and computational techniques have been developed to model such processes. In this work we studied a non-Markovian Random Walk (RW), in one dimension in which the development of the process is governed by decisions taken in the distant past. We used as tool of analysis, analytical and numerical procedures (Monte Carlo method). In this problem, the walker takes its decisions (go right or left) at a given time t, based on the decisions taken in the past, namely in a fraction f of the total time. As far as the decision making process is considered only the distant past is taken into account. This loss of recent memory leads the probability density function of the position to change from Gaussian to non-Gaussian and leads to the emergence of log-periodic oscillations in position, besides producing a change in the behavior of non-persistent to persistent, causing anomalous diffusion. This change is characterized by the Hurst exponent, and is found, surprisingly, in a region where there is negative feedback. The diagram of phases depending on the parameters f and p (fraction of old memory and feedback), shows the following phases: classical non persistence, classical persistence, log-periodic non persistence, log-periodic persistence, Gaussian and non Gaussian with respect to the position of the walker. / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Atualmente tem crescido o interesse por processos de difusão anômala, i.e., os super difusivos e sub-difusivos. O problema voltado para difusão normal já é bem conhecido, enquanto para difusões anômalas ainda existem vários problemas em abertos. Várias técnicas computacionais e modelos matemáticos têm sido desenvolvidos para modelar tais processos. Estudamos neste trabalho uma caminhada aleatória, não Markoviana em uma dimensão, em que o desenvolvimento do processo é regido por decisões tomadas em relação ao passado distante. Utilizamos como ferramenta de análise uma abordagem analítica e numérica (via método de Monte Carlo). Nesse problema, o caminhante toma suas decisões (entre ir para a direita ou para a esquerda), num determinado tempo t, com base nas decisões tomadas no passado, numa fração f do tempo transcorrido. Quando f<1 o passado recente é esquecido e apenas o passado distante é considerado. Essa perda de memória recente induz a função densidade de probabilidade da posição a passar de um regime Gaussiano para não Gaussiano e leva ao surgimento de oscilações log-periódicas na posição, além de produzir uma mudança no comportamento, de não persistente para persistente, ocasionando difusão anômala. Essa mudança é caracterizada pelo expoente de Hurst e ocorre também, surpreendentemente, numa região de feedback negativo. O diagrama de fases em função dos parâmetros f e p (fração de memória antiga e feedback), mostra as seguintes regiões: não persistência clássica; persistência clássica; não persistência log-periódica e persistência log-periódica; região Gaussiana e não Gaussiana da posição.