In this thesis, we study a class of reaction-diffusion equations of the form $\frac{\partial u}{\partial t} = \mathcal{L}u + \phi u - \tfrac{1}{k} u^{k+1}$ where $\mathcal{L}$ is the stochastic generator of a Markov process, $\phi$ is a function of the space variables and $k\in \mathbb{R}\backslash\{0\}$. An important example, in the case when $k > 0$, is equations of the FKPP-type. We also give an example from the theory of utility maximisation problems when such equations arise and in this case $k < 0$. We introduce a new representation, for the solution of the equation, as the optimal value of an optimal control problem. We also give a second representation which can be seen as a dual problem to the first optimisation problem. We note that this is a new type of dual problem and we compare it to the standard Lagrangian dual formulation. By choosing controls in the optimisation problems we obtain upper and lower bounds on the solution to the PDE. We use these bounds to study the speed of the wave front of the PDE in the case when $\mathcal{L}$ is the generator of a suitable Lévy process.

Orientador: Roberto André Kraenkel / Banca: Gilberto Corso / Banca: Cláudia Pio Ferreira / Banca: Fernando Fagundes Ferreira / Banca: Francisco Antonio Bezerra Coutinho / Resumo: Este trabalho é composto de estudos independentes, mas seus temas são conectados entre si. Ele foi feito baseando-se no estudo de equações de reação-difusão e em reação-difusão-advecção. Vários modelos foram utilizados para representar populações e apresentam características em comum. As populações representadas por esses modelos difundem, crescem e saturam de forma semelhante a equação de Fisher-Kolmogorov e Lotka-Volterra e foram modeladas usando condições de contorno de Dirichlet. Domínios limitados e ilimitados foram usados para que melhor representassem as devidas e diferentes aplicações de dados coletados em campo e publicados em periódicos. Esse trabalho também leva em conta a aplicabilidade à habitats fragmentados, isoladas e não-isoladas. Como foco principal temos o estudo do movimento de populações que vivem nesses habitats mostrando que a qualidade e distribuição deles afeta no movimento das populações / Abstract: This thesis consists on independent studies, but its subjects are interconnected. It has been based on the study of reaction-diffusion-advection equations. Several models were used to represent populations and have some characteristics in common. The populations represented by the models spread, grow, and saturate in a way similar to that described by the Fisher-Kolmogorov and Lotka-Volterra equations, and were modeled using Dirichlet boundary conditions. Limited and unlimited domains were used to better represent the necessary applications and different data collected in the field and published in journals. This work also takes into account the applicability to fragmented habitats, isolated and not isolated. As the main focus we study the movement of populations living in these habitats, showing that the quality and distribution affects them the movement of populations / Doutor

No description available.

acase@tulane.edu

No description available.

Thesis (Ph. D.)--Florida State University, 2005. / Advisor: Oliver Steinbock, Florida State University, College of Arts and Sciences, Dept. of Chemistry and Biochemistry. Title and description from dissertation home page (viewed Jan. 24, 2006). Document formatted into pages; contains xii, 123 pages. Includes bibliographical references.

Made available in DSpace on 2014-06-11T19:32:10Z (GMT). No. of bitstreams: 0 Previous issue date: 2013-04-05Bitstream added on 2014-06-13T19:25:53Z : No. of bitstreams: 1 azevedo_fs_dr_ift.pdf: 964901 bytes, checksum: 0ded483f0f1fa4571a038df930452981 (MD5) / Este trabalho é composto de estudos independentes, mas seus temas são conectados entre si. Ele foi feito baseando-se no estudo de equações de reação-difusão e em reação-difusão-advecção. Vários modelos foram utilizados para representar populações e apresentam características em comum. As populações representadas por esses modelos difundem, crescem e saturam de forma semelhante a equação de Fisher-Kolmogorov e Lotka-Volterra e foram modeladas usando condições de contorno de Dirichlet. Domínios limitados e ilimitados foram usados para que melhor representassem as devidas e diferentes aplicações de dados coletados em campo e publicados em periódicos. Esse trabalho também leva em conta a aplicabilidade à habitats fragmentados, isoladas e não-isoladas. Como foco principal temos o estudo do movimento de populações que vivem nesses habitats mostrando que a qualidade e distribuição deles afeta no movimento das populações / This thesis consists on independent studies, but its subjects are interconnected. It has been based on the study of reaction-diffusion-advection equations. Several models were used to represent populations and have some characteristics in common. The populations represented by the models spread, grow, and saturate in a way similar to that described by the Fisher-Kolmogorov and Lotka-Volterra equations, and were modeled using Dirichlet boundary conditions. Limited and unlimited domains were used to better represent the necessary applications and different data collected in the field and published in journals. This work also takes into account the applicability to fragmented habitats, isolated and not isolated. As the main focus we study the movement of populations living in these habitats, showing that the quality and distribution affects them the movement of populations

Orientador: Marcelo Martins dos Santos / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-23T21:53:25Z (GMT). No. of bitstreams: 1 Santos_RonaldoAntoniodos_D.pdf: 739129 bytes, checksum: f677894f21ef1223fced35636869835d (MD5) Previous issue date: 2013 / Resumo: Neste trabalho provamos a existência de solução global para um sistema não linear constituído de duas equações parabólicas acopladas a duas equações diferenciais ordinárias. Tal sistema modela um processo de combustão em um meio poroso com duas camadas, em que os efeitos de compressibilidade são desprezados, mas a troca de calor entre as camadas, bem como a propagação de calor por convecção são levadas em conta. Supondo que os dados iniciais são lipschitzianos, limitados e pertencentes a algum espaço , 1 < < ?, obtivemos solução clássica para o problema / Abstract: In this work we prove the existence of a global solution for a nonlinear system consisting of two parabolic equations coupled to two ordinary differential equations. Such a system models a combustion process in a porous medium with two layers in which compressibility effects are neglected, but heat transfer between the layers as well as heat conduction are taken into a account. We obtained a classical solution under the assumptions that the initial data is bounded, Lipschitz and belongs to some space, with 1 < < ? / Doutorado / Matematica / Doutor em Matemática

The growing flow of people and goods around the globe has allowed new, non-native species to establish and spread in already fragile ecosystems. The introduction of invasive species can have a detrimental impact on the already established species. Thus, it is important that we understand the mechanisms that facilitate or prevent invasion. Since reaction-diffusion invasion models produce travelling waves we can study invasion by looking at the mechanisms that allow for wave propagation failure, or wave-blocking. In this thesis we consider a perturbed reaction-diffusion model in which the perturbation resides in either the reaction or diffusion term. In doing so we exploit the underlying symmetry of our problem to define a region in the appropriate parameter space that leads to wave blocking. As a demonstrative example we apply our theory to the bistable equation and consider the effects of various perturbations.

The earth’s climate is increasing in temperature and as a result many species’ habitat ranges are shifting. The shift in habitat ranges threatens the local persistence of many species. Mathematical models that capture this phenomena of range shift do so by considering a bounded domain that has a time dependant location on the real line. The analysis on persistence conditions has been considered in both continuous-time and -space, and discrete-time, continuous-space settings. In both model types density was considered to be continuous across the boundaries. However it has been shown that many species exhibit particular behaviour at habitat edges, such as biased movement towards the more suitable habitat. This behaviour should be incorporated into the analysis to obtain more accurate persistence conditions. In this thesis persistence conditions are obtained for generalized boundary conditions with a continuous-time and -space model for a range-shifting habitat. It is shown that a high preference for the suitable habitat at the trailing edge can greatly reduce the size of suitable habitat required for species persistence. As well, for fast shifting ranges, a high preference at the trailing edge is crucial for persistence.