這篇文章中，我們探討離散型反應擴散方程u_t(x,t)=u(x+1,t)-2u(x,t)+u(x-1,t)+f(u(x,t))，其中 反應項f(u)=u^2(1-u)。在此， 我們證明此方程式存在一種全解其動態行為宛如兩個來自x軸兩端相向而行的行波。 / This paper deals with a discrete reaction-diffusion equation u_t(x,t)=u(x+1,t)-2u(x,t)+u(x-1,t)+f(u(x,t)), where f(u)=u^2(1-u). Here, we prove there exist entire solutions which behave as two traveling waves coming from both sides of x-axis.

This thesis is devoted to the study of parabolic systems of partial differential equations arising in mass action kinetics chemistry, population dynamics and electromigration theory. We are interested in the existence of global solutions, uniqueness of weak solutions, and in the fast-reaction limit in a reaction-diffusion system. In the first chapter, we study two cross-diffusion systems. We are first interested in a population dynamics model, where cross effects in the interactions between the different species are modeled by non-local operators. We prove the well-posedness of the corresponding system for any space dimension. We are then interested in a cross-diffusion system which arises as the fast-reaction limit system in a classical system for the chemical reaction C1+C2=C3. We prove the convergence when k goes to infinity of the solution of the system with finite reaction speed k to a global solution of the limit system. The second chapter contains new global existence results for some reaction-diffusion systems. For networks of elementary chemical reactions of the type Ci+Cj=Ck and under Mass Action Kinetics assumption, we prove the existence and uniqueness of global strong solutions, for space dimensions N<6 in the semi-linear case, and N<4 in the quasi-linear case. We also prove the existence of global weak solutions for a class of parabolic quasi-linear systems with at most quadratic non-linearities and with initial data that are only assumed to be nonnegative and integrable. In the last chapter, we generalize a global well-posedness result for reaction-diffusion systems whose nonlinearities have a "triangular" structure, for which we now take into account advection terms and time and space dependent diffusion coefficients. The latter result is then used in a Leray-Schauder fixed point argument to prove the existence of global solutions in a diffusion-electromigration system.

Biological organisms process information through the use of complex reaction networks. These can bea great source of inspiration for the tailoring of dynamic chemical systems. Using basic DNA biochemistry-the DNA-toolbox- modeled after the cell regulatory processes, we explore the construction ofspatio-temporal dynamics from the bottom-up.First, we design a monitoring technique of DNA hybridization by harnessing a usually neglectedinteraction between the nucleobases and an attached fluorophore. This fluorescence technique -calledN-quenching- proves to be an essential tool to monitor and troubleshoot our dynamic reaction circuits.We then go on a journey to the roots of the DNA-toolbox, aiming at defining the best design rulesat the sequence level. With this experience behind us, we tackle the construction of reaction circuitsdisplaying bistability. We link the bistable behavior to a topology of circuit, which asks for specificDNA sequence parameters. This leads to a robust bistable circuit that we further use to explore themodularity of the DNA-toolbox. By wiring additional modules to the bistable function, we make twolarger circuits that can be flipped between states: a two-input switchable memory, and a single-inputpush-push memory. Because all the chemical parameters of the DNA-toolbox are easily accessible,these circuits can be very well described by quantitative mathematical modeling. By iterating thismodular approach, it should be possible to construct even larger, more complex reaction circuits: eachsuccess along this line will prove our good understanding of the underlying design rules, and eachfailure may hide some still unknown rules to unveil.Finally, we propose a simple method to bring DNA-toolbox made reaction circuits from zerodimensional,well-mixed conditions, to a two-dimensional environment allowing both reaction anddiffusion. We run an oscillating reaction circuit in two-dimensions and, by locally perturbing it, areable to provoke the emergence of traveling and spiral waves. This opens up the way to the building ofcomplex, tailor-made spatiotemporal patterns.

Thesis (MSc (Logistics))--University of Stellenbosch, 2008. / The primary question considered in this study is whether a small population of a biological species introduced into a resource-heterogeneous environment, where it competes for these resources with an already established native species, will be able to invade successfully. A two-component autonomous system of reaction-diffusion equations with spatially inhomogeneous Lotka-Volterra competitive reaction terms and diffusion coefficients is derived as the governing equations of the competitive scenario. The model parameters for which the introduced species is able to invade describe the realized niche of that species. A linear stability analysis is performed for the model in the case where the resource heterogeneity is represented by, and the diffusion coefficients are, two-toned functions. In the case where the native species is not directly affected by the resource heterogeneity, necessary and sufficient conditions for successful invasion are derived. In the case where the native species is directly affected by the resource heterogeneity only sufficient conditions for successful invasion are derived. The reaction-diffusion equations employed in the model are deterministic. However, in reality biological species are subject to stochastic population perturbations. It is argued that the ability of the invading species to recover from a population perturbation is correlated with the persistence of the species in the niche that it occupies. Hence, invasion time is used as a relative measure to quantify the rate at which a species’ population distribution recovers from perturbation. Moreover, finite difference and spectral difference methods are employed to solve the model scenarios numerically and to corroborate the results of the linear stability analysis. Finally, a case study is performed. The model is instantiated with parameters that represent two different cultivars of barley in a hypothetical environment characterized by spatially varying water availability and the sufficient conditions for successful invasion are verified for this hypothetical scenario.

In recent years, there has been increased awareness that stochasticity in chemical reactions and diffusion of molecules can have significant effects on the outcomes of intracellular processes, particularly given the low copy numbers of many proteins and mRNAs present in a cell. For such molecular species, the number and locations of molecules can provide a more accurate and detailed description than local concentration. In addition to diffusion, drift in the movements of molecules can play a key role in the dynamics of intracellular processes, and can often be modeled as arising from potential fields. Examples of sources of drift include active transport, variations in chemical potential, material heterogeneities in the cytoplasm, and local interactions with subcellular structures. This dissertation presents a new numerical method for simulating the stochastically varying numbers and locations of molecular species undergoing chemical reactions and drift-diffusion. The method combines elements of the First-Passage Kinetic Monte Carlo (FPKMC) method for reaction-diffusion systems and the Wang—Peskin—Elston lattice discretization of the Fokker—Planck equation that describes drift-diffusion processes in which the drift arises from potential fields. In the FPKMC method, each molecule is enclosed within a "protective domain," either by itself or with a small number of other molecules. To sample when a molecule leaves its protective domain or a reaction occurs, the original FPKMC method relies on analytic solutions of one- and two-body diffusion equations within the protective domains, and therefore cannot be used in situations with non-constant drift. To allow for such drift in our new method (hereafter Dynamic Lattice FPKMC or DL-FPKMC), each molecule undergoes a continuous-time random walk on a lattice within its protective domain, and the lattices change adaptively over time. One of the most commonly used spatial models for stochastic reaction-diffusion systems is the Smoluchowski diffusion-limited reaction (SDLR) model. The DL-FPKMC method generates convergent realizations of an extension of the SDLR model that includes drift from potentials. We present detailed numerical results demonstrating the convergence and accuracy of our method for various types of potentials (smooth, discontinuous, and constant). We also present several illustrative applications of DL-FPKMC, including examples motivated by cell biology.

Neste trabalho estudamos a dinâmica assintótica de uma classe de equações diferenciais de reação-difusão definidas em abertos de \'R POT. 3\' arbitrários, limitados ou não, com condições de fronteira de Dirichlet. Utilizando a técnica de estimativas de truncamento, como nos artigos de Prizzi e Rybakowski, mostramos a existência de atratores globais / In this work we study the asymptotic behavior of a class of semilinear reaction-diffusion equations defined on an arbitrary open set of R3, bounded or not, with Dirichlet boundary conditions. Using the tail-estimates technic based on papers of Prizzi and Rybakowski, we prove existence of global attractors

Neste trabalho estudamos problemas de reação-difusão semilineares do tipo \'u IND..t(x, t) = \'DELTA\'u(x, t)+ f (u(x, t)), x \'PERTENCE A\' \'OMEGA\' \'PARTIAL\' U/\'PARTIAL\'V (x, t) = 0, x \'PERTENCE A\' \'PARTIAL\'\' OMEGA\'. Desenvolvemos uma teoria abstrata para a obtenção da continuidade da dinâmica assintótica de (P) sob perturbações singulares do domínio espacial W e aplicamos a uma série de exemplos dos assim chamados domínios finos / In this work we study semilinear reaction-diffusion problems of the type \'u IND.t(x, t) = \'DELTA\'u(x, t)+ f (u(x, t)), x \' PERTENCE A\' \'OMEGA\' \'PARTIAL\'u/\'ARTIAL\' v (x, t) = 0, x \"PERTENCE A\' \'PARTIAL\' \' OMEGA\' We develop a abstract theory to obtain the continuity of the asymptotic dynamics of (P) under singular perturbations of the spatial domain W and we apply that to many examples in thin domains

Sistemas de reação-difusão têm sido largamente estudados em diferentes contextos e através de diferentes métodos, motivados pela sua constante aparição em modelos de interação em contextos químicos, biológicos e ainda em fenômenos ecológicos. Neste trabalho nos propomos a estudar existência e unicidade - tanto do ponto de vista local como global - de soluções para uma classe de sistemas de reação-difusão acoplados, denidos em R^2, utilizando como ferramenta a teoria de semigrupos de operadores lineares. Apresentamos dois importantes exemplos: o modelo de Rosenzweig-MacArthur e um particular caso da classe de equações lambda-omega. Para o primeiro obtemos um resultado de existência e unicidade global utilizando um método de comparação envolvendo sub e super-soluções. Investigamos ainda a existência de soluções de ondas viajantes periódicas através do teorema de Bifurcação de Hopf. Já para o caso da equação lambda-omega obtemos a existência e unicidade de solucões, entretanto, a partir da aplicação da teoria de semigrupos de operadores lineares. / Reaction-diffusion systems have been widely studied in a broad variety of contexts in a large amount of disctinct approaches. It is due firstly by their constant appearance in interaction models in disciplines such as chemistry, biology and, more specific, ecology. The aim of this thesis is to provide an existence-uniqueness result - both from the local as well as from the global point of view - for solutions of a particular class of coupled reaction-diffusion systems defined over R^2. It is done applying the well established theory of semigroups of linear operators. Two remarkable examples of such systems are discussed: the Rosenzweig-MacArthur predator-prey model and a special case of lambda-omega class of equations. For the former one, an existence and uniqueness result is obtained through a comparison method - based on the notions of lower and upper solutions. Moreover, we investigate the existence of periodic travelling wave solutions via a Hopf bifurcation theorem. For the lambda-omega model another existence and uniqueness for solutions is obtained, on its turn, through the machinery obtained previously from the theory of semigroups for linear operators.

Reaction-diffusion equations have been used to study various phenomena across different fields. These equations can be posed on the whole real line, or on a subinterval, depending on the situation being studied. For finite intervals, we also impose diverse boundary conditions on the system. In the present thesis, we solely focus on the bistable reaction-diffusion equation while working on a bounded interval of the form $[0,L]$ ($L>0$). Furthermore, we consider both mixed and no-flux boundary conditions, where we extend the former to Dirichlet boundary conditions once our analysis of that system is complete. We first use phase-plane analysis to set up our initial investigation of both systems. This gives us an integral describing the transit time of orbits within the phase-plane. This allows us to determine the bifurcation diagram of both systems. We then transform the integral to ease numerical calculations. Finally, we determine the stability of the steady states of each system.

Dans cette thèse nous nous intéressons aux équations de réaction-diffusion et à leurs applications en sciences biologiques et médicales. Plus particulièrement on étudie l'existence ou la non-existence de phénomènes de propagation en milieux hétérogènes à travers l'existence d'ondes progressives ou plus généralement l'existence de fronts de transition généralisés. On obtient des résultats d'existence de phénomènes de propagation dans trois environnements différents. Dans un premier temps on étudie une équation de réaction-diffusion de type bistable dans un domaine extérieur. Cette équation modélise l'évolution de la densité d'une population soumise à un effet Allee fort dont le déplacement suit un processus de diffusion dans un environnement contenant un obstacle. On montre que lorsque l'obstacle satisfait certaines conditions de régularité et se rapproche d'un domaine étoilé ou directionnellement convexe alors la population envahit tout l'espace. On se questionne aussi sur les conditions optimales de régularité qui garantissent une invasion complète de la population. Dans un deuxième travail, nous considérons une équation de réaction-diffusion avec vitesse forcée, modélisant l'évolution de la densité d'une population quelconque qui se diffuse dans l'espace, soumise à un changement climatique défavorable. On montre que selon la vitesse du changement climatique la population s'adapte ou s'éteint. On montre aussi que la densité de population converge en temps long vers une onde progressive et donc se propage (si elle survit) selon un profile constant et à vitesse constante. Dans un second temps on étudie une équation de réaction-diffusion de type bistable dans des domaines cylindriques variés. Ces équations modélisent l'évolution d'une onde de dépolarisation dans le cerveau humain. On montre que l'onde est bloquée lorsque le domaine passe d'un cylindre très étroit à un cylindre de diamètre d'ordre 1 et on donne des conditions géométriques plus générales qui garantissent une propagation complète de l'onde dans le domaine. On étudie aussi ce problème d'un point de vue numérique et on montre que pour les cylindres courbés la courbure peut provoquer un blocage de l'onde pour certaines conditions aux bords. / In this thesis we are interested in reaction diffusion equations and their applications in biology and medical sciences. In particular we study the existence or non-existence of propagation phenomena in non homogeneous media through the existence of traveling waves or more generally the existence of transition fronts.First we study a bistable reaction diffusion equation in exterior domain modelling the evolution of the density of a population facing an obstacle. We prove that when the obstacle satisfies some regularity properties and is close to a star shaped or directionally convex domain then the population invades the entire domain. We also investigate the optimal regularity conditions that allow a complete invasion of the population. In a second work, we look at a reaction diffusion equation with forced speed, modelling the evolution of the density of a population facing an unfavourable climate change. We prove that depending on the speed of the climate change the population keeps track with the climate change or goes extinct. We also prove that the population, when it survives, propagates with a constant profile at a constant speed at large time. Lastly we consider a bistable reaction diffusion equation in various cylindrical domains, modelling the evolution of a depolarisation wave in the brain. We prove that this wave is blocked when the domain goes from a thin channel to a cylinder, whose diameter is of order 1 and we give general conditions on the geometry of the domain that allow propagation. We also study this problem numerically and prove that for curved cylinders the curvature can block the wave for particular boundary conditions.