Competitive-Consecutive and Competitive-Parallel reactions are both mixing sensitive reactions; the yield of desired product from these reactions depends on how fast the reactants are brought together. Recent experimental results have suggested that the mixing effect may depend strongly on the stoichiometry of the reactions. To investigate this, a 1-D, non-dimensional, reaction-diffusion model at the micro-mixing scale has been developed. Assuming constant mass concentration and diffusivities, systems of PDEs have been derived on a mass fraction basis for both types of reactions. A single general Damkhler number and specific dimensionless reaction rate ratios were derived for both reaction schemes. The resulting dimensionless equations were simulated to investigate the effects of mixing, reaction rate ratio and stoichiometry of the reactions. It was found that decreasing the striation thickness and the dimensionless rate ratio maximizes yield for both types of reactions and that the stoichiometry has a considerable effect on yield. All three variables were found to interact strongly. Phase plots showing the interactions between the three variables were developed.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.

Mathematical models displaying travelling waves appear in a variety of domains. These waves are often faced with different kinds of perturbations. In some cases, these perturbations result in propagation failure, also known as wave-blocking. Wave-blocking has been studied in the case of several specific models, often with the help of numerical tools. In this thesis, we will display a technique that uses symmetry and a center manifold reduction to find a criterion which defines regions in parameter space where a wave will be blocked. We focus on waves with low velocity and small symmetry-breaking perturbations, which is where the blocking initiates; the organising center. The range of the tools used makes the technique easily generalizable to higher dimensions. In order to demonstrate this technique, we apply it to the bistable equation. This allows us to do calculations explicitly. As a result, we show that wave-blocking occurs inside a wedge originating from the organising center and derive an expression for this wedge to leading order. We verify our results with some numerical simulations.

This thesis focuses on the impact of spatially heterogeneous environments on the spatio-temporal behavior of planktonic systems. Specific emphasis placed is on the influence of spatial variations in the strength of random or chaotic movements (diffusion) of the organisms. Interaction between different species is described by ordinary differential equations. In order to describe movements in space, reaction–diffusion or advection–reaction–diffusion systems are studied. Examples are given for different approaches of diffusive motion as well as for the possible effects on the localized biological system. The results are discussed based on their biological and physical meanings. In doing so, different mechanisms are shown which are able to explain events of fast plankton growth near turbulent flows. In general, it is shown that local variation in the strength of vertical mixing can have global effects on the biological system, such as changing the stability of dynamical solutions and generating new spatiotemporal behavior. The thesis consists of five chapters. Three of them have been published in international peer-reviewed scientific journals. Chapter 1. Introduction: This chapter gives a general introduction to the history of plankton modeling and introduces basic ideas and concepts which are used in the following chapters. Chapter 2. Fokker-Planck law of diffusion: The influence of spatially in- homogeneous diffusion on several common ecological problems is analyzed. Dif- fusion is modeled with Fick’s law and the Fokker–Planck law of diffusion. A discussion is given about the differences between the two formalisms and when to use the one or the other. To do this, the discussion starts with a pure diffusion equation, then it turns to a reaction–diffusion system with one logistically growing component which invades the spatial domain. This chapter also provides a look at systems of two reacting components, namely a trimolecular oscillating chemical model system and an excitable predator–prey model. Contrary to Fickian diffusion, spatial inhomogeneities promote spatial and spatiotemporal pattern formation in the case of Fokker–Planck diffusion. A slightly modified version of this chapter has been published in the Journal of Mathematical Biology (Bengfort et al., 2016). Chapter 3. Plankton blooms and patchiness: Microscopic turbulent motions of water have been shown to influence the dynamics of microscopic species. Therefore, the number, stability, and excitability of stationary states in a predator– prey model of plankton species can change when the strength of turbulent motions varies. In a spatial system these microscopic turbulent motions are naturally of different strength and form a heterogeneous physical environment. Spatially neighboring plankton communities with different physical conditions can impact each other due to diffusive coupling. It is shown that local variations in the physical conditions can influence the global system in the form of propagating pulses of high population densities. For this, three local predator–prey models with different local responses to variation in the physical environment are considered. The degree of spatial heterogeneity can, depending on the model, promote or reduce the number of propagating pulses, which can be interpreted as patchy plankton distributions and recurrent blooms. This chapter has been published in the Journal Ecological Complexity (Bengfort et al., 2014). Chapter 4. Advection–reaction–diffusion model: Here, some of the models introduced in chapter 1 and 2 are modified to perform two dimensional spatial simulations including advection, reaction and diffusion. These models include assumptions about turbulent flows introduced in chapter 1. Chapter 5. Competition: Some plankton species, such as cyanobacteria, have an advantage in competition for light compared to other species because of their buoyancy. This advantage can be diminished by vertical mixing in the surround- ing water column. A non–spatial model, based on ordinary differential equations, which accounts for this effect is introduced. The main aim is to show that vertical mixing influences the outcome of competition between different species. Hystersis is possible for a certain range of parameters. Introducing a grazing predator, the system exhibits different dynamics depending on the strength of mixing. In a diffusively coupled horizontal spatial model, local vertical mixing can also have a global effect on the biological system, for instance, destabilization of a locally stable solution, or the generation of new spatiotemporal behavior. This chapter has been published in the Journal Ecological Modelling (Bengfort and Malchow, 2016).

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In this thesis, the two-dimensional initial and boundary value problems (IBVPs) and the one-dimensional Cauchy problems defined by the nonlinear reaction- diffusion and wave equations are numerically solved. The dual reciprocity boundary element method (DRBEM) is used to discretize the IBVPs defined by single and system of nonlinear reaction-diffusion equations and nonlinear wave equation, spatially. The advantage of DRBEM for the exterior regions is made use of for the latter problem. The differential quadrature method (DQM) is used for the spatial discretization of IBVPs and Cauchy problems defined by the nonlinear reaction-diffusion and wave equations. The DRBEM and DQM applications result in first and second order system of ordinary differential equations in time. These systems are solved with three different time integration methods, the finite difference method (FDM), the least squares method (LSM) and the finite element method (FEM) and comparisons among the methods are made. In the FDM a relaxation parameter is used to smooth the solution between the consecutive time levels. It is found that DRBEM+FEM procedure gives better accuracy for the IBVPs defined by nonlinear reaction-diffusion equation. The DRBEM+LSM procedure with exponential and rational radial basis functions is found suitable for exterior wave problem. The same result is also valid when DQM is used for space discretization instead of DRBEM for Cauchy and IBVPs defined by nonlinear reaction-diffusion and wave equations.

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.

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.

A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.

The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.