This thesis aims to identify the parameter combinations necessary for the Brusselator and Oregonator models to exhibit oscillating behavior. Initially, the Brusselator model is reviewed, reproducing the results from [1]. Using Bendixson’s Criterion and the Poincaré-Bendixson Theorem, the existence of stable limit cycles is proven, and the Hopf-Bifurcation locations are derived analytically. The Brusselator model is then extended from a batch reactor to a Continuous Stirred-Tank Reactor to include the inflow of two components previously considered constant. An extensive eigenvalue investigation in Matlab is conducted to determine the parameter combinations that induce oscillating behavior, with Hopf-Bifurcation locations presented in a three-dimensional plot. Lastly, the Oregonator model is introduced and analysed. The existence of limit cycles is proven using similar methods, and an eigenvalue analysis yields the analytic expressions for the Hopf-Bifurcation locations. Periodic solutions are found in all three models, and the necessary parameter combinations are identified.

In dit verslag is een eolisch sedimenttransport model geformuleerd en het model van De Vries geïmplementeerd en geanalyseerd. Na de afleiding van de eendimensionale advectie-diffusievergelijking vanuit de driedimensionale continuïteitsvergelijking is eerst het model van De Vries zonder diffusie onderzocht. In het model kan er onderscheid gemaakt worden tussen 3 verschillende situaties: Geen (toevoer van) sediment, Sediment-gelimiteerd en Erosie-gelimiteerd. In elke situatie zijn voor lange tijd\-schalen de toestanden van het model afgeleid vanuit de advectievergelijking. In de resultaten kwam naar boven dat de theoretische toestanden sterk overeenkwamen met de toestanden in de numerieke simulatie. Verder is het effect van numerieke diffusie van het model van De Vries geconstateerd en de grootte van dit effect afgeleid. Om het effect van numerieke diffusie te verminderen is diffusie in het model geïntroduceerd samen met het Lax-Wendroff numerieke schema. Dit schema gebruikt de gewogen upwind methode samen met de gewogen centrale Euler methode om de nauwkeurigheid van het model te verbeteren. In de resultaten is aangetoond dat numerieke diffusie kan worden gecompenseerd door het juist kiezen van dit gewicht. Uit een stabiliteitsanalyse blijkt dat de Lax-Wendorff methode een betere stabiliteitcriterium heeft dan de upwind methode. Hierdoor kan een grotere tijdstap gekozen in een numerieke simulatie. De maximale tijdstap is getest door het model te runnen met verschillende tijdstappen rondom het theoretische maximum. Uit de resultaten bleek dat het model op tijdstap die een fractie kleiner was dan de theoretische maximale tijdstap, stabiele oplossingen simuleerde. Zodra de tijdstap de maximale waarde behaalde werden de oplossingen instabiel. Door het toevoegen van diffusie aan het model is een extra voorgeschreven randvoorwaarde nodig. Door implementatie van verschillende randvoorwaarden aan het einde van het domein is gebleken dat de randvoorwaarde 'extrapoleren tweede orde' het best is. De dimensie-analyse van de advectie-diffusievergelijking wijst uit de eolischsedimenttransport gedomineerd wordt door advectief transport en het effect van diffusie verwaarloosd mag worden.

Due to climate change, the sea temperature is rising. This temperature change has an effect on the phytoplankton population. Phytoplankton is responsible for more than 50 percent of the oxygen production on earth, and is therefore crucial for life on earth. In this report, the research question is: is the temperature increase of the water important for the development in space and time of the plankton population in a 2D model of a coastal area of an ocean?

To investigate this, first a 2D model for a coastal area of an ocean is formed. As a basis for this model the 0D model of Steele and Henderson is used to describe the predator-prey model for zooplankton and phytoplankton. Due to this model a repeating pattern in time arises, which is called a limit cycle. This predator-prey
model is expanded to a 2D model by applying convection (v) and diffusion (D) to it, representing the current and dispersion in the water. The chosen parameters and boundary conditions are inspired by the coastal area of the west coast of Portugal.

To evaluate the effect of the water temperature rising, first, the effect of diffusion and convection are studied by answering the following questions: is the convection important for the development in space and time of the plankton population in a 2D model of a coastal area of an ocean? And: is the diffusion important for the development in space and time of the plankton population in a 2D model of a coastal area of an ocean?

The diffusion is dominant in the direction perpendicular to the coast. Due to the diffusion the limit cycle of the predator-prey model is suppressed. Its range gets smaller until the plankton population is completely constant for D > 2.5 m^2/s. This critical value is included in the realistic value of D, which varies from 1 to 2000 m^2/s.

Due to convection, the limit cycle corresponding to the boundary condition occurs in the direction parallel to the coast, making every point in the domain steady state. The higher the velocity v corresponding to the convection is, the less oscillations in phytoplankton density there are in the domain, thus the larger the wavelength of the oscillation is. For v > 0.06 m/s the current is so fast that the predator-prey model has little time to develop before it reaches the right boundary, making the left boundary value more important. This critical value is included in the realistic value of v, which varies from 0.03 to 0.28 m/s.

The model experiences both diffusion and convection more or less equally when the Péclet number vH^2/DL≈1, resulting in D/v≈50 m. If this number is significantly larger than 50 m, diffusion dominates the solution. If the number is significantly smaller than 50 m, convection dominates the solution.

Now the main research question can be answered. By increasing the temperature of the water the growth rate of phytoplankton increases. Due to this increase, the timescale of the predator-prey (τpp ) decreases. The value of D and v for which the predator-prey model and diffusion and convection all influence the solution
equally depends on the timescale with the factor 1/τpp . A smaller time scale means that the predator-prey model contributes more to the model. However, this change is very small in comparison to the scale of the realistic values. In conclusion, in the formed 2D model the temperature increase of the water is not important for the development of the plankton population in space and time.

The results of this report are influenced by the assumptions and approximations made. In this study, several processes, both physical and biological, are described with constants and simplifications. Improving the models of these processes is left for further research.

In this thesis we apply the numerical method of Morphological Geometric Active Contours as proposed by Alvarez, Baumela and Marquez-Neila to microscopic images of plant cells. The goal is to find all plant cells in the images, and then to find their cell walls. This is done in order to calculate certain properties of these cells automatically, such as area, perimeter and ellipticity. As the name suggests, mathematical morphology plays a big part in morphological GAC. This thesis describes the theory behind morphological GAC, mathematical morphology and the implementation in Python of the actual algorithm. We made two adaptations compared to the original model. These are a change in the conditions of the balloon force, and an extra step at the end of each iteration. The change in the balloon force was made to keep edges stronger. The extra step was added to improve smoothness of the contours, since the smoothing parameter had no effect.