The Generalized Lotka-Volterra (GLV) model, widely employed in population dynamics, serves to characterize how interactions between species influence system equilibria. This model can be extended to the economic domain to quantify wealth inequality among individuals. In this thesis, we explore the impact of the underlying random network structure on the final distribution of abundances, with a focus on homogeneous degree settings, particularly regular random graphs. While the GLV model has been extensively studied in competitive and predator-prey scenarios, the mutualistic (or cooperative) case has received less attention due to the divergence of species abundances. In the context of this divergence, we investigate the hierarchy of infinities by revisiting the GLV model and introducing a framework wherein species undergo relative extinction when compared to others. For large interaction strengths, we observe the emergence of effective competition within a cooperative system, resulting in localization, a phenomenon in which a few highly connected species dominate and accumulate most of the wealth. As interaction strength decreases, the surviving component follows a geometrical evolution, which we analytically quantify in this work. Finally, we discuss the implications of this framework for wealth inequality through the Gini index and extend the analysis to heterogeneous degree settings using the Erdos-Renyi random graph mode

Cancer affects a countless number of lives across the world each day. Mathematical oncology develops and studies mathematical models of cancer and its treatment. This thesis focuses on spatiotemporal modeling in mathematical oncology, developing an agent-based model for prostate cancer, with the aim of gaining insights into how the different tumor cells respond to varying testosterone levels and different treatment strategies.

We began by analyzing non-spatial population models, the replicator dynamics and Lotka-Volterra dynamics, proving their equivalence under certain conditions. The study then transitioned to spatial agent-based modeling on a discrete lattice, simulating the interactions between testosterone-dependent and testosterone-independent tumor cells. Through this, we identified a possible phase transition in the testosterone level in the bloodstream, which could influence which tumor cells dominates the grid.

A continuum limit of the discrete model was derived, leading to partial differential equations that describe the tumor's spatial behavior. We applied mathematical tools like non-dimensionalization and linear stability analysis to gain deeper insights into the dynamics of the model. Additionally, we simulated three treatment strategies: (1) testosterone removal from the blood with Lupron, (2) Lupron combined with Abiraterone to stop the testosterone producing cancer cell to grow, and (3) Lupron and Abiraterone alongside high-dose testosterone injections to simulate extinction therapy.

The flexibility of our model allows for its application to other hormonal cancers, and our findings support the promising potential of hormonal manipulation in controlling tumor growth and composition, especially extinction therapy. The mathematical analysis together with simulations provide unique insights into the tumor dynamics. Future research directions include changing assumptions, expanding the model to three dimensions and integrating patient data for more accurate simulations.

A cellular automaton for simulating territories is presented. In it, cells have a certain amount of markings of two different groups. The amount of markings for each group gets higher based on the amounts of that group in neighboring cells and the amount of markings of the opposite group in the current cell to simulate an avoidance tendency. The markings also decay at a certain rate. Depending on the parameters, the simulation can end up in a mixed state, where there are no clear territories, or a segregated state, where both groups have a large amount of connected cells where they are dominant. Small changes in these parameters can change the outcome significantly. Unless an unconsidered combination of parameters changes it, this model is not the most realistic. It could however have uses in more influence-based processes, such as the spread and boundaries of languages or religion.

In 1952, Alan M. Turing presented a reaction-diffusion model that described formation of skin patterns. The patterns he predicted have later been found in various natural phenomena, such as in skins of fish or even in vegetation around termite hills. His patterns have even been taken to the micro-level. In 1984, David A. Young proposed a discretisation of this model, which enabled computer simulation. Both Turing and Young had only looked at two-dimensional patterns, until Martin Skrodzki and Konrad Polthier took the patterns to the third dimension in 2017. In this paper, these 3D simulations are generalised to produce patterns with more than two substances. We want to see whether Turing-like patterns also emerge there. To increase simulation speeds, a Graphics Processing Unit (GPU) implementation is described for this multi-species extension. Furthermore, we begin to investigate to what extent an order parameter can be defined, to analyse the formation of 3D structures. We found that our multi-species extension produced Turing-like structures that look similar to the ones found by former models. Our GPU simulation provided a significant performance increase. We also found that our order parameter can distinguish between well-mixed, well-segregated, and fully dominated states. However, it is yet unclear whether it can also be used to classify shapes.

This paper describes the process and evaluation methods by which we adapted a Reaction-Diffusion model and an order parameter to monitor its segregation state in a 2D Cellular Automaton model. The model simulates Turing pattern formations, whose behavior will be studied by an order parameter formulated for this project. This order parameter acts as an indicator of the mixture state of the world. The implementation process of such a cellular automaton and an order parameter is described in detail, as well as the evaluation of this model and its order parameter. Different evaluation methods will be performed on this order parameter to show it as a valid indicator of the segregation state of the world in this model, as well as showcase different interesting properties and limitations of the parameters. Such methods will involve plots like Evolution, Phase-Transition, or Linear Regression plots, and the paper will conclude by summing up all the work taken into this project to confirm that this order parameter, with its formulation in this paper, is a valid and interesting one to use.

This research paper proposes a discrete agent-based model to simulate territorial development among micro-organisms. The model involves two species that interact through marker signals left behind by agents as they move through a three-dimensional lattice. The study builds on previous research that established a phase transition from a well-mixed to a well-segregated state for two-dimensional lattices. This research extends the finding to three dimensions and confirms that the properties observed in the two-dimensional model are also present in the three-dimensional model. We conclude that the addition of more mass or the ratio between gamma and lambda behaves similarly to the two-dimensional model. However, the three-dimensional model needs a larger mass to reach the same critical point as the two-dimensional one.

In several experiments, enzymes have shown an in increase in diffusivity in the presence of their substrate. The enhancement in diffusivity ranged from as low as 28% for urease to 80% in the case of alkaline phosphatase. There are two main competing theories. One asserts that catalytically driven boosts propel the enzyme forward in ‘leaps’, while the other argues that phoretic activity due to attractive or repulsive surface interactions on the enzyme are responsible for the enhanced diffusivity. At the moment of writing, no consensus has been reached on the mode of enhancement. A novel agent-based lattice model for the diffusion of enzymes was derived in this Bachelor’s Thesis in accordance to the phoretic theory of enhanced diffusion. The model that was developed is a stochastic model which describes the interactions at the particle level. The model showed that phoresis produces enhanced diffusion in the order that was expected for the enzyme urease. Furthermore, the model showed pattern formation in the case of repulsive interactions between enzyme and substrate. The instability of this phase transition was investigated using linear stability analysis on the continuum limit of the lattice model. This yielded a restriction on the strength of the interactions for which pattern formation could occur.