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.

This work is a bachelor thesis featuring a collection of explorations of the Kuramoto model in the presence of noise in one and two dimensions. The model was explored analytically and numerically. It contains a study of literature and new analytical derivations. An attempt to link the fraction of oscillators to the order parameter resulted in experience through the process of deriving the result, but a plot of the first few terms of this expansion showed it not being close. The assumption on which the expansion was based however was shown to be valid in simulations. More successfully an upper bound on the probability of synchronization was found. This probability was found on a ring consisting of N oscillators, and verified of a ring of 3 oscillators. The results was a bound better than one that was already calculated, though the method of verification used was not ideal. An analysis of the stability of solutions on this 3-ring resulted a visual representation of entrainment and an amount of stable solutions possible, which was 1. This result was in accordance with the more general result following from the investigation of a system of coupled rings with arbitrary sizes of either ring. We then expanded this system to conclude that the stability of the entire system could be split into the stability of either ring and the possibility of having a solution for the linking phase difference. We elaborated on general practices when numerically analyzing SDEs and integrations methods. The Kuramoto model proved to be kind in the sense that the Euler-forward and Milstein scheme coincided. In a more physics-centered approach we studied the Ising and XY model. This prompted literature study on the Kosterlitz Thouless transition, through analysing the spin-wave Hamiltonian and the energy of free vortex of a pair of vortices. We also then explored conservation of the vortex-pair behaviour in a noise including Kuramoto model, which verified a duality in partition functions as described in ref. [1].