We investigate the migratory behaviour of juvenile loggerhead sea turtle (Caretta caretta) hatchlings within a region of the North Atlantic Gyre. To do so, we develop and compare an individual based (IB) model and a partial differential equation (PDE) model, specifically an advection-diffusion equation derived from a position-jump process. We compare their behaviour and show that they yield similar results, but that there are still some differences between the two. Using the IB model we assess survival probabilities of hatchlings in recent years (2016-2023), revealing that survival rates are statistically significantly (𝑝-value = 2.62 ⋅ 10^−6) not constant over time in this region. However, there could be global events that influence the survival probability, as years 2019 and 2023 have a large deviation in survival probability.

Vascular calcification, the deposition of calcium in the vessel wall, is associated with several vascular diseases, including atherosclerosis, diabetes mellitus, and hypertension.
Fluid-structure interaction (FSI) is recommended to simulate blood flow incorporating vascular calcification. However, FSI applied to a three-dimensional (3D) model takes several days to simulate.
To reduce the computational complexity, 1D reduced order models (ROMs) are often used instead.

Reduced order modeling decreases the computational complexity of a model by removing dimensions of the coordinate system within a model. The cylindrical coordinate system is used in hemodynamics, especially in ROMs. The 1D ROM for hemodynamics is obtained by removing the azimuthal dimension (accomplished by assuming axial symmetry for all properties within arteries) and the radial dimension (accomplished by applying a predefined velocity profile to blood flow) from the 3D model. However, incorporating vascular calcification can make the geometry of arteries and flow within arteries asymmetric. A 2D ROM can increase the accuracy of the 1D ROM by including one of the two removed dimensions. Research regarding 2D blood flow mainly focuses on including the radial dimension, which cannot implement asymmetric calcification since axisymmetry is assumed.

This study obtains a 2D ROM for blood flow by removing the dimension corresponding to the radial distance from the three-dimensional model and by assuming that axial velocity is continuous in the neighborhood near the artery's origin. The 2D ROM obtains axisymmetric velocity by only allowing a single velocity profile. However, enabling a family of velocity profiles can make flow within arteries asymmetric. Hence, this study contributes to hemodynamics by studying blood flow that allows a family of velocity profiles.

A non-physiological steady-state solution has been obtained analytically, in which the volumetric flow rate vanishes, and numerical methods are developed to simulate the 2D ROM, which incorporates dimensional (Godunov) splitting, linear approximate solvers, and high-resolution methods. Jump-discontinuities within the mechanical properties of the vascular walls are smoothened for the 2D simulations. Numerical methods for the 2D ROM yield significant errors within the smoothening region for simulations with coarse grids.
The numerical method obtains the non-physiological steady-state solutions for arteries without calcification and has a relative error of O(Δx^1.500) for arteries with axisymmetric calcification. The 2D ROM cannot numerically obtain the non-physiological steady-state solution for arteries with asymmetric calcification due to the numerical errors within the smoothening range.

 3D and 2D numerical simulations with pulsatile blood flow are compared. The 3D simulation without calcification has a significantly higher diastolic pressure, larger inner wall radii, and larger volumetric flow rates than the 2D simulation. The differences in blood flow observed between pulsatile blood flow without calcification and with calcification match decently between the 3D simulations and the 2D simulations, except for locations within the smoothening region.

In this thesis, Boltzmann's H-theorem is studied and applied to prove a general convergence to equilibrium for the adiabatic piston paradox, governed by a specially-derived kinetic Fokker-Planck equation. We review general results in kinetic theory on the Boltzmann collision operator, and rates of convergence to equilibrium. Furthermore, we turn our attention to simulations of the piston paradox, and apply the algorithm of Sigureirsson et al. for particle-particle collision dynamics to the piston setting, determining empirically the optimal parameters.

In this thesis, wealth distribution in a closed economic system is examined by studying the simple inclusion process (SIP). The simple inclusion process is a model coming from statistical physics that models the jumping of particles in a graph. In the model particles have attraction among each other and each site also has a characteristic attraction parameter, denoted by the variable α. We regard the simple inclusion process as an agent based model for the economy for which sites represent agents and the particles represent wealth transferring from agent to agent. For the model, invariant measures are found that represent possible long term distributions of wealth in the system. We extend the model by looking at α as a parameter drawn from its own underlying probability distribution, ψ(α). In the view of wealth distribution, α can be seen as the wealth attraction an agent has. We set out to look for conditions on the distribution of α for which we obtain asymptotic power law behaviour of the resulting wealth distribution. It is shown that if a higher-order moment of ψ(α) diverges, the resulting wealth distribution will have a weak asymptotic power law lower bound. By setting stricter conditions on the distribution of α, we obtain that the wealth distribution will be asymptotically equal to a power law.

The Leslie model considers age-specific birth and survival rates to describe how a population size and age distribution changes over time. This thesis investigates the long-term dynamics of the Leslie model for population growth, utilizing mathematical theorems such as Perron-Frobenius, Doeblin’s theorem, and Branching processes.

The Perron-Frobenius theorem guarantees the existence of a dominant eigenvalue. This dominant eigenvalue and its corresponding eigenvector represent the long-term behaviour of a population; The dominant eigenvalue indicates the long-term population growth, and the corresponding eigenvector indicates the long-term age distribution. Furthermore, the Perron-Frobenius theorem implies that a population asymptotically reaches a stable age distribution that is independent of its initial age structure. Once this stable age distribution is reached, the population continues to grow exponentially, exhibiting Malthusian behaviour. Doeblin’s theorem, although not directly applicable to the complete Leslie model, provides valuable insights into the long-term behaviour of Markov chains. As Doeblin‘s theorem can not be applied to the complete Leslie model, the reproduction process of the Leslie model is formulated as a Branching process. Introducing the Leslie model as a Branching process allows for the consideration of demographic stochasticity. Simulations reveal that for larger populations, the Branching process closely mirrors the Leslie model, while disparities become more pronounced in smaller populations. These results illustrate the impact of probabilistic factors in population dynamics, as well as the strength of the Leslie model for larger populations.