The analysis of equilibria of complex systems is challenging due to the high dimensionality and non-linear interactions. There are different kinds of complex systems such as cooperative systems in which all interactions help growth and mixed-weight systems, in which some interactions help growth, while others hinder it. In this thesis, we focus on the correlation between the equilibria of cooperative and mixed-weight systems. We simplified the equilibrium equations by combining the attributes of a system into a one-dimensional equation. This reduced equation is easy to compute and provides an upper bound on the equilibrium value of each node. Although this bound may exceed many actual equilibrium values, it still defines the subspace in which all equilibria must lie. For cooperative systems, we presented a theorem that provides constraints on two vectors. If these vectors satisfy the given conditions, then there exists an equilibrium between the components. We also discussed methods to find such vector pairs. We applied this theorem to relate the equilibria of mixed-weight and cooperative systems. The equilibria of the mixed-weight system are always less than or equal to some equilibrium in the cooperative system. We introduced a framework for classifying cooperative equilibria. On any subset of nodes, an equilibrium may have entries that are maximal compared to all other equilibria on that subset. This leads to a single equilibrium that is the largest at every entry, called the principal equilibrium, which is component-wise maximal. The principal equilibrium upper bounds all equilibria of the mixed-weight system. Finally, we discussed the inherent difficulty of translating cooperative equilibria into the mixed-weight system, which stems from high dimensionality and non-linearity. We stated the conditions that mixedweight equilibria must satisfy and provided constraints determining if a cooperative-system equilibrium remains valid when competitive interactions are added. This concludes the comparison by showing that the principal equilibrium provides a component-wise upper bound for all equilibria of the mixed-weight system.

Modeling plankton communities has been an import topic in mathematical biology for quite some time. Previous research mostly comes in two flavors. On one hand we have large global models, which try and recreate measured data, but often lose track of what are the root causes of phenomena. On the other hand we have smaller models, where the (mathematical) reasoning behind phenomena are tried to be understood, but they lose some of their applicability. We try to bridge the gap between these two, by exploring how far results after simplification carry over to a more general case. We do this by investigating a size structured plankton model as proposed by (Poulin & Franks, 2010). We first simplify the interaction between phyto- and zooplankton, for which we are able to find analytical stationary solutions. Using numerical methods we are able to show stable stationary and limit cycle behavior. Furthermore we are able to show how much diversity remains, and how this is linked to the analytical solutions. Then we are able to show that these structures remain in place after allowing more complex interactions, and identify how far this remains true. Using this knowledge we are able to give quick insights into more complex models.

This thesis investigates the performance of quantum walk search algorithms (QWSAs) on
two-dimensional rectangular lattices and on bond percolated two-dimensional square lat-
tices. The focus is on how structural disorder, modelled by static and dynamic percolation,
affects the success probability and optimal runtime of the algorithm. In the unpercolated
case, the principal eigenvalue technique provides asymptotic expressions for runtime and
success probability. These approximations agree well with simulations on square grids,
but underestimate the runtime for rectangular grids. This seems to be a consequence of
breaking the grid symmetry, which enhances higher-frequency spectral components and re-
sults in interference effects that delay the optimal runtime. The introduction of dynamic
percolation causes a rapid drop in success probability, especially for larger grids. This
models strong decoherence, where the short timescale of structural change disrupts coher-
ent amplitude buildup. Static percolation degrades performance more gradually, with a
sharp decline only near the percolation threshold, where global connectivity is lost. These
results show that QWSAs are adversely affected by changing graph structures over time,
highlighting the importance of sufficiently long coherence times. This study provides a
way to test how well quantum search algorithms perform in disordered environments and
future work could extend this to other types of graphs or quantum walk algorithms.

Accurately characterizing the properties of semiconductors at atomic resolution is crucial for advancing semiconductor technology. One of the key challenges in quantitatively interpreting Scanning Tunneling Spectroscopy (STS) is the influence of tipinduced band bending (TIBB) during semiconductor measurements. This thesis presents a numerical method that self-consistently solves the one-dimensional Poisson equation to correct for TIBB in adatom-covered semiconductors. The model uses the Block-SOR-Newton method to solve the nonlinear system of equations that arises from the discretization of the Poisson equation. Simulations demonstrate that the model can accurately describe the effect of the STM tip voltage on band bending. However, numerical instabilities were observed for high doping concentrations and surface state energies close to the Fermi level, attributed to overshooting of the Newton method. Potential solutions, such as using a Newton-Krylov method and adaptive grid refinement, were proposed to address these instabilities. Future work includes extending the model to non-equilibrium situations by introducing the full set of semiconductor equations and expanding to three dimensions to account for the STM tip geometry. These advancements would provide a useful tool for correcting STS data, ultimately deepening our understanding of semiconductor physics at the atomic scale.

Models governed by systems of ordinary differential equations (ODEs) often produce complex and unpredictable behaviors. To address this, we can use dimension reduction techniques, which simplify these models, allowing for the retention of specific behaviors while greatly decreasing the cost of numerical solutions and, in some cases, enabling analytical derivations of sufficient conditions for the existence of nonzero fixed points of the model’s ODEs. This thesis reviews the state-of-the-art reduction theories and extends the established proofs by Wu et al., by providing necessary assumptions, lemmas, and a novel proof of an important proposition used in their work. We additionally verify and confirm Wu et al.’s findings and predictions for a cooperative version of the Cowan-Wilson model, which describes a population of neurons’ firing activity. We derived a one-dimensional reduction, inspired by Laurence et al., for a generalized Cowan-Wilson model, which we introduced in this thesis. Unlike the original, this generalized model can produce oscillatory behavior without external stimulus. A valuable finding is that the method of reduction does not depend on the specific form of the Cowan-Wilson function, allowing it to be applied to a broader class of nonlinear dynamics. Our reduction is able to predict system behavior effectively, given the network yields a unique reduction. However, the reduction parameter was not unique in approximately half of the networks studied, which saw a 66% increase in average error, suggesting these networks are inherently multi-dimensional. This opens the door for future research into the existence of a multi-dimensional reduction framework that could mitigate this discrepancy.

In this thesis the stored energy and its fluctuations of a central spin battery with nearest-
neighbour interactions between the battery spins are investigated. Using analytical ex-
pressions, it is shown that for 2 battery spins and equal strength in the flip-flop interaction
g and nearest-neighbour interaction J, the fluctuations are minimal whenever the bat-
tery is maximally charged when taking at least four charge spins. Similarly, whenever
the formed envelopes of the energy have a zero, the fluctuations have a global maximum.
In the same limit, it could also be seen that an increase of the charge spins Nc and
spin-ups m, resulted in a higher global maximum of the stored energy. Furthermore for
2 battery spins, taking the limit J ≫ g results in a situation where the battery cannot
be charged at all, whereas taking the limit g ≫ J results in a central spin battery where
no nearest-neighbour interactions are present; its stored energy as a function of time is
a single cosine function, that is always able to reach its theoretical maximum.
Similar results were found for systems with more than 2 battery spins. Increasing J with
constant g resulted in a decrease of the global maximum of the energy, dropping from its
theoretical maximum to its minimum. Opposite behaviour could be seen when increasing
g with constant J. Whenever the global maximum of the energy crossed the line E = 0,
the fluctuations at the same moment in time formed a peak.

Angiogenesis, i.e. the formation of blood vessels from existing ones, plays a vital role in bone or wound healing. The expansion of vascularization facilitates the healing process through the delivery of oxygen and nutrients to the injured site and through the removal of waste products. Clinical observations indicate that impaired angiogenesis can impede the healing process, or can result in non-healing outcomes.
The computational model developed in this thesis predicts tip/stalk cell patterning, marking the initial phase of sprouting angiogenesis. Growth factors signal endothelial cells to differentiate into tip and stalk cells. Tip cells branch from the existing vessel, leading the sprout, while stalk cells proliferate and follow behind, forming the newly emerged blood vessel. Understanding tip/stalk cell patterning is vital to ensure successful angiogenesis, as an excess or deficiency in tip cells leads to improper healing.
Despite several experimental studies and mathematical models exploring the signaling pathways behind tip cell selection, there is a noticeable gap regarding the effect of extracellular matrix (ECM) stiffness on this process. Given that alterations in stiffness occur in various physiological and pathological processes, comprehension of this effect is clinically relevant. This thesis aims to address the existing gap by investigating the specific influence of ECM stiffness on tip/stalk cell patterning.
A computational model is created that simulates a vessel sprout under stimulation of growth factors. This model is able to predict the cell patterning over various ECM stiffness levels, and highlights the relevance of incorporating ECM stiffness in the investigation of angiogenic treatments.
Enhancing the models’ accuracy and validating the ECM stiffness-dependent model predictions requires additional experimental data. However, further development of the model has great potential for deepening our understanding of angiogenesis dynamics and for facilitating the investigation of treatment strategies.

Staphylococcus aureus and Pseudomonas aeruginosa are two species of bacteria that are involved in numerous conditions, including lung infections and chronic wound infections. The aim of this project was to study the short-term interactions that occur when P. aeruginosa first encounters an established S. aureus colony, which it then seeks to break apart whilst mixing with S. aureus. Limoli et al. have studied these interactions using experiments, and have thus identified several key aspects involved in these interactions, such as the mechanisms that P. aeruginosa employs to approach the S. aureus colony. The means by which we intended to study interactions between S. aureus and P. aeruginosa is a model that was made by previous members of the Idema group and that was based on the experiments by Limoli et al. In this report, we discuss this model and the biological background relevant to it. We also document the problems that we encountered while trying to run simulations using an existing implementation of this model.

Quantum algorithms have shown much potential in solving complex problems more efficiently than our current classical algorithms, particularly in search problems with a large and complex search space. When leveraging the principles of superposition and interference, quantum walk based algorithms can lead to a faster convergence on a target state. The thesis begins with a theoretical framework for the study of classical and quantum walks on graphs. The usage in search algorithms is discussed together with the effect of a modified Hamiltonian that introduces a bias toward the target state. This is analyzed using perturbation theory showing a decrease of the ground state energy. Ultimately, this thesis focuses on the optimization of quantum random walk search algorithms using resetting techniques. Simulations of quantum random walks on simple graph structures, such as path and cycle graphs are used to provide concrete examples. Results show that resetting not only improves the probability of finding the target state, but also has a faster convergence. Furthermore, decoherence effects are shown to be reduced when using resetting techniques.

While many people believe in egoism and protectionism, the concept of "win-win cooperation" is widely accepted worldwide. Investigating how cooperation behavior emerges and evolves can help us understand the interaction mechanism in human society.

In evolutionary game theory (EGT), the traditional methods are the Replicator equation and the Moran Model. However, these methods have limitations as they do not adequately consider the impact of network topology.

This thesis aims to investigate the cooperation behavior in different network topologies using analysis and modelling methods.

Networks in EGT are high-level representations of intricate systems, capturing the individuals in a real complex system and the relationships between them as nodes and connected edges. Each node can adopt one of two strategies: Cooperation or Defection. Nodes can also alter their strategies based on max-payoff strategy updating rule. Cooperation behavior is analyzed through the cooperation percentage during the generation.

We began by examining well-mixed populations. We then explored homogeneous network games. The results show that in well-mixed and circle networks, defection is the only stable state. However, in grid networks, there exists an infinite coexistence of cooperation and defection, which has a threshold on the size of the grid networks.

Lastly, we expanded our investigation to heterogeneous network games. Through numerical simulations, we demonstrated that both cooperation and defection can be stable states. We also discovered that heterogeneity does not directly promote cooperation, but rather indirectly influences it through the "celebrity effect".

The Kuramoto model (KM) is a well known mathematical model of coupled oscillators that is frequently used to study synchronization phenomena. In this bachelor thesis we investigate the effects of noise on synchronization in Kuramoto-type networks.

In the first part we follow the methods of Maggi and Paoluzzi [1], but include detailed in between steps, to obtain an analytical expression for the critical coupling strength, $k_c$, of the KM in the thermodynamic limit under the influence of time-correlated noise (i.e non-white noise). The coupling strength, $k$, is a parameter in the KM that essentially determines to what extent oscillators influence each other. When $k > k_c$ we start to see synchronization. Our numerical simulations agree with the results found in [1], in that the analytical expression for $k_c$ holds up for low values of correlation time, but quickly breaks down as correlation time increases.

In the second part of this thesis we consider a Kuramoto-type adaptive dynamical network that is also investigated in Fialkowski et al. [2]. The dynamical phenomena that are observed in [2] are also present in our simulations. We explain these dynamical phenomena with the help of a variety of plots. Subsequently, the Kuramoto-type adaptive dynamical network is expanded to include white noise terms in the coupling dynamics. We find, through the use of simulation, that under the same conditions as in [2], synchronization is observed for significantly lower values of coupling strength. This result is explained qualitatively. Simulations also show, that for specific values of noise strength, the
hysteric behaviour observed in [2] is not present.

Other conclusions, like the degree to which the noise can reduce the coupling strength required for full synchronization, or beyond what value of noise strength full synchronization can no longer occur, are unable to be drawn. Additional simulation work and a further analytical work is recommended for an ensuing study.

Quantum random walks are the quantum analogs of classical random walks and appear to be promising tools to design fast quantum algorithms. Therefore it is important to study their time-related features and see how these differ compared to the classical case. For the discrete-time quantum walk on the line it has been shown that the probability to be absorbed by an absorbing boundary equals 2/π in contrast to the classical case where this probability equals 1, hence a quantum walk may continue forever without getting absorbed. It is also shown that mixing times of discrete-time quantum walks on the hypercube scale with n, the dimension of the hypercube, which is faster than O(n log(n)) for the classical random walk. So the quantum walk might offer a slight speed-up compared to the classical case. Finally, it is shown that the mixing times of the continuous-time quantum walk on a 2-layer multiplex graph depend on the eigenvalue gaps of the corresponding Laplacian matrix L. When the strength of the connections between the layers of a multiplex graph becomes very large, the eigenvalues of the Laplacian matrix converge. Thus the mixing times of the continuous-time quantum walk on 2-layer multiplex graphs converge.

Random walks have been used in a number of different fields for a long time. With the rise of of quantum random walks a lot of these applications have been made more efficient. Recently there have been a lot of improvements in the realization of these quantum random walks in real world systems.
In this research, the effect of uncertainty in the measurement time and measurement frequency are studied on three problems that make use of a quantum random walk. These problems are the network centrality problem, the graph isomorphism problem and the spatial search problem.
These effects are studied by first looking at the theory behind these problems after which the theoretical results of these three problems are calculated. These theoretical results will then be compared to simulated results that have a certain level of uncertainty in the measurement time or frequency.
For both the network centrality problem and the spatial search problem we concluded that only a small error is made when uncertainties are introduced in the system. This implies that these problems are solvable in realizations of the quantum random walk in real world systems. For the Graph isomorphism problem we saw that the error that is made when uncertainties are introduced was large which implies that this problem would only be solvable for small uncertainties.

The detection of non-Abelian exchange statistics is an open challenge which holds important promises for the advent of topological quantum computation. A recent work proposes to rely on the edges to reveal the braiding statistics of nonAbelian anyons in the bulk, in an entirely deterministic dynamical process. A time-dependent gap in a Josephson junction couples two co-propagating Majorana fermions, and as the gap closes, a pair of edge-vortices is injected into the edges. Because these defects have the same non-Abelian statistics, they are braided with vortices in the bulk. Conveniently, the fusion of the edge-vortices results in a quantized unit of charge at the exit. However, this process is so far only predicted in the adiabatic limit. In this work, this assumption is relaxed by means of a full manybody evolution of the superconducting ground state in the Bogoliubov-de-Gennes formalism. Beyond revealing the collective nature of the edge-vortex excitation, we demonstrate that the quantization of charge still holds if the system does not return to the ground state. Furthermore, the effect of path length difference between the edge-vortices confirms the theoretical predictions done in another work on the subject. At fast injections, we reveal weak oscillations in current contributed by the bound states in the junction which average to zero and are removed in the short junction limit. This work is concluded with a preliminary evaluation of the manybody parity operator, which indicates that the edge-vortex may encode the parity of the bulk vortices. This opens the possibility for sequential qubit manipulations on the edge-vortex.

Random walks have been applied in a many different fields for a long time. More recently, classical random walks are being used in wide variety of computer algorithms used to solve complex computational problems like 2-SAT, 3-SAT and the estimation of the volume of complex bodies. In this thesis we look into the quantum version of the familiar random walk, distinguishing between the discrete- and continuous-time quantum randomwalk. Some general properties of random walks are studied throughout this thesis. First we look at the behaviour of both the classical and the random walk on a simple 1-dimensional lattice. Then, to investigate the speed and efficiency of algorithms that use quantumwalks, we analyse quantumrandom walks in graphs, as graphs do a good job of representing the decision trees of algorithms. The graph we look at specifically is the ndimensional hypercube. The criterion we use to see how fast a quantumwalk can traverse certain types of graph is the hitting time. The hitting time is the expected amount of time a random walk takes to reach a certain vertex in a graph for the first time. Literature has shown that classical random walks have a hitting time that scales exponentially with the dimension n of the hypercube. We find that quantum random walks offer a significant speed-up to its classical counterpart. Generally they have a polynomial hitting time, that depends on the Hamming distance between the start and sink vertex. Furthermore we find that quantum walks can have interesting properties such as a non-unitary hitting probability, meaning the quantum walker will never visit the sink vertex. Finally we see that, in some simple, symmetric cases, the hitting time of the quantum walk even is sub-linear, scaling almost like a square root, rather than a polynomial.

Interference of light can be used to determine the concentration of a gas, called gas sensing. The absorption of the light by the gas molecules is measured based on the phase change of the light. In this report, a hollow core photonic crystal fiber is treated. The gas sample is inserted into the hollow core. Photonic crystal fibers possess the characteristic to have electromagnetic modes confined to the core with low attenuation. This means that there can be a high interference rate between the gas and light, which is desirable for gas sensing.
First, less complicated optical fibers were studied. The analytical solution of the electric and magnetic field for the TE and TM modes of the simple fiber were derived and for the TE modes of the step-index fiber. These results were compared to simulation done with COMSOL Multiphysics. Photonic crystal fiber with circular core was then simulated with COMSOL Multiphysics, but the results did not match with the literature. Therefore another photonic crystal fiber was simulated with a star-shaped core. The simulation results found modes concentrated in the core, with low attenuation.
It was attempted to get similar results with a three layer step-index fiber, by varying the imaginary refractive index of the middle layer. The attenuation of the three layer fiber was much higher than that of the photonic crystal fiber for all simulations. This indicates that the three layer step-index fiber does not support propagation modes that are confined to the core. Further research could be done by studying the effect of the radius of the middle layer and the real part of the refractive index.

The transverse flute can be studied on many aspects. In this thesis the sound waves that can be generated inside the flute are studied on their shape and frequencies. To achieve this, two mathematical models of the sound waves inside a flute are developed. To develop these models, solutions to the second dimensional wave equation are calculated analytically and numerically. The exact model is found within a rectangular parallelepipid region with homogeneous boundary conditions. By the method of separation of variables the second-dimensional wave equation is solved and a function of the air pressure inside the flute is calculated. This model can be used to visualise the waves inside the flute. It allows approximate true frequencies of measured musical notes in the flute. The second model, the numerical model is found in the same domain as the exact model, but with non-homogeneous boundary conditions.These are more realistic boundary conditions and therefore this model might be more accurate than the exact model. By the method of finite difference the numerical second-dimensional wave equation is solved. A system of two numerical equations is found. One of the equations calculates the pressure of the air inside the flute by inserting the previous pressure and velocity of the wave inside the flute. The other one calculates the velocity of the sound wave inside the flute by inserting the previous pressure of the sound wave inside the flute. The visualisation of this model however is not yet an accurate model to visualise waves inside the transverse flute. Therefore a conclusion is made that the exact model best describes the sound waves that can be generated inside the transverse flute.

Synchronisation refers to the tendency of coupled oscillators to move in phase with one another over time, departing from their own natural frequency. This phenomenon occurs in almost all branches of science, engineering, and social life. Synchronisation is a well understood concept in classical mechanics. Quantum synchronisation involves attempting to extend this concept to quantum mechanics, where there are number of phenomena such as damping that are more difficult to realise. This thesis focuses on two things. Firstly, it explores the bridge between classical and quantum synchronisation. Both the equivalencies as well as the difficulties that must be overcome. This is initially done by looking at a simple classical model for two coupled oscillators and extending this to a quantum model, and subsequently by exploring the derivation of a master equation for a damped quantum harmonic oscillator with non-linear damping. Secondly, it explore the synchronisation of four coupled quantum van der Pol oscillators. To this purpose, the six different systems (tree, chain, loop, tower, spade, all-to-all) with four coupled classical van der Pol oscillators first analysed and the Arnold tongues are determined, showing for what parameters synchronisation is expected to occur. The equivalent systems with four quantum van der Pol oscillators are explored to determine whether the classical Arnold tongues give a good indication of when quantum synchronisation is expected to occur. An alternative measure for synchronisation is also investigated for these systems based on the relative entropy of coherence. The Arnold tongues for each of the four classical oscillator systems were determined. For some systems (chain and loop) the Arnold tongues were much smaller than for the three oscillator system while for others (tree, all-to-all) they were only slightly smaller. The quantum Arnold tongues showed very similar behaviour to the classical Arnold tongues. Furthermore, when investigating the correlations between different pairs of oscillators, it was found that the correlation decreased as the number of connections separating two oscillators increased. In the chain system, an expected symmetry was found. The relative entropy of coherence is found to be a good measure of synchronisation, better than the more frequently used complex correlator. This is because the relative entropy can take into account the entire system, while the correlator describes the synchronisation between two individual oscillators in the system.

Janus particles are colloidal particles for which one half of the surface has different attributes than the other half. One property of a spherical dielectric particle with half of its surface covered by a layer of another dielectric or metal is that it has a non-uniform scattering pattern when exposed to light. However, the angle with which the light is shone on the particle has a large effect on the scattering pattern produced. Thus it is important that we are able to orient these Janus particles. The orientation can be controlled if we apply an electric field to the particle for example. The movement of colloidal particles with an electric field is widely studied and this field is called dielectrophoresis. For a Janus particle, the calculations for the force and torque become complicated. The movement and rotation of these particles have been studied, however, no analytic solution has been found. In this report, we derive a semi-analytic description of the force and torque due to an external electric field on a spherical Janus particle. For this, first the potential due to an external electric field is determined and then the force and torque are calculated with two methods: the dipole approximation and the Maxwell Stress Tensor method. In the dipole approximation, there is no force on the Janus particle. But, there is a torque on the particle in the dipole approximation. Due to this torque, the Janus particle will orient itself such that its cap points in the direction perpendicular to the applied field. For the torque calculated with the Maxwell Stress Tensor, we get a similar result as in the dipole approximation. On the other hand, according to the calculations with the stress tensor, there is a relatively small force on the particle.