The earth’s climate is increasing in temperature and as a result many species’ habitat ranges are shifting. The shift in habitat ranges threatens the local persistence of many species. Mathematical models that capture this phenomena of range shift do so by considering a bounded domain that has a time dependant location on the real line. The analysis on persistence conditions has been considered in both continuous-time and -space, and discrete-time, continuous-space settings. In both model types density was considered to be continuous across the boundaries. However it has been shown that many species exhibit particular behaviour at habitat edges, such as biased movement towards the more suitable habitat. This behaviour should be incorporated into the analysis to obtain more accurate persistence conditions. In this thesis persistence conditions are obtained for generalized boundary conditions with a continuous-time and -space model for a range-shifting habitat. It is shown that a high preference for the suitable habitat at the trailing edge can greatly reduce the size of suitable habitat required for species persistence. As well, for fast shifting ranges, a high preference at the trailing edge is crucial for persistence.

The increase in antibiotic resistance continues to pose a public health risk as very few new antibiotics are being produced, and bacteria resis- tant to currently prescribed antibiotics is growing. Within a typical hospital setting, one may find patients colonized with bacteria resistant to a single an- tibiotic, or, of a more emergent threat, patients may be colonized with bacteria resistant to multiple antibiotics. Precautions have been implemented to try to prevent the growth and spread of antimicrobial resistance such as a reduction in the distribution of antibiotics and increased hand washing and barrier pre- ventions; however, the rise of this resistance is still evident. As a result, there is a new movement to try to re-examine the need for the development of new antibiotics. In this paper, we use mathematical models to study the possible benefits of implementing a new antibiotic in this setting; through these models, we examine the use of a new antibiotic that is distributed in various ways and how this could reduce total resistance in the hospital. We compare several dif- ferent models in which patients colonized with both single and dual-resistant bacteria are present, including a model with no additional treatment proto- cols for the population colonized with dual-resistant bacteria as well as models including isolation and/or treatment with a new antibiotic. We examine the benefits and limitations of each scenario in the simulations presented.

In this thesis an age structured model of fish and harvesting of fish are studied. This includes the basic properties as well as the analysis of equilibria and their stability for various recruitment functions. The only nonlinearity of this model lies in the juvenile stage in form of the recruitment function. Recruitment is considered as recruitment into the first year fertile class. The Ricker recruitment function will mostly be used . But we also discuss the Beverton-Holt recruitment and compare its impact. The main result is that monotone recruitment functions lead to a globally stable equilibrium. For Ricker recruitment functions periodic solutions are possible, but this model for cod and herring always leads to stable equilibria. Harvesting occurs in all classes separately with no harvesting in the first class. We describe harvesting in form of harvesting effort, resulting in an increase of mortality. This approach allows to model the interference of natural mortality and fishing mortality. Harvesting cost will be considered as a linear function of effort and the corresponding maxima are determined. This has the advantage to keep the number of parameters low. This makes the model more tractable and easier to apply to other stocks or species of fish. Optimization of the harvest is determined and analyzed. It is based on harvesting intensity and mesh size. We also show that harvesting increases stability of the static model. This model is extended to a discrete optimal control model in order to determine optimal sustainable policies. The solutions are almost of the fastest approach type to the equilibrium type solution. Periodic or pulse solutions do not arise. Near optimal strategies are analyzed and other feasible strategies are also considered. In last part of this thesis the problem of extinction of population is investigated. The effect of various factors as well as natural factors on marine fish populations are studied, which are described by the model in order to get some idea on the size of various factors. Most fishing takes place near the break even intensity. It is defined as the fishing mortality at which cost equals profit. As a further aspect recovery scenarios are studied. Concrete numerical studies for optimal solutions are determined for cod and herring. Care is taken to take into account all possible realistic parameters but also to keep the model tractable for other species.

This project investigates the relationship between social interactions among marketparticipants and the emergence of cyclical trends in stock markets. Two models aredeveloped to capture the interaction dynamics among individuals within the market:a basic model and a model with a price mechanism. By conducting numerical examplesand systematic simulations, we examine the behavior of these models. Ourfindings reveal that the basic model does not generate cyclical trends in the stockmarket. However, the model incorporating the price mechanism demonstrates theability to create such trends.

An abundant Paddlefish Polyodon spathula population exists in a 0.8 ha pool below a water control structure at Sam D. Hamilton Noxubee National Wildlife Refuge, Mississippi. Managers were concerned that regulated flows from the structure were causing an ecological trap if Paddlefish were being attracted from the larger river downstream during the spawning period, but conditions were not suitable to facilitate reproduction. Between February 2016 to April 2018, 117 Paddlefish were identified and daily abundance was estimated between 18 and 75 fish. Telemetry study of 59 fish suggests a mixed population structure where some remain in the pool year-round and other emigrate seasonally, cued by rising spring discharge and water temperature. Reproduction was not documented which suggests a critical component needed for spawning may be missing, at least during this study. Therefore, given the need to remove Paddlefish from the pool, translocation and flow releases may be effective management strategies.

This thesis extends an existing cell population modelling framework to investigate two different hypotheses for what drives the phenomenon of cell sorting, which is the spontaneous self-reorganization of cell populations. This behaviour cause cells to find their way back into their original configuration after they have been scrambled. Original tissue function may also be regained. The modelling framework is called discrete Laplacian cell mechanics (DLCM), and models cell movement on a lattice as a result of pressure differences. The first hypothesis suggests that cells exhibit type-specific adhesion properties which cause cells of the same type to adhere more to each other than to cells of a different kind. The other, more recent, hypothesis explains cell sorting behaviour as a consequence of interfacial tension, where cells of different types exhibit larger tension between them compared to cells of the same type. Adhesion is implemented as a passive force between cells of the same type, which counteract the pressure-driven events, while interfacial tension is implemented as pressure sources arising due to contact with cells of a different type. This thesis investigates whether these additions on the scale of individual cells can be sufficient to induce sorting behaviour on the cell population scale. Subsequently the suitability of implementing these effects in the DLCM framework can be evaluated. Starting from a scrambled cell configuration of two types, the results show that differential adhesion can result in the cell population sorting into smaller clusters, with the addition of Brownian motion improving the sorting ability significantly. Differential interfacial tension as it is implemented here demonstrates the effect of dissociation between cells of different type, but this is not sufficient to achieve sorting. The behaviour can be likened to a form of localized Brownian motion where more unsorted areas are prone to more movement events. Therefore, differential tension is not deemed suitable within the DLCM framework on its own. The cohesive effect of differential adhesion together with the dissociative effect of differential interfacial tension proved to work well together, acheiving a high degree of sorting both overall and compared to the case of only differential adhesion with some Brownian motion. Full separation into one distinct cell mass for each cell type present could not be achieved.

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This thesis consists of four papers, three papers about modelling animal populations and one paper about an area integral estimate for solutions of partial differential equations on non-smooth domains. The papers are: I. Å. Brännström, Single species population models from first principles. II. Å. Brännström and D. J. T. Sumpter, Stochastic analogues of deterministic single species population models. III. Å. Brännström and D. J. T. Sumpter, Coupled map lattice approximations for spatially explicit individual-based models of ecology. IV. Å. Brännström, An area integral estimate for higher order parabolic equations. In the first paper we derive deterministic discrete single species population models with first order feedback, such as the Hassell and Beverton-Holt model, from first principles. The derivations build on the site based method of Sumpter & Broomhead (2001) and Johansson & Sumpter (2003). A three parameter generalisation of the Beverton-Holtmodel is also derived, and one of the parameters is shown to correspond directly to the underlying distribution of individuals. The second paper is about constructing stochastic population models that incorporate a given deterministic skeleton. Using the Ricker model as an example, we construct several stochastic analogues and fit them to data using the method of maximum likelihood. The results show that an accurate stochastic population model is most important when the dynamics are periodic or chaotic, and that the two most common ways of constructing stochastic analogues, using additive normally distributed noise or multiplicative lognormally distributed noise, give models that fit the data well. The latter is also motivated on theoretical grounds. In the third paper we approximate a spatially explicit individual-based model with a stochastic coupledmap lattice. The approximation effectively disentangles the deterministic and stochastic components of the model. Based on this approximation we argue that the stable population dynamics seen for short dispersal ranges is a consequence of increased stochasticity from local interactions and dispersal. Finally, the fourth paper contains a proof that for solutions of higher order real homogeneous constant coefficient parabolic operators on Lipschitz cylinders, the area integral dominates the maximal function in the L2-norm.

Introduction: Cigarette smoking is associated with lung cancer, cardiovascular disease, and chronic respiratory conditions. It is responsible for high mortality and morbidity risk in the US population. Smokers find sudden quitting difficult and it is reported that a large number of unassisted quitting attempts are eventually unsuccessful. Electronic cigarette or e-cig is a novel battery-driven, nicotine delivery product, currently being used as a smoking cessation tool by current and former smokers. Since its resemblance to a conventional cigarette, and its non-combustible nature, e-cig use has risen exponentially in the last few years. To address such public health issues, the US FDA is working on formulating regulations to manufacture, market, and distribute e-cigs has called for research evidence on the long term use of e-cig use. Objective: The objective of this study was to develop and validate a Discrete Event Simulation model to simulate the electronic cigarette (e-cig) use behavior, and to estimate the long term e-cig use prevalence, in different groups of the US population. Methods: The model population was generated from analyzing the National Health Interview Survey data from 2011-2013. The population was categorized into current, recent former, late former and never smokers. Population birth rates and death rates were applied using the 2012 US Census Bureau data. Model parametrization, transition probabilities and e-cig related risks were obtained and applied using cross sectional survey and longitudinal e-cig studies done on US population. The model was run for the period of 15 years and e-cig use prevalence at the end of the simulation period was estimated. Each simulation was replicated 100 times using Monte Carlo simulation approach. Model validation was performed by the use of null and extreme input values (internal validation), examining programing codes (debugging), verification by tobacco science and system analysis experts (structural and technical validation), comparison of model’s first year results with CDC reports (external validation). Conclusion: Total projected e-cig prevalence in the US population at the end of simulation of period was found to be around 19%. The results showed a gradual reduction in the number of conventional cigarette smokers and an increase in the e-cig users over the simulation period. Highest e-cig users were old, male, white and had less than high school level education. Sensitivity analyses of various model parameters showed that the e-cig prevalence was most sensitive to the impact and timing of policy implementation. As a novel nicotine delivery system, e-cigs are rapidly gaining acceptance in the US and recent reports have shown an exponential rise in the popularity of e-cig among minors and young adults. Our research provides empirical evidence that can be used by the scientific community and regulatory bodies to formulate regulations for marketing and sales of e-cigs in various sections of the population, where the prevalence is expected to rise in future. Our study can also guide the policy makers to introduce relevant policies at specific time points when the e-cig use is expected to rise.