In this dissertation, we are interested in the dynamics of spatially distributed populations. In particular, we focus on persistence conditions and minimal traveling periodic wave speeds for stage-structured populations in heterogeneous landscapes. The model includes structured populations of two age groups, juveniles and adults, in patchy landscapes. First, we present a stage-structured population model, where we divide the population into pre-reproductive and reproductive stages. We assume that all parameters of the two age groups are piecewise constant functions in space. We derive explicit formulas for population persistence in a single-patch landscape and in heterogeneous habitats. We find the critical size of a single patch surrounded by a non-lethal matrix habitat. We derive the dispersion relation for the juveniles-adults model in homogeneous and heterogeneous landscapes. We illustrate our results by comparing the structured population model with an appropriately scaled unstructured model. We find that a long pre-reproductive state typically increases habitat requirements for persistence and decreases spatial spread rates, but we also identify scenarios in which a population with intermediate maturation rate spreads fastest. We apply sensitivity and elasticity formulas to the critical size of a single-patch landscape and to the minimal traveling wave speed in a homogeneous landscape. Secondly, we use asymptotic techniques to find an explicit formula for the traveling periodic wave speed and to calculate the spread rates for structured populations in heterogeneous landscapes. We illustrate the power of the homogenization method by comparing the dispersion relation and the resulting minimal wave speeds for the approximation and the exact expression. We find an excellent agreement between the fully heterogeneous speed and the homogenized speed, even though the landscape period is on the same order as the diffusion coefficients and not as small as the formal derivation requires. We also generalize this work to the case of structured populations of n age groups. Lastly, we use a finite difference method to explore the numerical solutions for the juveniles-adults model. We compare numerical solutions to analytic solutions and explore population dynamics in non-linear models, where the numerical solution for the time-dependent problem converges to a steady state. We apply our theory to study various aspects of marine protected areas (MPAs). We develop a model of two age groups, juveniles and adults, in which only adults can be harvested and only outside MPAs, and recruitment is density dependent and local inside MPAs and fishing grounds. We include diffusion coefficients in density matching conditions at interfaces between MPAs and fishing grounds, and examine the effect of fish mobility and bias movement on yield and fish abundance. We find that when the bias towards MPAs is strong or the difference in diffusion coefficients is large enough, the relative density of adults inside versus outside MPAs increases with adult mobility. This observation agrees with findings from empirical studies.

Abstract in English / In this dissertation we apply linear evolution equations to the Newtonian derivative, Caputo time fractional derivative and $-time fractional derivative. It is notable that the most utilized fractional order derivatives for modelling true life challenges are Riemann- Liouville and Caputo fractional derivatives, however these fractional derivatives have the same weakness of not satisfying the chain rule, which is one of the most important elements of the match asymptotic method [2, 3, 16]. Furthermore the classical bounded perturbation theorem associated with Riemann-Liouville and Caputo fractional derivatives has con rmed not to be in general truthful for these models, particularly for solution operators of evolution systems of a derivative with fractional parameter ' that is less than one (0 < ' < 1) [29]. To solve this problem, we introduce the derivative with new parameter, which is de ned as a local derivative but has a fractional order called $-derivative and apply this derivative to linear evolution equation and to support what we have done in the theory, we utilize application to population dynamics and we provide the numerical simulations for particular cases. / Mathematical Sciences / M.Sc. (Applied Mathematics)

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New analytical and technological tools have the potential to yield unprecedented insights into the life histories of migratory species. I used Bayesian population models and Global Positioning System-acceleration tracking devices to understand the demographic mechanism and likely drivers underpinning the Greenland White-fronted Goose (Anser albifrons flavirostris) population decline. I used a 27-year capture-mark-recapture dataset from the main wintering site for these birds (Wexford, Ireland) to construct multistate models that estimated age- and sex-specific survival and movement probabilities and found no sex-bias in emigration or ‘remigration’ rates (chapter 2). These formed the foundation for an integrated population model, which included population size and productivity data to assess source-sink dynamics through estimation of age-, site-, and year-specific survival and movement probabilities, the results of which suggest that Wexford is a large sink and that a reduction in productivity (measured as recruitment rate) is the proximate demographic mechanism behind the population decline (chapter 3). Low productivity may be due to environmental conditions on breeding areas in west Greenland, whereby birds bred at youngest ages when conditions were favourable during adulthood and the breeding year (chapter 4), and possibly mediated by links with the social system, as birds remained with parents into adulthood, forfeiting immediate reproductive success, although a cost-benefit model showed the ‘leave’ strategy was marginally favoured over the ‘stay’ strategy at all ages (chapter 5). Foraging during spring does not appear to limit breeding, as breeding and non-breeding birds did not differ in their proportion of time feeding or energy expenditure (chapter 6). Two successful breeding birds were the only tagged individuals (of 15) to even attempt to nest, suggesting low breeding propensity has contributed to low productivity. Although birds wintering in Ireland migrated further to breeding areas than those wintering in Scotland, there were no differences in feeding between groups during spring migration (chapter 7). These findings suggest that Greenland White-fronted Geese are not limited until arrival on breeding areas and the increasingly poor environmental conditions there (chapter 8). More broadly, these findings demonstrate the application of novel tools to diagnose the cause of population decline.

Presence of coastal aquaculture activities in marine landscapes is growing. However, there is insufficient knowledge on the subsequent ecological interactions between these activities and marine fish communities. The overall aim of this thesis was to evaluate the direct and indirect ecological effects of aquaculture activities on marine fish communities in Scotland. A combination of empirical and modelling approaches was employed to collect evidence of how aquaculture activities affect marine fish communities at the individual, population and ecosystem levels around coastal sea cages. The two fish farms evaluated in this research provided the wild fish sampled near the sea cages with a habitat rich in food resources which is reflected in an overall better biological condition. Results of the stomach content analysis indicated that mackerel (Scomber scombrus), whiting (Merlangius merlangus) and saithe (Pollachius virens) sampled near sea cages consumed wasted feed which was also reflected in their modified FA profiles. The overall effects of the two fish farms were more pronounced in young whiting and saithe than in mixed aged mackerel sampled near the sea cages. The phase space modelling approach indicated that the overall potential for fish farms to act at the extremes as either population sources (a habitat that is rich in resources and leads to an overall improved fitness) or ecological traps (a habitat that appears to be rich in resources but is not and leads to an overall poor fitness) are higher for juvenile whiting than for mackerel. Based on the empirical evidence and literature the two fish farms are more likely to be a population source for wild fishes. Using an ecosystem modelling approach indicated that fish farming impacts the food web in a sea loch via nutrient loading. Mussel farming relies on the natural food resources and has the potential to affect the food web in a sea loch via competing with zooplankton for resources which can affect higher trophic levels. The presence of both activities can balance the overall impact in a sea loch as compared to the impact induced if each of these activities were present on their own. Both activities have the potential do induce direct and indirect effects on the wild fish and the entire sea loch system. The results of this PhD identified several gaps in data and thus could be used to improve future sampling designs. It is important to evaluate the cumulative effect of the presence of aquaculture activities in terms of nutrient loading and physical structure in the environment. Using a combination of empirical and modelling approaches is recommended to gain further insight into the ecological impacts of aquaculture activities on wild fish communities. Results of this PhD study could lead to more informed decisions in managing the coastal aquaculture activities. Establishing coastal fish farms as aquatic sanctuaries can be of an advantage to increase fish production and conserve species that are endangered provided that no commercial and recreational fishing is allowed nearby. It would be useful to have long term monitoring of the fish stocks around the cages and if there is any production at the regional level. Additionally, information on behaviour, migration patterns should be collected to understand the impacts of aquaculture activities on fish stocks. From an aquaculture perspective, ecologically engineered fish farms in addition to careful site selection in new aquaculture developments may improve nutrient loading into the ecosystem.

During the summers of 2007-2009, I studied the population growth and reproductive and foraging ecology of the Razorbill (Alca torda) at Matinicus Rock (MR), Maine. This medium-sized marine bird in the family Alcidae (auks) was extirpated from the Gulf of Maine in the late 19th century by hunting, collecting, and colony disturbance. Following legislation protecting seabirds and their nesting habitats, the Razorbill has recolonized probable former nesting habitat in the Gulf of Maine during the past several decades. Six small colonies comprise the Maine population, which is listed as threatened and forms the southern extension of the species breeding distribution. In Chapter 1, I present a population model of the MR breeding colony, based on studies of population growth and reproductive success, and supplemented with previously collected data from the National Audubon Society Seabird Restoration Program (Project Puffin), with whom I collaborated. I also describe chick diet (supplemented with Project Puffin data) and draw connections between diet and reproductive success. I found that reproductive success was too low to account for the observed population growth rate, and conclude that the colony is a sink population supported by substantial immigration. Because annual fledging success was positively associated with prey quality, I suggest that substandard chick diet may contribute to the sink population dynamic via diet-driven depressed fledging success. In Chapter 2, I report on the foraging behavior of chick-rearing Razorbills fitted with bird-borne data-loggers at MR in 2008-2009. I describe diving behavior including depth, duration, and profile shape of dives, as well as diel patterns. Diving activity was restricted to daylight hours, and dives were shallowest and most frequent in the evening. Though generally similar to diving behavior reported at four European and Canadian colonies, Razorbills at MR performed three times as many dives per day as at the Gannet Islands, Labrador, and the mean dive depth was greater than three of four previous studies. Deeper and more frequent dives may indicate higher foraging effort and lower prey availability. Reproductive success would suffer if parents cannot buffer chicks against the effects of low prey availability through increased foraging effort or other behavioral modifications. Together, the pieces of our research indicate that prey availability may be negatively affecting reproduction and population growth at MR. Rapid colony growth cannot be explained by local reproductive success, and is likely the result of substantial immigration from other colonies. Chick diet is varied and includes multiple high-quality forage fish species, yet chicks also consume poor-quality prey (larval fish and euphausiids) that may signal periods of very poor prey availability. Frequency and depth of dives made by chick-provisioning adults are also suggestive of parents allocating extra effort to foraging, relative to other colonies.

Integrated data analyses are becoming increasingly popular in studies of wild animal populations where two or more separate sources of data contain information about common parameters. Here we develop an integrated population model using abundance and demographic data from a study of common guillemots (Uria aalge) on the Isle of May, southeast Scotland. A state-space model for the count data is supplemented by three demographic time series (productivity and two mark-recapture-recovery (MRR)), enabling the estimation of prebreeder emigration rate - a parameter for which there is no direct observational data, and which is unidentifiable in the separate analysis of MRR data. A Bayesian approach using MCMC provides a flexible and powerful analysis framework. This model is extended to provide predictions of future population trajectories. Adopting random effects models for the survival and productivity parameters, we implement the MCMC algorithm to obtain a posterior sample of the underlying process means and variances (and population sizes) within the study period. Given this sample, we predict future demographic parameters, which in turn allows us to predict future population sizes and obtain the corresponding posterior distribution. Under the assumption that recent, unfavourable conditions persist in the future, we obtain a posterior probability of 70% that there is a population decline of >25% over a 10-year period. Lastly, using MRR data we test for spatial, temporal and age-related correlations in guillemot survival among three widely separated Scottish colonies that have varying overlap in nonbreeding distribution. We show that survival is highly correlated over time for colonies/age classes sharing wintering areas, and essentially uncorrelated for those with separate wintering areas. These results strongly suggest that one or more aspects of winter environment are responsible for spatiotemporal variation in survival of British guillemots, and provide insight into the factors driving multi-population dynamics of the species.

I detta tvärvetenskapliga arbete studeras från den matematiska sidan tre klassiska populationsmodeller: Malthus tillväxtmodell, Verhulsts logistiska modell och Lotka-Volterras jägarebytesmodell. De klassiska modellerna jämförs med stokastiska. De stokastiska modeller som studeras är födelsedödsprocesser och deras diffusionsapproximation. Jämförelse görs med medelvärdesbildade simuleringar. Det krävs många simuleringar för att kunna genomföra jämförelserna. Dessa simuleringar måste utföras i datormiljö och det är här den datalogiska aspekten av arbetet kommer in. Modellerna och deras resultathantering har implementerats i både MatLab och i C, för att kunna möjliggöra en undersökning om skillnaderna i tidsåtgången mellan de båda språken, under genomförandet av ovan nämnda jämförelser. Försök till tidsoptimering utförs och även användarvänligheten under implementeringen av de matematiska problemen i de båda språken behandlas. Följande matematiska slutsatser har dragits, att de medelvärdesbildade lösningarna inte alltid sammanfaller med de klassiska modellerna när de simuleras på stora tidsintervall. I den logistiska modellen samt i Lotka-Volterras modell dör förr eller senare de stokastiska simuleringarna ut när tiden går mot oändligheten, medan deras deterministiska representation lever vidare. I den exponentiella modellen sammanfaller medelvärdet av de stokastiska simuleringarna med den deterministiska lösningen, dock blir spridningen stor för de stokastiska simuleringarna när de utförs på stora tidsintervall. Datalogiska slutsatser som har dragits är att när det kommer till att implementera få modeller, samt resultatbearbetning av dessa, som ska användas upprepade gånger, är C det bäst lämpade språket då det visat sig vara betydligt snabbare under exekvering än vad MatLab är. Dock måste hänsyn tas till alla de svårigheter som implementeringen i C drar med sig. Dessa svårigheter kan till stor del undvikas om implementeringen istället sker i MatLab, då det därmed finns tillgång till en uppsjö av väl lämpade funktioner och färdiga matematiska lösningar. / In this interdisciplinary study, three classic population models will be studied from a mathematical view: Malthus’ growth, Verhulst’s logistic model and Lotka-Volterra’s model for hunter and prey. The classic models are being compared to the stochastic ones. The stochastic models studied are the birthdeath processes and their diffusion approximation. Comparisons are made by averaging simulations. It requires numerous simulations to carry out the comparisons. The simulations must be carried out on a computer and this is where the computer science emerges to the project. The models, along with the handling of the results, have been implemented in both MatLab and in C in order to allow a comparison between the two languages whilst executing the above mentioned study. Attempts to time optimization and an evaluation concerning the user-friendliness regarding the implementation of mathematical problems will be performed. Mathematic conclusions, which have been drawn, are that the averaging solutions do not always coincide with the traditional models when they are being simulated over large time. In the logistic model and in Lotka-Volterra’s model the stochastic simulations will sooner or later die when the time is moving towards infinity, whilst their deterministic representation keeps on living. In the exponential model, the mean values of the stochastic simulations and of the deterministic solution coincide. There is, however, a large spread for the stochastic simulations when they are carried out over a large time. Computer scientific conclusions drawn from the study includes that when it comes to implementing a few models, along with the handling of the results, to be used repeatedly, C is the most appropriate language as it proved to be significantly faster during execution. However, all of the difficulties during the implementation of mathematical problems in C must be kept in mind. These difficulties can be avoided if the implementation instead takes place in MatLab, where a numerous of mathematical functions and solutions will be available.

<p>I detta tvärvetenskapliga arbete studeras från den matematiska sidan tre klassiska populationsmodeller: Malthus tillväxtmodell, Verhulsts logistiska modell och Lotka-Volterras jägarebytesmodell. De klassiska modellerna jämförs med stokastiska. De stokastiska modeller som studeras är födelsedödsprocesser och deras diffusionsapproximation. Jämförelse görs med medelvärdesbildade simuleringar.</p><p>Det krävs många simuleringar för att kunna genomföra jämförelserna. Dessa simuleringar måste utföras i datormiljö och det är här den datalogiska aspekten av arbetet kommer in. Modellerna och deras resultathantering har implementerats i både MatLab och i C, för att kunna möjliggöra en undersökning om skillnaderna i tidsåtgången mellan de båda språken, under genomförandet av ovan nämnda jämförelser. Försök till tidsoptimering utförs och även användarvänligheten under implementeringen av de matematiska problemen i de båda språken behandlas.</p><p>Följande matematiska slutsatser har dragits, att de medelvärdesbildade lösningarna inte alltid sammanfaller med de klassiska modellerna när de simuleras på stora tidsintervall. I den logistiska modellen samt i Lotka-Volterras modell dör förr eller senare de stokastiska simuleringarna ut när tiden går mot oändligheten, medan deras deterministiska representation lever vidare. I den exponentiella modellen sammanfaller medelvärdet av de stokastiska simuleringarna med den deterministiska lösningen, dock blir spridningen stor för de stokastiska simuleringarna när de utförs på stora tidsintervall.</p><p>Datalogiska slutsatser som har dragits är att när det kommer till att implementera få modeller, samt resultatbearbetning av dessa, som ska användas upprepade gånger, är C det bäst lämpade språket då det visat sig vara betydligt snabbare under exekvering än vad MatLab är. Dock måste hänsyn tas till alla de svårigheter som implementeringen i C drar med sig. Dessa svårigheter kan till stor del undvikas om implementeringen istället sker i MatLab, då det därmed finns tillgång till en uppsjö av väl lämpade funktioner och färdiga matematiska lösningar.</p> / <p>In this interdisciplinary study, three classic population models will be studied from a mathematical view: Malthus’ growth, Verhulst’s logistic model and Lotka-Volterra’s model for hunter and prey. The classic models are being compared to the stochastic ones. The stochastic models studied are the birthdeath processes and their diffusion approximation. Comparisons are made by averaging simulations.</p><p>It requires numerous simulations to carry out the comparisons. The simulations must be carried out on a computer and this is where the computer science emerges to the project. The models, along with the handling of the results, have been implemented in both Mat- Lab and in C in order to allow a comparison between the two languages whilst executing the above mentioned study. Attempts to time optimization and an evaluation concerning the user-friendliness regarding the implementation of mathematical problems will be performed.</p><p>Mathematic conclusions, which have been drawn, are that the averaging solutions do not always coincide with the traditional models when they are being simulated over large time. In the logistic model and in Lotka-Volterra’s model the stochastic simulations will sooner or later die when the time is moving towards infinity, whilst their deterministic representation keeps on living. In the exponential model, the mean values of the stochastic simulations and of the deterministic solution coincide. There is, however, a large spread for the stochastic simulations when they are carried out over a large time.</p><p>Computer scientific conclusions drawn from the study includes that when it comes to implementing a few models, along with the handling of the results, to be used repeatedly, C is the most appropriate language as it proved to be significantly faster during execution. However, all of the difficulties during the implementation of mathematical problems in C must be kept in mind. These difficulties can be avoided if the implementation instead takes place in MatLab, where a numerous of mathematical functions and solutions will be available.</p>