The goal of the present work was to analyze the Gompertz functions in order to describe the growth of hillbilly broilers Color Plume (CPK) subjected to different levels of digestible lysine. On the experiment were used 432 chickens from one day old, was divided into three stages: initial, growth and finale respectively corresponding to the ages from 1 to 21, from 22 to 77 days of age and from 78 to 84 days of age. During the initial phase, the birds were reared on shavings beds with access to water, food and a heating source until after 21 days of age, the birds had access to paddocks. After 21 days of age, the birds which were in terms of the medium weight were distributed randomly in four treatments of 27 birds, with four replicates each one and 10 birds per replication (5 males and 5 females) were chipped with microchip brand Animall TAG and dimensions 12x2 mm. Treatments were represented by different lysine levels (0.850, 0.970, 1.090; 1.210%) on the experimental diet. The diets used were based on corn and bran soybean, and were formulated to meet the nutritional requirements at each stage of creation, as suggested by Rostagno et al. (2011), except lysine. Diets and birds were weighed weekly to determine the performance (weight gain, feed intake and feed conversion). Gompertz curves adjusted showed good correlation coefficients above 90%, and good values of Durbin Watson test accurately, estimating the average weight gain, and the consumption ration of mixed batch of free range broiler chickens. After the tests comparison between Gompertz parameters of the model for each level of lysine was determined that: the only parameter that varied was A, since B = 0.0329 and C = 45.819 parameters presented not significant differences. At the level of digestible lysine 0.850%, A = 4870.3, 0.970% A = 4894.9, 4784.3 A = 1.09%, 1.21% A = 4740. Among them, the parameter A in 0.970% level, differed from the others, ie, showing a higher body weight at maturity. Based the performance results, it is recommended for free range broiler CPK chickens of both sexes, to use a level of digestible lysine up to 0.970% on the period 22-77 days of age. After comparison tests between the Gompertz parameters of the model for each lysine level was determined for female that: the only parameter that varied was A, since B = 0.0356 and C = 45.54 parameters presented not significant differences. In males it has been concluded that only a single equation may represent the growth in four levels of lysine. The lysine levels did not influence (P> 0.05) carcass variables, except the yield of abdominal fat, than for females improved quadratically up to the level of 1.064%, for males there was a quadratic effect. / O presente trabalho objetivou-se utilizar o modelo não linear de Gompertz para descrever o crescimento de frangos caipira da linhagem Colorcpk Plumé (CPK), de ambos os sexos e entre os sexos, submetidos a diferentes níveis de lisina digestível. Foram utilizados 432 frangos com um dia de idade, o experimento foi dividido em três fases: inicial, crescimento e final correspondendo respectivamente às idades de 1 a 21, 22 a 77 e 78 a 84 dias de idade. Na fase inicial as aves foram criadas sobre cama de maravalha com acesso a água, ração e fonte de aquecimento e uma única dieta. Aos 21 dias de idade, as aves em função do peso médio foram distribuídas num delineamento inteiramente casualizado, em quatro tratamentos com quatro repetições de 27 aves por unidade experimental, e dentre elas, 10 aves de cada repetição (5 machos e 5 fêmeas) foram chipadas com microchips da marca Animall TAG e dimensões de 12x2 mm. Os tratamentos foram representados pelos diferentes níveis de lisina digestível (0,850; 0,970; 1,090; 1,210%) na dieta experimental na fase de crescimento. As dietas foram à base de milho e farelo de soja e formuladas para atender as exigências nutricionais em cada fase de criação, conforme sugerido de Rostagno et al. (2011), exceto lisina digestível. As rações e as aves foram pesadas semanalmente para determinação do desempenho (ganho de peso, consumo de ração e conversão alimentar), as aves chipadas também eram pesadas individualmente a cada 7 dias. Aos 84 dias de idade as aves chipadas foram abatidas, totalizando 160 aves. As características de carcaça avaliadas foram calculadas em relação ao peso corporal após jejum. As curvas de Gompertz ajustadas apresentaram bons coeficientes de determinação, acima de 90%, e bons valores do teste de Durbin Watson, estimando com boa acurácia o peso médio, ganho de peso e consumo de ração dos frangos caipira. Após os testes de comparação entre os parâmetros do modelo de Gompertz para cada nível de lisina para o lote misto, determinou-se que somente o parâmetro A deveria variar, pois os parâmetros B=0,0329 e C=45,819 não apresentaram diferenças significativas. No nível de lisina digestível 0,850% A=4870,3; 0,970% A=4894,9; 1,09% A=4784,3; 1,21% A=4740; dentre eles o parâmetro A no nível 0,970% diferiu dos demais, ou seja, maior peso corporal a maturidade. Com base nos resultados de desempenho, recomenda-se, para aves caipiras da linhagem CPK de ambos os sexos o nível de lisina digestível na ração de 0,970% no período de 22 a 77 dias de idade. Após os testes de comparação entre os parâmetros do modelo de Gompertz para cada nível de lisina, determinou-se para fêmeas que somente o parâmetro A deveria variar, pois os parâmetros B=0,0356 e C=42,54 não apresentaram diferenças significativas. Para os machos concluiu-se que apenas uma única equação pode representar o crescimento nos quatro níveis de lisina digestível. Os níveis de lisina digestível não influenciaram (P>0,05) as variáveis de carcaça, exceto o rendimento de gordura abdominal, que para as fêmeas melhorou de forma quadrática até o nível estimado de 1,064%.

Studier av livslängder och dödssannolikheter är avgörande för livförsäkring. Betalningar gällande livförsäkringar är helt beroende av om en individ lever eller ej, eller befinner sig i olika hälsotillstånd. För att kunna prissätta premier korrekt och avsätta reserver är det därför av stort intresse att modellera livslängden på ett så korrekt sätt som möjligt. Försäkringsbranschen använder idag historiskt beprövade och välfungerande modeller som går så långt bak i tiden som 200 år. Det finns modeller ännu längre bak i tiden, men de modeller som används idag är främst Gompertz (1826), Makeham (1860) och Lee-Carter (1992). Även om dessa modeller presterar bra är det alltid nödvändigt att undersöka om det kan finnas alternativa modeller som modellerar dödligheten bättre. I detta examensarbete tillämpas affina korträntemodeller för modellering av dödlighetsintensiteten som ligger till grund för flertalet intressanta aktuariella storheter. Då dessa modeller introducerar stokastisk dödlighet kan osäkerheten och beroendet över tid därmed beskrivas. De korträntemodeller som undersöks i arbetet och som är vanligt förekommande inom den finansiella teorin; är Ornstein-Uhlenbeck, Feller och Hull-White. Dessa modeller jämförs sedan mot varandra vad gäller modellerad dödlighetsintensitet samt förväntad återstående livslängd och ettårig dödssannolikhet. En aspekt av stokastisk dödlighetsmodellering som ej återfinns i befintlig litteratur men som undersöks i detta examensarbete är modellering av dödlighet över tid då detta är en av de mest väsentliga aspekterna inom det livförsäkringsmatematiska arbetet. Till sist i valideringssyfte utvärderas samtliga korträntemodeller genom back-testing. Den andra huvudsakliga delen av arbetet består i att generera resultat för samma storheter som ovan baserat på DUS-metoden för att på så sätt jämföra en kommersiell metod mot en mer teoretisk mindre beprövad sådan. Resultaten visar på en stor potential hos flera av korträntemodellerna kontra DUS både vad gäller modellering över åldrar och kalenderår. Däremot är inte resultaten helt felfria för enstaka kalenderår där stora spikar uppstår på grund av parametermässig felanpassning. Modelleringen av korträntemodellerna över tid var över förväntan då modellerna inte är konstruerade för att fånga avtagande trender. Detta är något som kan betraktas som en stor flexibilitet hos korträntemodellerna då de står sig väl mot Lee-Cartermodellen som används i DUS, både vad gäller ålders- och tidsmodellering av dödlighet. / Studies of life expectancy and death probabilities are crucial for life insurance. Payments for life insurance are completely dependent on whether an individual is alive or not, or is in various health conditions. In order to be able to price premiums correctly and set aside reserves, it is therefore of great importance to model life expectancy in the most accurate way possible. The insurance industry today uses historically proven well-functioning models that go as far back in time as 200 years. There are models even further back in time, but the models used today are mainly Gompertz (1826), Makeham (1860) and Lee-Carter (1992). Although these models perform well, it is always necessary to investigate whether there may be alternative models that model mortality better. In this thesis, affine short-term interest rate models are applied for modeling the force of mortality that forms the basis for most interesting actuarial variables. As these models introduce stochastic mortality, the uncertainty and dependence over time can thus be described. The three short-term interest rate models examined in this project, which are common in financial theory; are Ornstein-Uhlenbeck, Feller and Hull-White. These models are then compared against each other in terms of the modeled force of mortality as well as the expected remaining life expectancy and the one-year probability of death. One aspect of stochastic mortality modeling that is not found in the existing literature but which is examined in this thesis is the modeling of mortality over time as this is one of the most important aspects in the life insurance mathematical industry. Finally, for validation purposes, all short-term interest rate models are evaluated using back-testing. The second main part of the work consists of generating results for the same quantities as above based on the DUS method in order to compare a commercial method with more theoretical and less approved ones. The results show a great potential in several of the short-term interest rate models versus DUS both in terms of modeling over ages and calendar years. However, the results are not completely impeccable for individual calendar years where large spikes occur due to inaccurate parameter calibration. The satisfactory modeling of the short-term interest rate models over time was above the expectations as the models are not designed to capture decreasing trends. This is something that can be considered a great flexibility of the short-term interest rate models as they are more or less as accurate as the Lee-Carter model used in DUS, both in terms of age and time modeling of mortality.

Because of its specific qualities, lithium is a key element for making lithium-ion batteries, which is of great relevance since the battery sector is the fastest growing user of lithium to date and with the transition towards a fully electrified transportation sector due to the current climate agenda and an even bigger increase in demand for lithium is anticipated for the coming decades. The two main sources for lithium to date are mines and brines, with brine water composing around 87 % of the world’s lithium reserves in terms of contained lithium. Salars, dominantly large salt flats in South America, provide the type of brine water that has a high concentration of lithium. These salars occur primarily in Argentina, Bolivia and Chile (the ABC-triangle), where solid lithium-compounds occur dominantly as evaporate minerals such as lithium carbonates. Lithium from salars is recovered with natural evaporation and salar-derived lithium-salts generally have a higher purity than lithium derived from pegmatite mining. However, the process of lithium extraction from salars much slower than mining lithium from pegmatite, although large amounts of energy are required and large amounts of waste rock material is produced in the process. Lithium also occurs in geothermal waters and in seawater, but at far lower concentrations then in pegmatites or salar brines, which thus requires large processing efforts to be economically attractive.The purpose of this study is to provide a modern update based on the lithium production data in metric tons from recent years and interpolate the lithium production for the remaining part of the century. Then I will assess the difference between the projections from the study by Vikström et al. 2014, that were previously used to estimate possible future production rates annually, and current production and availability estimates. To achieve this, the present study will compare the different sources of lithium all over the world, mainly hard rock and brine, with a focus on the geological aspects and the EU from a financial aspect. This thesis thus aims to assess future production trends based on most recent data on geological availability, which will be retrieved from The United States Geological Survey, and earlier forecasts with the purpose to investigate if and how previous forecasts will need to change in respect to various resource availability. The production data ranging from 1900 to 2010 from the previous study by Vikström et al., on which the previous forecast production is based on, is updated with data from 2011 to 2019. With the addition of the more recent data, a similar forecast projection will be made using the same mathematical models (logistic, Gompertz and Richards).

台灣地區的人口隨著生育率及死亡率的雙重下降而呈現快速老化，其中生育率的降低影響尤為顯著。民國50年時，台灣平均每位婦女生育5.58個小孩，到了民國70年卻只生育1.67個小孩，去年（民國90年）生育率更創歷年新低，只有1.4。死亡率的下降可由平均壽命的延長看出，民國75年時男性為70.97歲，女性為75.88歲；到了民國90年，男性延長到72.75歲，女性延長到78.49歲。由於生育率的變化幅度高於死亡率，對人口結構的影響較大，因此本文分成兩個部份，主要在研究台灣地區15至49歲婦女生育率的變化趨勢，再將研究結果用於台灣地區未來人口總數及其結構的預測。 　　本研究第一部分是生育率的研究，引進Gamma函數、Gompertz函數、Lee-Carter法三種模型及單一年齡組個別估計法，以民國40年至84年（西元1951年至1995年）的資料為基礎，民國85年至89年（西元1996年至2000年）資料為檢測樣本，比較模型的優劣，尋求較適合台灣地區生育率的模型，再以最合適的模型預測民國91年至140年（西元2002年至2051年）的生育率。第二部分是人口推估，採用人口變動要素合成方法（Cohort Component Projection Method）推估台灣地區未來50年的人口總數及其結構，其中生育率採用上述最適合台灣地區的模型、死亡率則引進國外知名的Lee-Carter法及SOA法（Society of Actuaries），探討人口結構，並與人力規劃處的結果比較之。 / Both the fertility rate and mortality rate have been experiencing dramatic decreases in recent years. As a result, the population aging has become one of the major concerns in Taiwan area, and the proportion of the elderly (age 65 and over) increases promptly from 2.6% in 1965 to 8.8% in 2001. The decrease of fertility rate is especially significant. For example, the total fertility rate was 5.58 in 1961, and then decreases dramatically to 1.67 in 1981 (1.4 in 2001), a reduction of almost 70% within 20 years. 　　The goal of this paper is to study the population aging in Taiwan area, in particular, the fertility pattern. The first part of this paper is to explore the fertility models and decide which model is the most suitable based on age-fertility fertility rates in Taiwan. The models considered are Gamma function, Gompertz function, Lee-Carter method and individual group estimation. We use the data from 1951 to 1995 as pilot data and 1996 to 2000 as test data to judge which model fit well. The second part of this study is to project the Taiwan population for the next 50 years, i.e. 2002-2051. The projection method used is Cohort Component Projection method, assuming the population in Taiwan area is closed. We also compare our projection result to that by Council for Economic Planning and Development, the Executive Yuan of the Republic of China.