abstract: There have been many studies on the dynamics of infectious diseases considering the age structure of the population. This study analyzes the dynamics when the population is stratified by size. This kind of models are useful in the spread of a disease in fisheries where size matters, for microorganism populations or even human diseases that are driven by weight. A simple size structured SIR model is introduced for which a threshold condition, R0, equilibria and stability are established in special cases. Hethcote's approach is used to derive, from first principles, a parallel ODE size-structure system involving n-size classes.The specific case of n = 2 is partially analyzed. Constant effort harvesting is added to this model with the purpose of exploring the role of controls and harvesting. Different harvesting policies are proposed and analyzed through simulations. / Dissertation/Thesis / Ph.D. Applied Mathematics for the Life and Social Sciences 2012

The effects of habitat complexity and predation risk on community assembly in small standing waters are reviewed. Both lethal and nonlethal predator effects are discussed. Emphasis is put on the role of omnivorous predators which do not fit into the standard food chain model. The review is complemented by a manuscript intended for publication in Freshwater Biology. It reports the results of a mesocosm experiment focusing on the effects of both lethal and nonlethal predation risk, induced by the invasive crayfish species Orconectes limosus, and habitat complexity on the abundance and biomass of macroinvertebrates in a naturally assembled community.

The Caledonian pinewoods are a habitat of crucial environmental and cultural importance, and the sole home of many rare species. However, they have seen steady decline in recent centuries, through the establishment of hunting estates and forestry plantations. A recent trend in management is the attempted transformation of existing plantations (dense communities with a regular spatial structure and low variance in size and age) towards a state mimicking the perceived natural condition, which has a lower density, irregular spatial pattern, high variance in size and age. This presents a problem for traditional forestry practices, which were conceived primarily with “even-aged” plantation populations in mind. The shift towards management of an uneven-aged structure requires a more in-depth consideration of individual trees’ lifecycles and their effect upon long-term population dynamics. In recent years, great advances in computational and mathematical models for spatially interacting populations have been made. However, certain complications have prevented them from being utilised to their full potential for the purposes of forest management. Forest communities are not only spatially structured; the size of each tree plays a role in its ability to acquire resources for growth and survival. Existing models of population dynamics are discussed, and their extension to incorporate both size- and spatially- structured interactions is presented. The key aspects of populations’ structural development are studied. Data from both plantation and semi-natural Scots Pine stands in Scotland allow parameterisation of a stochastic individual-based model, which in turn provides insights into the behaviour of real populations, and the importance of spatial effects and heterogeneity in individuals. A partial differential equation (moment) approximation to the stochastic model is presented. While this is analytically intractable, numerical integration and heuristic analysis of the equations enable clearer identification of the drivers of population structure. Many results are concordant with existing models of both qualitative forest stand development and theoretical dynamics of spatially-structured populations, while others are specific to joint size-space structure. This deeper understanding of the population dynamics allows robust recommendations for diverse uneven-aged stand management objectives to be made. Approaches to accelerating the transformation of plantation stands towards a “natural” state (using two key operations: thinning – removal of trees, and planting) are investigated. Finally, approaches to so-called “continuous cover forestry” – the practice of maintaining a quasi-natural state while also obtaining economic value from a forest – are also considered. In both cases, the model’s simplicity enables clearer conclusions than would be possible using other approaches.

La structure de taille et la distribution spatiale du zooplancton ont été étudiées dans deux écosystèmes : une baie de la côte Ouest de la péninsule antarctique et le golfe du Lion en mer méditerranée occidentale. L'acquisition des spectres a été permise par l'utilisation de deux capteurs optiques : le ZooScan / ZooProcess et le Laser Optical Plankton Counter (LOPC). L'impact de différents types de forçages sur les spectres de taille des communautés zooplanctoniques a été mis en évidence dans les deux écosystèmes côtiers, notamment grâce aux caractéristiques des spectres de biomasse normalisée. Le long de la péninsule Antarctique, la fin de l'automne est une période charnière à tous les niveaux trophiques avec la baisse de la production primaire et l'agrégation du krill (Euphausia superba) dans les baies continentales. Les données ADCP ont permis de localiser dans une des baies le plus grand banc de krill échantillonné depuis 20 ans. L'étude du comportement alimentaire du krill en réponse à la baisse de la production primaire a montré l'impact du krill sur les spectres de taille des communautés mésozooplanctoniques, et en particulier sur les espèces de petites tailles. Un autre type de forçage a été mis en avant dans le golfe du Lion, où les structures physiques très variables induites par les apports du Rhône, le courant Liguro-Provençal et les vents influent directement sur la distribution spatiale du zooplancton. Des sous-régions ont été identifiées à partir de corrélations entre des paramètres physiques tels que la stratification de la colonne d'eau et des paramètres biologiques tels que la concentration en chl-a ou la pente des spectres de biomasse normalisée. / Zooplankton size structure and spatial distribution were investigated in a bay along the West Antarctic Peninsula (WAP) and in the Gulf of Lion in Western Mediterranean Sea. Zooplankton size spectra were determined with the use of two optical sensors, the Laser Optical Plankton Counter (LOPC) and the ZooScan / ZooProcess system. Using features of the biomass size spectrum is was possible to identify different forcing processes that affected zooplankton size spectrum and spatial distribution in the two ecosystems. In WAP, late fall is a transition time at all the trophic levels, with the decrease of primary production and the aggregation of Antarctic krill (Euphausia Superba) in continental bays. ADCP data permitted to find in a bay the largest Antarctic krill swarm reported in the last twenty years. Study of krill feeding behavior in response to the decrease of primary production showed impact of krill on mesozooplankton size spectrum and especially a decrease of the small size species abundance. A different type of forcing was found in the Gulf of Lion, where zooplankton spatial distribution was affected by physical structures such as the inputs by the Rhône river, the Liguro-Provençal current and winds. The correlation between physical and biological parameters e.g. water masses stratification, chl-a concentration, slope of the normalized biomass size spectrum (NBSS), allowed the identification of three different regions in the Gulf of Lion. These potential habitats show different zooplankton size structure, with strong NBSS slopes close to the coast and weaker NBSS slopes in the zone of the Rhône plume influence.

MSc., Faculty of Science, University of the Witwatersrand, 2011 / Little is known about the longevity of grassland suffrutex plants and the relationship between longevity and plant morphological structures and seedling growth strategies. The aim of this study was to determine the longevity of four grassland suffrutex plant species, namely Berkheya insignis, Callilepis laureola, Protea insignis and Tephrosia kraussiana. Seed viability, seedling growth, morphology, habitat and population structure and demography were also assessed and related to plant age. The rhizomes, seeds (if available) and canopies of B. insignis, C. laureola, P. simplex and T. kraussiana were removed from a site near Port Edward in the Pondoland region, KwaZulu-Natal, South Africa in April 2008. Seeds were available for P. simplex and T. kraussiana only. Seed germination and viability were tested in the field and laboratory using germination trials and tetrazolium tests. Greenhouse and field grown seedlings were used to monitor seedling growth and to record seedling morphology. The aerial and rhizome morphologies of adult plants excavated from the field were also recorded. The largest of these rhizomes were aged using radiocarbon dating. Attempts were made to develop morphological surrogates for plant age as no method currently exists. Plant density, demographies, number of inflorescences and various environmental variables of wild populations of B. insignis, C. laureola, P. simplex and T. kraussiana were sampled from the Red Desert Nature Reserve and the Umtamvuna Nature Reserve. Species morphologies varied, however important similarities suggest convergent evolution. These included a single vertical main stem rhizome with multiple side branches, early rhizome development in seedlings, and high root: shoot ratios. Seed viability was high in the laboratory but low seedling emergence was observed in the field. Seedling growth was rapid for T. kraussiana and slow for P. simplex. The population stage structure for all four species comprised primarily adults with few or no juveniles. Wild population plant height and main stem diameter followed normal distributions. Number of stems, basal area, canopy area and the number of floral structures had distributions favouring the smaller size classes. Soil P, N, K and organic carbon were important soil nutrients in a PCA analysis of the habitats of the four species. Radiocarbon dating yielded the following ages: B. insignis: 49-51 years, C. laureola: 49-50 years, P. simplex: 49-51 years and T. kraussiana: 51 years. There was a significant relationship between rhizome mass and canopy area, basal area, height, number of stems and main stem diameter. Since rhizome mass had a positive relationship with age- a relationship between age and aerial structures is likely. Therefore, creating surrogates for age may be possible. Overall, these species have moderate longevity, are poor seed producers with possibly slow population growth and are closely associated with soil nutrients. Therefore, these and other suffrutex species are particularly vulnerable to habitat destruction and climate change. The results of this study indicate that there needs to be a greater focus on below ground growth during ecological assessments in order to better understand the ecology of our diverse grassland biome plants.

In this thesis I have investigated the effects of invading top and intermediate consumers in a size-structured fish community, using a combination of field studies, a lake invasion experiment and smaller scale pond and aquaria experiments. The lake invasion experiment was based on introductions of an intermediate consumer, ninespine stickleback (Pungitius pungitius L.), in to allopatric populations of an omnivorous top predator, Arctic char (Salvelinus alpinus L.). The invasion experiment was performed in two tundra lakes and in two birch forest lakes to investigate the effect of climate on the invasion success. I found that the effect of sticklebacks on char was size dependent. Small char suffered reduced growth from resource competition with sticklebacks whereas the maximum size of adult char increased from the addition of a larger prey resource, stickleback. The negative effect of sticklebacks on the growth of small char suggests that sticklebacks may be a better resource competitor than char, which was also supported by the pond and aquaria experiments. The pond experiments also suggested that char were more efficient cannibals than interspecific predators on sticklebacks. Cannibalism in char may limit the recruitment of char and decrease both their predatory and competitive effect on coexisting species and thereby also promote the coexistence of char and sticklebacks. The successful invasion by sticklebacks and their subsequent increases in density suggest that the absence of sticklebacks in char lakes in this region is not caused by biotic interactions with char. Instead, it may be suggested that co-occurrence of sticklebacks and char in the region is limited by dispersal. The char – stickleback system resembles an intraguild predation system with char as the top consumer and stickleback as the intermediate consumer. The effects of the stickleback invasion is also contrasted with a field study of a northern pike (Esox lucius L.) invasion into a system with coexisting char and stickleback, where pike can be viewed as the top consumer and char as the intermediate consumer both feeding on sticklebacks. In this case pike excluded char. The identity of the invading species and the relative strength of the predatory and competitive interactions in the two contrasting systems are discussed in relation to coexistence in intraguild predation systems. I found that the identity of the invading species is of crucial importance for the response at the ecosystem level, and that the inherent size dependency of competitive and predatory interactions in fish communities is important for attaining a mechanistical understanding of the effects of invasive species in lake ecosystems.

Most animals grow substantially during their lifetime and change in competitive ability, predatory capacity and their susceptibility to predation as they grow. This thesis addresses the implications of this on regulation and dynamics within populations as well as between population interactions. In size-structured populations either reproduction or maturation may be more limiting. If juveniles are competitively superior, the competitive bottleneck will be in the adults and reproduction will be limiting. Mortality will in this case result in overcompensation in juvenile biomass through increased reproduction. Compensation in biomass was demonstrated in Daphnia pulex populations subjected to size-independent mortality, where juvenile biomass did not decrease when a substantial harvest was imposed due to increase per capita fecundity. This supported that juveniles were superior competitors and that population cycles seen in Daphnia are juvenile-driven. Compensatory responses in biomass may lead to that predators facilitate eachothers existence by feeding on a common prey, a phenomenon coined emergent facilitation. In an experimental test of the mechanism behind emergent facilitation it was demonstrated that the invertebrate predator Bythotrephes longimanus was favoured by thinning of its prey Holopedium gibberum. The thinning mimicked fish predation and targeted large individuals while Bythotrephes preferrs small prey. Size dependent predation also occurs within populations, i.e. cannibalism, were large individuals feed on smaller conspecifics. Two populations of the common guppy (Poecilia reticulata) originating from different environments were demonstrated to differ in cannibalistic degree. Cannibalism was also affected by the presence of refuges and females and juveniles from one population were better adapted to structural complexity than the other. The effects of these differences in cannibalism on population regulation and dynamics were studied in long term population experiments. Both populations were regulated by cannibalism in the absence of refuges, and displayed cannibal-driven cycles with suppression of recruitment and high population variability. The presence of refuges decreased density dependence and population variability and harvesting of large females in the absence of refuges led to population extinctions in the more cannibalistic population. The less cannibalistic population had higher population biomass and stronger density-dependence in the presence of refuges. When refuges were present, cohort competition increased and cycles with short periodicity were seen. Large individuals were not only cannibals, but could successfully prey on other species. Small and large guppies were allowed to invade resident populations of Heterandria formosa. Small invaders failed while large invaders succeeded as predation from large invaders broke up the competitive bottleneck that the resident population imposed on juveniles of the invader.

O tamanho corporal é uma importante característica, de indivíduos até comunidades e diz respeito ao crescimento, reprodução, mortalidade e alimentação. Pode ser utilizado para avaliar alterações no funcionamento e estrutura dos ecossistemas. Este trabalho é dividido em dois capítulos. No primeiro, fizemos uma revisão bibliográfica para investigar quais fatores de alteração ambiental afetam o tamanho corporal de peixes. Utilizamos uma metodologia semelhante a uma revisão sistemática. Identificamos oito fatores antrópicos de alteração ambiental que influenciam o tamanho corporal de peixes, no entanto os resultados da revisão não apontaram um padrão de resposta do tamanho corporal e poucos artigos selecionados na revisão utilizaram mais de uma métrica para medir o tamanho e este fato pode ser fundamental para um melhor entendimento dos efeitos decorrentes das alterações ambientais. No segundo, avaliamos como a variação do tamanho corporal em grupos tróficos em um sítio responde às alterações resultantes da agricultura quantificadas em quatro escalas (local, riparia até 1km a montante, riparia toda montante e em subbacia). No segundo capítulo investigamos como o tamanho corporal entre grupos tróficos de peixes responde as alterações decorrentes da agricultura em 4 escalas de paisagem em 52 riachos do Pampa Os resultados sugerem que a variação no tamanho corporal depende da medida de distribuição de tamanhos utilizada, do grupo trófico e da escala onde a perturbação é avaliada. A maioria dos grupos tróficos apresentou uma relação negativa com a escala local, geralmente com a redução do tamanho corporal médio ou com uma maior representação de tamanhos pequenos (left skewed). Enquanto em escalas riparia ou de bacia, tamanhos intermediários prevaleceram e tamanhos grandes e pequenos foram desfavorecidos (valores de curtose positivos). Nós ressaltamos a importância de se considerar as características tróficas das espécies para avaliações de alterações ambientais, uma vez que os artigos selecionados na revisão poucas vezes discutiam questões a respeito do efeito das alterações na estrutura trófica das assembléias de peixes. Além disso, a avaliação das escalas de agricultura reforça a importância da conservação da vegetação riparia. O uso de diferentes medidas de distribuição é importante para entender os efeitos da agricultura no tamanho corporal, uma vez que curtose, skewness e coeficiente de variação podem variar sem haver uma mudança no tamanho corporal médio dos peixes. / Body size is an important feature from individuals to communities and it concerns to growth, reproducing, mortality and diet. Body size can be used to evaluate variation on the ecosystem structure. This study was divided in two chapters. In the first, we performed a literature review to investigate which factors of environmental variations affect body size of fishes. We identified eight anthropic factors of environmental disturbance that influence fishes body size, however the results did not show any pattern related to body size variation and few selected papers used more than one size metric what could be fundamental to better understanding the effects caused by environmental changes. In the second chapter, we evaluated how the body size variation in trophic groups responds to agriculture in four landscape scales in 52 streams of the Pampas. The results suggested that body size variation depends on distributional size measures, on trophic group and on scale. Most of the trophic groups have negative relationship with to local scale, usually with mean body size reduction or with higher concentration of small sizes (left skewed), while in the riparian and basin scales fishes with mean values prevailed, and fishes with bigger and smaller sizes had disadvantage (positive values of kurtosis). We highlight the importance of considering trophic features of species to evaluate environmental changes, once previously selected papers rarely discussed the effect of changes on trophic structure of fish communities. Besides, evaluation of agriculture scales reinforce the importance of conservation of riparian vegetation. The use of different distributional measures is important to understand the effects of agriculture on body size, once kurtosis, skewness and coefficient of variation could variate without having changes on mean body size of fishes.

O tamanho corporal é uma importante característica, de indivíduos até comunidades e diz respeito ao crescimento, reprodução, mortalidade e alimentação. Pode ser utilizado para avaliar alterações no funcionamento e estrutura dos ecossistemas. Este trabalho é dividido em dois capítulos. No primeiro, fizemos uma revisão bibliográfica para investigar quais fatores de alteração ambiental afetam o tamanho corporal de peixes. Utilizamos uma metodologia semelhante a uma revisão sistemática. Identificamos oito fatores antrópicos de alteração ambiental que influenciam o tamanho corporal de peixes, no entanto os resultados da revisão não apontaram um padrão de resposta do tamanho corporal e poucos artigos selecionados na revisão utilizaram mais de uma métrica para medir o tamanho e este fato pode ser fundamental para um melhor entendimento dos efeitos decorrentes das alterações ambientais. No segundo, avaliamos como a variação do tamanho corporal em grupos tróficos em um sítio responde às alterações resultantes da agricultura quantificadas em quatro escalas (local, riparia até 1km a montante, riparia toda montante e em subbacia). No segundo capítulo investigamos como o tamanho corporal entre grupos tróficos de peixes responde as alterações decorrentes da agricultura em 4 escalas de paisagem em 52 riachos do Pampa Os resultados sugerem que a variação no tamanho corporal depende da medida de distribuição de tamanhos utilizada, do grupo trófico e da escala onde a perturbação é avaliada. A maioria dos grupos tróficos apresentou uma relação negativa com a escala local, geralmente com a redução do tamanho corporal médio ou com uma maior representação de tamanhos pequenos (left skewed). Enquanto em escalas riparia ou de bacia, tamanhos intermediários prevaleceram e tamanhos grandes e pequenos foram desfavorecidos (valores de curtose positivos). Nós ressaltamos a importância de se considerar as características tróficas das espécies para avaliações de alterações ambientais, uma vez que os artigos selecionados na revisão poucas vezes discutiam questões a respeito do efeito das alterações na estrutura trófica das assembléias de peixes. Além disso, a avaliação das escalas de agricultura reforça a importância da conservação da vegetação riparia. O uso de diferentes medidas de distribuição é importante para entender os efeitos da agricultura no tamanho corporal, uma vez que curtose, skewness e coeficiente de variação podem variar sem haver uma mudança no tamanho corporal médio dos peixes. / Body size is an important feature from individuals to communities and it concerns to growth, reproducing, mortality and diet. Body size can be used to evaluate variation on the ecosystem structure. This study was divided in two chapters. In the first, we performed a literature review to investigate which factors of environmental variations affect body size of fishes. We identified eight anthropic factors of environmental disturbance that influence fishes body size, however the results did not show any pattern related to body size variation and few selected papers used more than one size metric what could be fundamental to better understanding the effects caused by environmental changes. In the second chapter, we evaluated how the body size variation in trophic groups responds to agriculture in four landscape scales in 52 streams of the Pampas. The results suggested that body size variation depends on distributional size measures, on trophic group and on scale. Most of the trophic groups have negative relationship with to local scale, usually with mean body size reduction or with higher concentration of small sizes (left skewed), while in the riparian and basin scales fishes with mean values prevailed, and fishes with bigger and smaller sizes had disadvantage (positive values of kurtosis). We highlight the importance of considering trophic features of species to evaluate environmental changes, once previously selected papers rarely discussed the effect of changes on trophic structure of fish communities. Besides, evaluation of agriculture scales reinforce the importance of conservation of riparian vegetation. The use of different distributional measures is important to understand the effects of agriculture on body size, once kurtosis, skewness and coefficient of variation could variate without having changes on mean body size of fishes.

O tamanho corporal é uma importante característica, de indivíduos até comunidades e diz respeito ao crescimento, reprodução, mortalidade e alimentação. Pode ser utilizado para avaliar alterações no funcionamento e estrutura dos ecossistemas. Este trabalho é dividido em dois capítulos. No primeiro, fizemos uma revisão bibliográfica para investigar quais fatores de alteração ambiental afetam o tamanho corporal de peixes. Utilizamos uma metodologia semelhante a uma revisão sistemática. Identificamos oito fatores antrópicos de alteração ambiental que influenciam o tamanho corporal de peixes, no entanto os resultados da revisão não apontaram um padrão de resposta do tamanho corporal e poucos artigos selecionados na revisão utilizaram mais de uma métrica para medir o tamanho e este fato pode ser fundamental para um melhor entendimento dos efeitos decorrentes das alterações ambientais. No segundo, avaliamos como a variação do tamanho corporal em grupos tróficos em um sítio responde às alterações resultantes da agricultura quantificadas em quatro escalas (local, riparia até 1km a montante, riparia toda montante e em subbacia). No segundo capítulo investigamos como o tamanho corporal entre grupos tróficos de peixes responde as alterações decorrentes da agricultura em 4 escalas de paisagem em 52 riachos do Pampa Os resultados sugerem que a variação no tamanho corporal depende da medida de distribuição de tamanhos utilizada, do grupo trófico e da escala onde a perturbação é avaliada. A maioria dos grupos tróficos apresentou uma relação negativa com a escala local, geralmente com a redução do tamanho corporal médio ou com uma maior representação de tamanhos pequenos (left skewed). Enquanto em escalas riparia ou de bacia, tamanhos intermediários prevaleceram e tamanhos grandes e pequenos foram desfavorecidos (valores de curtose positivos). Nós ressaltamos a importância de se considerar as características tróficas das espécies para avaliações de alterações ambientais, uma vez que os artigos selecionados na revisão poucas vezes discutiam questões a respeito do efeito das alterações na estrutura trófica das assembléias de peixes. Além disso, a avaliação das escalas de agricultura reforça a importância da conservação da vegetação riparia. O uso de diferentes medidas de distribuição é importante para entender os efeitos da agricultura no tamanho corporal, uma vez que curtose, skewness e coeficiente de variação podem variar sem haver uma mudança no tamanho corporal médio dos peixes. / Body size is an important feature from individuals to communities and it concerns to growth, reproducing, mortality and diet. Body size can be used to evaluate variation on the ecosystem structure. This study was divided in two chapters. In the first, we performed a literature review to investigate which factors of environmental variations affect body size of fishes. We identified eight anthropic factors of environmental disturbance that influence fishes body size, however the results did not show any pattern related to body size variation and few selected papers used more than one size metric what could be fundamental to better understanding the effects caused by environmental changes. In the second chapter, we evaluated how the body size variation in trophic groups responds to agriculture in four landscape scales in 52 streams of the Pampas. The results suggested that body size variation depends on distributional size measures, on trophic group and on scale. Most of the trophic groups have negative relationship with to local scale, usually with mean body size reduction or with higher concentration of small sizes (left skewed), while in the riparian and basin scales fishes with mean values prevailed, and fishes with bigger and smaller sizes had disadvantage (positive values of kurtosis). We highlight the importance of considering trophic features of species to evaluate environmental changes, once previously selected papers rarely discussed the effect of changes on trophic structure of fish communities. Besides, evaluation of agriculture scales reinforce the importance of conservation of riparian vegetation. The use of different distributional measures is important to understand the effects of agriculture on body size, once kurtosis, skewness and coefficient of variation could variate without having changes on mean body size of fishes.