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>Magister Scientiae - MSc / In principle, an enrichment of resources in predator-prey systems show prompts destabilisation of a framework, accordingly, falling trophic communication, a phenomenon known to as the \Paradox of Enrichment" [54]. After it was rst genius postured by Rosenzweig [48], various resulting examines, including recently those of Mougi-Nishimura [43] as well as that of Bohannan-Lenski [8], were completed on this problem over numerous decades. Nonetheless, there has been a universal none acceptance of the \paradox" word within an ecological eld due to diverse interpretations [51]. In this dissertation, some theoretical exploratory works are being surveyed in line with giving a concise outline proposed responses to the paradox. Consequently, a quantity of di usion-driven models in mathematical ecology are evaluated and analysed. Accordingly, piloting the way for the spatial structured pattern (we denote it by SSP) formation in nonlinear systems of partial di erential equations [36, 40]. The central point of attention is on enrichment consequences which results toward a paradoxical state. For this purpose, evaluating a number of compartmental models in ecology similar to those of [48] will be of great assistance. Such displays have greater in uence in pattern formations due to diversity in meta-population. Studying the outcomes of initiating an enrichment into [9] of Braverman's model, with a nutrient density (denoted by n) and bacteria compactness (denoted by b) respectively, suits the purpose. The main objective behind being able to transform [9]'s system (2.16) into a new model as a result of enrichment. Accordingly, n has a logistic- type growth with linear di usion, while b poses a Holling Type II and nonlinear di usion r2 nb2 [9, 40]. Five fundamental questions are imposed in order to address and guide the study in accordance with the following sequence: (a) What will be the outcomes of introducing enrichment into [9]'s model? (b) How will such a process in (i) be done in order to change the system (2.16)'s stability state [50]? (c) Whether the paradox does exist in a particular situation or not [51]? Lastly, (d) If an absurdity in (d) does exist, is it reversible [8, 16, 54]? Based on the problem statement above, the investigation will include various matlab simulations. Therefore, being able to give analysis on a local asymptotic stability state when a small perturbation has been introduced [40]. It is for this reason that a bifurcation relevance comes into e ect [58]. There are principal de nitions that are undertaken as the research evolves around them. A study of quantitative response is presented in predator-prey systems in order to establish its stability properties. Due to tradeo s, there is a great likelihood that the growth rate, attack abilities and defense capacities of species have to be examined in line with reviewing parameters which favor stability conditions. Accordingly, an investigation must also re ect chances that leads to extinction or coexistence [7]. Nature is much more complex than scienti c models and laboratories [39]. Therefore, di erent mechanisms have to be integrated in order to establish stability even when a system has been under enrichment [51]. As a result, SSP system is modeled by way of reaction-di usion di erential equations simulated both spatially and temporally. The outcomes of such a system will be best suitable for real-world life situations which control similar behaviors in the future. Comparable models are used in the main compilation phase of dissertation and truly re ected in the literature. The SSP model can be regarded as between (2018-2011), with a stability control study which is of an original.

We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).

<p>We investigate a certain epidemics model, with and without noise. Some parameter analysis is performed together with computer simulations. The model was presented in Iacus (2008).</p>

O uso de ferramentas matemáticas e computacionais se faz cada vez mais presente nos estudos da dinâmica predador-presa devido à complexidade das populações que sofrem influências externas relativas à interação com o ambiente. O objetivo deste estudo foi investigar as equações de Lotka-Volterra e aplicá-las para modelar e descrever a dinâmica da interação entre o inseto Cryptolaemus montrouzieri (predador) e sua presa, a cochonilha-do-carmim (Dactylopius opuntiae) visando identificar cenários que conduzam ao combate da cochonilha. Esta é uma praga que tem atacado severamente as áreas de cultivo da palma forrageira, utilizada para alimentação de caprinos, ovinos e bovinos causando prejuízos econômicos, ambientais e sociais à região do semiárido nordestino. A destruição dos palmais devido ao ataque da cochonilha obriga os agricultores a se desfazerem dos seus rebanhos, migrarem para outras regiões ou devastar áreas da Caatinga para realizar novos plantios de palma. Diante do exposto, verifica-se a necessidade de prover suporte para uma cultura sustentável da palma forrageira, essencial para o gado leiteiro. Para fins de controle de pragas, órgãos como a Embrapa e o IPA têm recomendado o controle biológico da cochonilha através da introdução de seu inimigo natural (C. montrouzieri) que tem se mostrado eficiente no combate à praga. Neste trabalho foram realizadas simulações da evolução das populações de D. opuntiae e de C. montrouzieri utilizando as equações que representam o modelo Lotka-Volterra e suas variantes. Os resultados desta pesquisa têm tornado possível sugerir a quantidade de predadores a ser inserida no ambiente natural para buscar combater e controlar a infestação de pragas que tem causado danos à região do Semiárido do Nordeste. O uso do modelo não apenas possibilita sugerir a população de predadores para interagir com a espécie presa numa relação trófica, como também permite o monitoramento da conformidade da evolução populacional das espécies estudadas / The use of mathematical and computational tools have been found within studies of the dynamics of predator-prey due to the complexities of the population evolution that suffer external influences of their interaction with the environment. Within this context, this work aims at investigating the Lotka-Volterra model and apply them to capture and describe the dynamics of the interaction between the Cryptolaemus montrouzieri (predator) and its prey, the cochineal-carmine (Dactylopius opuntiae) in order to identify scenarios that aim at reducing the amount of ladybug. This a pest that has infested a large area in the semiarid region of the Northeast Brazil where the cactus pear has been used as a base of supply for ruminants. As a result, environmental damages as well as social and economic losses have occurred. This pest infestation has caused farmers migration to other areas. Given the above, there is a need to provide support for a sustainable cultivation of cactus pear. For the purpose of pest control, agencies such as Embrapa and IPA have recommended biological control of ladybug by introducing natural enemies such as C. montrouzieri that has been effective at reducing the pest. In that sense, within this work simulations of population evolution for D. opuntiae and C. montrouzieri have been made by using the Lotka-Volterra equations as well as its variants. Results of this research has made possible to suggest the amount of predators that might be inserted into the natural environment aiming at reducing and control pest infestation in a case study of the semiarid region of the Northeast Brazil. The use of the model not only allows suggesting the population of predators for species within a trophic interaction, but also allows the monitoring of compliance of the evolution of the species under study

Made available in DSpace on 2017-06-01T18:20:36Z (GMT). No. of bitstreams: 1 dissertacao_mirella_renata.pdf: 1480280 bytes, checksum: 15dfff5f39ca1626a0e638550e849fec (MD5) Previous issue date: 2012-05-08 / The use of mathematical and computational tools have been found within studies of the dynamics of predator-prey due to the complexities of the population evolution that suffer external influences of their interaction with the environment. Within this context, this work aims at investigating the Lotka-Volterra model and apply them to capture and describe the dynamics of the interaction between the Cryptolaemus montrouzieri (predator) and its prey, the cochineal-carmine (Dactylopius opuntiae) in order to identify scenarios that aim at reducing the amount of ladybug. This a pest that has infested a large area in the semiarid region of the Northeast Brazil where the cactus pear has been used as a base of supply for ruminants. As a result, environmental damages as well as social and economic losses have occurred. This pest infestation has caused farmers migration to other areas. Given the above, there is a need to provide support for a sustainable cultivation of cactus pear. For the purpose of pest control, agencies such as Embrapa and IPA have recommended biological control of ladybug by introducing natural enemies such as C. montrouzieri that has been effective at reducing the pest. In that sense, within this work simulations of population evolution for D. opuntiae and C. montrouzieri have been made by using the Lotka-Volterra equations as well as its variants. Results of this research has made possible to suggest the amount of predators that might be inserted into the natural environment aiming at reducing and control pest infestation in a case study of the semiarid region of the Northeast Brazil. The use of the model not only allows suggesting the population of predators for species within a trophic interaction, but also allows the monitoring of compliance of the evolution of the species under study / O uso de ferramentas matemáticas e computacionais se faz cada vez mais presente nos estudos da dinâmica predador-presa devido à complexidade das populações que sofrem influências externas relativas à interação com o ambiente. O objetivo deste estudo foi investigar as equações de Lotka-Volterra e aplicá-las para modelar e descrever a dinâmica da interação entre o inseto Cryptolaemus montrouzieri (predador) e sua presa, a cochonilha-do-carmim (Dactylopius opuntiae) visando identificar cenários que conduzam ao combate da cochonilha. Esta é uma praga que tem atacado severamente as áreas de cultivo da palma forrageira, utilizada para alimentação de caprinos, ovinos e bovinos causando prejuízos econômicos, ambientais e sociais à região do semiárido nordestino. A destruição dos palmais devido ao ataque da cochonilha obriga os agricultores a se desfazerem dos seus rebanhos, migrarem para outras regiões ou devastar áreas da Caatinga para realizar novos plantios de palma. Diante do exposto, verifica-se a necessidade de prover suporte para uma cultura sustentável da palma forrageira, essencial para o gado leiteiro. Para fins de controle de pragas, órgãos como a Embrapa e o IPA têm recomendado o controle biológico da cochonilha através da introdução de seu inimigo natural (C. montrouzieri) que tem se mostrado eficiente no combate à praga. Neste trabalho foram realizadas simulações da evolução das populações de D. opuntiae e de C. montrouzieri utilizando as equações que representam o modelo Lotka-Volterra e suas variantes. Os resultados desta pesquisa têm tornado possível sugerir a quantidade de predadores a ser inserida no ambiente natural para buscar combater e controlar a infestação de pragas que tem causado danos à região do Semiárido do Nordeste. O uso do modelo não apenas possibilita sugerir a população de predadores para interagir com a espécie presa numa relação trófica, como também permite o monitoramento da conformidade da evolução populacional das espécies estudadas