Modelling dispersal processes in impala-cheetah-lion ecosystems with infection in the lions

The study involved the predator-prey interaction of three species namely the predator (Cheetah Acinonyx jubatus), the super-predator (Lion Panthera leo), and their common prey (Impala Aepyceros melampus). The study area is the Kruger National Park. The predator being an endangered species, faces a survival problem. It is frequently killed by the super-predator to reduce competition for prey. The super-predator also scares away the predator o_ its kills. The prey forms the main diet of the predator. The plight of the predator motivated the author to formulate disease and reaction-diffusion models for the species interactions. The purpose of the models were to predict and explain the effect of large competition from the super-predator on the predator population. Important parameters related to additional predator mortality due to presence of super-predator, the disease incidence rate and induced death rate formed the focal points of the analysis. The dynamics of a predator-prey model with disease in super-predator were investigated. The super-predator species is infected with bovine Tuberculosis. In the study, the disease is considered as biological control to allow the predator population to regain from low numbers. The results highlight that in the absence of additional mortality on the predator by the super-predator, the predator population survives extinction. Furthermore, at current levels of disease incidence, the super-predator population is wiped out by the disease. However, the super-predator population survives extinction if the disease incidence rate is low. Persistence of all populations is possible in the case of low disease incidence rate and no additional mortality imparted on the predator. Furthermore, a two-species subsystem, prey and predator, is considered as a special case to determine the effect of super-predator removal from the system, on the survival of the predator. This is treated as a contrasting case from the smaller parks. The results show that the predator population thrives well in the total absence of its main competitor, with its population rising to at least twice the initial value. A reaction-diffusion three-species predator-prey model was formulated and analysed. Stability of the temporal and the spatio-temporal systems, existence and non-existence of stationary steady state solutions were studied. Conditions for the emergence of stationary patterns were deduced. The results show that by choosing the diffusion coeffcient d2 > _D 2 suffciently large, a non-constant positive solution is generated, that is, stationary patterns emerge, depicting dispersal of species. Predators were observed to occupy habitats surrounding prey. However, super-predators were observed to alternate their habitats, from staying away from prey to invading prey habitat. In the investigation, strategies to determine ways in which the predator species could be saved from extinction and its population improved were devised, and these included isolation of the predator from the super-predator

Modelling the dynamics of Breast Cancer disease with hormone therapy and surgery controls

In this study, we discussed a mathematical model that incorporates important interactions between normal cells, tumor cells, immune cells, and estrogen. The mathematical model was revised to include two control measures; namely surgery and hormone therapy to minimize the number of tumor cells. The model was mathematically analyzed with the premise that the two control measures are positive constants. Locally and globally analyses were performed using a variety of analytical methods to investigate the stability of the breast cancer model. Furthermore, an optimal control problem was formulated and used to determine the best strategy for reducing the number of tumor cells by incorporating hormone therapy and surgery, based on the well-known Pontryagin’s Maximum Principle. The numerical results indicates combining both optimal control measures (surgery and hormone therapy) simultaneously is more efficacious than using single control measure separately in decreasing the number of tumor cells.Thesis (MSc) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics, 202

Modelling the dynamics of Breast Cancer disease with hormone therapy and surgery controls

In this study, we discussed a mathematical model that incorporates important interactions between normal cells, tumor cells, immune cells, and estrogen. The mathematical model was revised to include two control measures; namely surgery and hormone therapy to minimize the number of tumor cells. The model was mathematically analyzed with the premise that the two control measures are positive constants. Locally and globally analyses were performed using a variety of analytical methods to investigate the stability of the breast cancer model. Furthermore, an optimal control problem was formulated and used to determine the best strategy for reducing the number of tumor cells by incorporating hormone therapy and surgery, based on the well-known Pontryagin’s Maximum Principle. The numerical results indicates combining both optimal control measures (surgery and hormone therapy) simultaneously is more efficacious than using single control measure separately in decreasing the number of tumor cells.Thesis (MSc) -- Faculty of Science, School of Computer Science, Mathematics, Physics and Statistics, 202