Abstract
In studying some biological problems, the descriptive features of biological systems are delivered by differential equations which can be studied quantitatively by both analytical and computational methods. The obtained results can be interpreted back to the biological level, then explain some phenomena and also provide some strategies. In this thesis, both of ecological and epidemiological problems have been studied. In the first chapter, I studied the nonlinear dynamics of some ecological models to understand how seasonal harvesting affects ecological systems. I proposed a predator- prey model with a non-monotone functional response and seasonal prey harvesting, which is described by a periodic perturbation. Then various possible bifurcations in a predator-prey system have been studied when it is perturbed by constant and seasonal harvesting efforts. Ecologically and economically, the results are helpful in the precise and careful management of the renewable resource and prevention of overexploitation. In Chapter 2, a mathematical model was proposed to study the transmission dynamics of West Nile virus taking into account the local interactions between birds and mosquitoes as well as the transmission from mosquitoes to human. Based on this model, we discussed the existence of the disease-free and endemic equilibria and calculated the basic reproduction number that is shown to be a threshold of the disease dynamics. To validate the model, we used it to simulate the WNv human data of infected cases and accumulative deaths from 1999 to 2013 in the states of New York, Florida, Texas and California by minimizing Chi2. Some suggestions about larval and adult mosquito control have been compared and discussed in details by the sensitivity analysis of the basic reproduction number in terms of model parameters. In Chapter 3, we proposed a multi-patch model for the transmission dynamics of rabies among dogs and from dogs to humans to study how the movement of dogs induces the geographically inter-provincial spread of rabies in Mainland China. The two- patch model is used to simulate the human rabies data to investigate the inter- provincial spread of rabies between Guangxi and Guizhou, Sichuan and Shaanxi, and Fujian and Hebei, respectively. In order to reduce and prevent geographical spread of rabies in China, our results suggest that the management of dog market and trade need to be regulated and transportation of dogs need to be better monitored and under constant surveillance. In the last chapter, I discussed some possibilities in the future study.