Wang, Xiaoying

This study focuses on the world dynamics of a cholera model that includes delayed bacterial shedding and water disinfection. From the method of the next generation matrix, a basic reproduction number is found that sets a threshold of disease persistence. It is shown that the disease disappears if $R_0<1$, which means that the disease-free equilibrium is globally asymptotically stable. The system is not destabilized by the delay, which leads to periodic oscillations. The numerical simulations validate the theoretical analysis, which illustrates the importance of delay and disinfection in cholera prevention and control.

Author Keywords: Basic reproduction number, Cholera, Delay differential equations, Disinfection, Lyapunov function, Stability analysis

This thesis is focused on studying the population dynamics of a predator-prey system in a patchy environment, taking anti-predation responses into consideration. Firstly, we conduct mathematical analysis on the equilibrium solutions of the system. Using techniques from calculus we show that particular steady state solutions exist when the parameters of the system meet certain criteria. We then show that a further set of conditions leads to the local stability of these solutions. The second step is to extend the existing mathematical analysis by way of numerical simulations. We use octave to confirm the previous results, as well as to show that more complicated dynamics can exist, such as stable oscillations. We consider more complex and meaningful functions for nonlinear dispersal between patches and nonlinear predation, and show that the proposed model exhibits behaviours we expect to see in a population model.

Author Keywords: Anti-predation response, Asymptotic stability, Dispersal, Patch model, Population dynamics, Predator-prey

This thesis is focused on studying the population dynamics of a predator-prey system in a patchy environment, taking anti-predation responses into consideration. Firstly, we conduct mathematical analysis on the equilibrium solutions of the system. Using techniques from calculus we show that particular steady state solutions exist when the parameters of the system meet certain criteria. We then show that a further set of conditions leads to the local stability of these solutions. The second step is to extend the existing mathematical analysis by way of numerical simulations. We use octave to confirm the previous results, as well as to show that more complicated dynamics can exist, such as stable oscillations. We consider more complex and meaningful functions for nonlinear dispersal between patches and nonlinear predation, and show that the proposed model exhibits behaviours we expect to see in a population model.

Author Keywords: Anti-predation response, Asymptotic stability, Dispersal, Patch model, Population dynamics, Predator-prey