Case studies in mathematical modelling for biological conservation

Abstract

The use of mathematical modelling as a tool for investigating selected topics in conservation biology is the focus of this thesis. A continuous system of partial and ordinary differential equations model the age structured population dynamics of a cohort of endemic, threatened New Zealand North Island brown kiwi, Apteryx mantelli. Critical predation and recruitment rates of immature birds are estimated. Stoats, Mustela erminea, are the main predator of immature kiwi. A refinement to the model allows the calculation of acceptable stoat densities. In order to reduce stoats to this critical density, a linear system of ordinary differential equations, representing an acute secondary poisoning regime, is solved. An optimal secondary poisoning scheme, which minimises the number of prey poisoned and the amount of poison used, is found. The minimum area required for pest control is estimated by simulating the dispersal of sub-adult kiwi using a discrete random walk approach. Simulations and a discrete age structured model are used to investigate pulsed management strategies for both kiwi and kokako, Callaeas cinerea wilsoni. Finally, a two dimensional discrete random walk is generalised and a continuous diffusion equation is derived. A diffusion equation is incorporated into a S1 R (Susceptible, Infected, Recovered) model representing the natural spread of Rabbit Haemorrhagic Disease from a point source in rabbit, Oryctolagus cuniculus cuniculus, populations. The speed for the virus, dependant on certain model parameters, is found and the minimum initial population density, below which the wave of infection will not travel, is estimated. All specific models discussed throughout the thesis are generic by nature and can be applied to a diverse range of subjects.