Aspects of Mass Extinctions

Species go extinct as a natural process over periods of millions of years. Mass extinction events correspond to times when the rate of this back-ground extinction is greatly enhanced. Extinction events all differ in duration, intensity, diversity of species affected and cause. There are many mechanisms by which extinction occurs and individual extinctions often have multiple contributing mechanisms. Large igneous province volcanism is often implicated as a major contributor towards extinction. This thesis considers various aspects of mass extinction and, relatedly, volcanism. The aim is to show how applied mathematical techniques may be utilised to further understanding of extinction and associated phenomena. Firstly, we consider plumes from a classical fluid dynamics perspective. We focus on the prescription of plumes of finite width and choose an eddy viscosity that vanishes with vertical velocity at the plume edge. Analysis concludes that such a choice of eddy viscosity implies finite width and that the entrainment condition can be determined as a consequence of model formulation, rather than formally prescribed. Our resulting similarity profiles lack the Gaussian tail exhibited experimentally; we suggest including molecular kinematic viscosity may reconcile this discrepancy. We outline a methodology utilising the method of strained coordinates and demonstrate how it may be applied to problems of this kind. Next, we consider the effect increased volcanism may have on the global ecosystem. Mercury is used as a proxy to indicate times of enhanced volcanic activity. A model of the global mercury cycle is created, and it is shown that increased volcanic emissions may lead to spikes in mercury concentrations comparable to that of sedimentary records of the geological past. Oceanic anoxic events correspond to times in which sections of the world’s oceans became depleted in oxygen. We create a model of the processes under which anoxia occurs and show that a sufficient increase in volcanic activity can lead to a state of anoxia during which anaerobic oxidation processes occur. Finally, we consider how extinction occurs in continuous population models. The standard pathways to extinction in such models are through competition or extreme periodicity in predator-prey systems. We discuss a model of competition by which the competitive mechanism is purely in the form of resource depletion. We show the model leads to stable coexistence and discuss how to bring about extinction, with comparison to the classical case of competitive exclusion. We also analyse a predator-prey system which exhibits extreme oscillations and describe how populations may reach extinction levels before, due to the continuous nature of the model, reactivation occurs. We modify the model by the addition of a threshold term which allows actual extinction. We then show that with the addition of spatial variation, modelled via diffusion, localised patches of extinction will extend and propagate as a travelling wave: the wave of extinction