Mathematical Sciences - Masters by Research Theses
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Item Rate-induced tipping in a moving habitat(University College Cork, 2021-03) MacCárthaigh, Ruaidhrí; Wieczorek, Sebastian; Hasan, CrisAs our planet's surface warms, its ecosystems are shifting to cooler and more suitable climates to survive [1]. This issue raises some important research questions. How successfully can life on earth adapt to these changes? How fast can populations be pushed to migrate before they fail to adapt and collapse to extinction? This thesis addresses the problem of species adapting to shifting habitats, using the framework of ``critical transitions'' or ``tipping points''. Specifically, it explores the persistence of a single species in a fast-moving habitat, and how the nature of its population growth affects its adaptability. The spatiotemporal evolution of the species concerned is described by partial differential equations (PDEs) in the form of nonlinear one-dimensional reaction-diffusion equations and reaction-convection-diffusion equations. Two distinct growth models are constructed and the moving habitat is incorporated into each model with the addition of a continuous non-autonomous term. The first model is a quadratic monostable model of logistic growth limited by the carrying capacity of the habitat, similar to models used in previously published work on the topic of species in a moving habitat [2,3,4]. The second model is made cubic and bistable by incorporating an additional limitation to population growth at low population densities, known as the Allee effect. The spatial distribution of the populations are computed as standing wave solutions in the static habitat and travelling pulse solutions in the moving habitat, drifting at a constant habitat speed. A critical habitat speed is found for both the monostable and bistable models, above which no physical travelling pulse solution exists, meaning that the population within the habitat dies out. In the monostable Logistic Model, a transcritical bifurcation occurs between its travelling pulse solution and its zero (extinction) solution, which gives rate-induced tipping that can be reversed with a clear indication of an eventual local extinction as this critical speed is approached. In contrast, in the bistable Allee Model a saddle-node bifurcation of travelling pulses occurs, which produces an rate-induced tipping point, which typically cannot be reversed in nature, at which the population in the habitat abruptly collapses to extirpation, without any clear indication of the imminent collapse. This provides a stark ecological insight into how the increasing rate of climate change may give little to no advanced warning of extinction for some species that are observed to have an Allee effect and other complex features influencing their population dynamics.