My PhD work focused upon developing mathematical models of wild bird population dynamics and analyzing them using nonlinear dynamical systems tools.
Existence of chaos in a seasonally forced SIR model.
It has been long observed in computer simulations that seasonally forced SIR compartmental models can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal variation increases. I developed and analyzed a seasonally forced SIR model to describe the spread of highly pathogenic avian influenza in a seabird population. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host- pathogen system. Seabirds have unusual life histories in that they have low reproductive output, deferred breeding and long life expectancy. In addition, they form large colonies which may contain thousands, and sometimes millions, of birds on inaccessible mainland cliffs and offshore islands. I have proved that chaotic solutions of this seasonally forced system exist using methods based on the concept of topological hyperbolicity. The approach elucidates the geometry of the chaos in the phase space of the model, thereby providing a potential mechanism for the persistence of the infection. The work shows that theoretically, recurrent epidemics of highly infectious pathogens in seabird populations are possible and it underscores that it is recruitment of susceptibles that drive recurrent epidemics in species with K-strategist life histories. I collaborated with Alexei Pokvrovskii, Tom Kelly, Michael O’Callaghan, Dmitrii Rachinskii and Andrei Korobeinikov on this project.
Mechanisms behind increase in woodpigeon in the British Isles.
Woodpigeons are multi-brooded, obligate herbivores whose abundance in the British Isles has rapidly increased over the past forty years. Prior to this increase, their numbers were stable and most likely regulated by the effects of intraspecific competition. It is unclear what has driven their increase in abundance. Using an age-structured hybrid model with density-dependent mortality and a discrete birth pulse, I explored the impacts of two hypotheses for the increase–reduced intraspecific competition for food and increased reproductive success. The two mechanisms predict contrasting population age profiles at equilibrium. The model indicates that an increase in reproductive success leads to an age profile dominated by juveniles whereas reduction in intraspecific competition leads to a population structure dominated by adults. However the overall population abundances under the former mechanism increase by ~300% whereas under reduced competition, an increase of a mere ~25% was predicted. These results tentatively suggest that reproductive success drove the population increase. I also used the model to compare the impacts of various control measures. An annual shooting season that follows the period of density-dependent mortality is predicted to be the most effective control strategy because it simultaneously removes adult and juvenile woodpigeons. I worked on this project with Alexei Pokvrovskii, Tom Kelly, Michael O’Callaghan, Dmitrii Rachinskii and Denis Flynn.
Global stability of a family of SIR and SIRS models.
I developed a simple Lyapunov function for a large family of SIR and SIRS epidemiological models, thereby establishing the global stability of the endemic equilibrium in the absence of seasonal and stochastic forces. This is a stronger statement than establishing local stability, achieved via a linear stability analysis that verifies that small perturbations from the equilibrium will return to it. Establishing global stability of the equilibrium is to show rigorously that all trajectories (perturbations from the equilibrium), no matter where they start from in the phase space, so long as S, I and R are positive and R0>1, will eventually approach the endemic equilibrium of the model. This work was done in collaboration with Alexei Pokvrovskii, Tom Kelly, Michael O’Callaghan and Andrei Korobeinikov.