Periodic solutions of integro-differential equations which arise in population dynamics

The problem of the existence and stability of periodic solutions of infinite-lag integra-differential equations is considered. Specifically, the integrals involved are of the convolution type with the dependent variable being integrated over the range (- ∞,t), as occur in models of population growth. It is shown that Hopf bifurcation of periodic solutions from a steady state can occur, when a pair of eigenvalues crosses the imaginary axis. Also considered is the existence of traveling wave solutions of a model population equation allowing spatial diffusion in addition to the usual temporal variation. Lastly, the stability of the periodic solutions resulting from Hopf bifurcation is determined with aid of a Floquet theory.

The first chapter is devoted to linear integro-differential equations with constant coefficients utilizing the method of semi-groups of operators. The second chapter analyzes the Hopf bifurcation providing an existence theorem. Also, the two-timing perturbation procedure is applied to construct the periodic solutions. The third chapter uses two-timing to obtain traveling wave solutions of the diffusive model, as well as providing an existence theorem. The fourth chapter develops a Floquet theory for linear integro-differential equations with periodic coefficients again using the semi-group approach. The fifth chapter gives sufficient conditions for the stability or instability of a periodic solution in terms of the linearization of the equations. These results are then applied to the Hopf bifurcation problem and to a certain population equation modeling periodically fluctuating environments to deduce the stability of the corresponding periodic solutions.

Engineered underdominance as a method of insect population replacement and reproductive isolation

Insect vector-borne diseases, such as malaria and dengue fever (both spread by mosquito vectors), continue to significantly impact health worldwide, despite the efforts put forth to eradicate them. Suppression strategies utilizing genetically modified disease-refractory insects have surfaced as an attractive means of disease control, and progress has been made on engineering disease-resistant insect vectors. However, laboratory-engineered disease refractory genes would probably not spread in the wild, and would most likely need to be linked to a gene drive system in order to proliferate in native insect populations. Underdominant systems like translocations and engineered underdominance have been proposed as potential mechanisms for spreading disease refractory genes. Not only do these threshold-dependent systems have certain advantages over other potential gene drive mechanisms, such as localization of gene drive and removability, extreme engineered underdominance can also be used to bring about reproductive isolation, which may be of interest in controlling the spread of GMO crops. Proof-of-principle establishment of such drive mechanisms in a well-understood and studied insect, such as Drosophila melanogaster, is essential before more applied systems can be developed for the less characterized vector species of interest, such as mosquitoes. This work details the development of several distinct types of engineered underdominance and of translocations in Drosophila, including ones capable of bringing about reproductive isolation and population replacement, as a proof of concept study that can inform efforts to construct such systems in insect disease vectors.

From lakes to lungs : assessing microbial activity in diverse environments

All major geochemical cycles on the Earth’s surface are mediated by microorganisms. Our understanding of how these microbes have interacted with their environments (and vice versa) throughout Earth's history, and how they will respond to changes in the future, is primarily based on studying their activity in different environments today. The overarching questions that motivate the research presented in the two parts of this thesis -- how do microorganisms shape their environment (and vice versa)? and how can we best study microbial activity in situ? -- have arisen from the ultimate goal of being able to predict microbial activity in response to changes within their environments both past and future.

Part one focuses on work related to microbial processes in iron-rich Lake Matano and, more broadly, microbial interactions with the biogeochemical cycling of iron. Primarily, we find that the chelation of ferrous iron by organic ligands can affect the role of iron in anoxic environmental systems, enabling photomixotrophic growth of anoxygenic microorganisms with ferrous iron, as well as catalyzing the oxidation of ferrous iron by denitrification intermediates. These results imply that the ability to grow photomixotrophically on ferrous iron might be more widespread than previously assumed, and that the co-occurrence of chemical and biological processes involved in the coupled biogeochemical cycling of iron and nitrogen likely dominate organic-rich environmental systems.

Part two switches focus to in situ measurements of growth activity and comprises work related to microbial processes in the Cystic Fibrosis lung, and more broadly, the physiology of slow growth. We introduce stable isotope labeling of microbial membrane fatty acids and whole cells with heavy water as a new technique to measure microbial activity in a wide range of environments, demonstrate its application in continuous culture in the laboratory at the population and single cell level, and apply the tool to measure the in situ activity of the opportunistic pathogen Staphylococcus aureus within the environment of expectorated mucus from cystic fibrosis patients. We find that the average in situ growth rates of S. aureus fall into a range of generation times between ~12 hours and ~4 days, with substantial heterogeneity at the single-cell level. These data illustrate the use of heavy water as a universal environmental tracer for microbial activity, and highlight the crucial importance of studying the physiology of slow growth in representative laboratory systems in order to understand the role of these microorganisms in their native environments.