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.

Effects of TI-299423 on Neuronal Nicotinic Acetylcholine Receptors

Nicotinic acetylcholine receptors (nAChRs) are pentameric, ligand-gated, cation channels found throughout the central and peripheral nervous system, whose endogenous ligand is acetylcholine, but which can also be acted upon by nicotine. The subunit compositions of nAChR determine their physiological and pharmacological properties, with different subunits expressed in different combinations or areas throughout the brain. The behavioral and physiological effects of nicotine are elicited by its agonistic and desensitizing actions selectively on neuronal nAChRs. The midbrain is of particular interest due to its population of nAChRs expressed on dopaminergic neurons, which are important for reward and reinforcement, and possibly contribute to nicotine dependence. The α6-subunit is found on dopaminergic neurons but very few other regions of the brain, making it an interesting drug target. We assayed a novel nicotinic agonist, called TI-299423 or TC299, for its possible selectivity for α6-containing nAChRs. Our goal was to isolate the role of α6-containing nAChRs in nicotine reward and reinforcement, and provide insight into the search for more effective smoking cessation compounds. This was done using a variety of in vitro and behavioral assays, aimed dually at understanding TI-299423’s exact mechanism of action and its downstream effects. Additionally, we looked at the effects of another compound, menthol, on nicotine reward. Understanding how reward is generated in the cholinergic system and how that is modulated by other compounds contributes to a better understand of our complex neural circuitry and provides insight for the future development of therapeutics.