Mathematical modeling for contrasting dynamics of a plant herbivore interaction

Abstract

The Nicholas-Bailey model was designed to study population dynamics of host-parasite systems. The model was first developed by Nicholson and Bailey (1935) and applied to parasites (Encarsia Formosa) and hosts (Trialeurodes vaporariorum). These types of models are presented by discrete-time equations for biological systems that involve two species, e.g. a parasite population and its hosts. In this dissertation, we develop and then investigate a revised version of Nicholson-Bailey's discrete host-parasite model. Additionally, we incorporate and analyze the Allee effect dynamics in this newly constructed model. In Chapter one of this dissertation, we outline some background and literature. Second, we provide basic definitions of ordinary differential equations. We define several core concepts of dynamical systems including stability and instability analysis, manifold analysis, stable and unstable manifold, invariant manifold, center manifold, bifurcation, and the Lambert W function. Then we provide some known results and theorems that are useful in this research investigation. Third, we study the dynamics behavior of the newly developed system of a host-parasite model with four positive parameters in the first closed quadrant. A re-scaling procedure will be then applied to reduce the model to a two-parameter model that reproduces the entire dynamics of the original model. The model always possesses two boundary steady states and a third interior steady state may exist for particular conditions imposed on the parameters. Moreover, by applying the linearized stability function, we find thresholds for which the system is stable or unstable. We then study locally the long-term stability of steady states and center manifold theory based on the separating boundary curves for non-hyperbolic steady states, that is analyzing steady states when crossing from stable to unstable regions. We then analyze the stability for one or two parameter bifurcation (co-dimension one or two) depending on a different range of parameters, by considering the linearization of the model about each of the steady states. We show a period-doubling bifurcation occurs once the eigenvalue crosses these thresholds, leading to chaos. Numerical simulations support the results and conclusions. Fourth, we introduce the density dependence of the Allee effect and population dynamics into the model by adding a parameter to the modified system of the Nicholson-Bailey model. We then study the local stability of its steady states. Multiple bifurcation analyses of the system, including the period-doubling behavior and Neimark-Sacker bifurcation, will be analyzed. We then identify regions where the Allee effect system ultimately leads to chaos. Finally, the modified systems of the Nicholson-Bailey model and the Allee effect model are compared by analyzing different short-term and long-term dynamical behaviors and results acquired from the two systems.