Mathematical Modeling of Plant-Herbivore Interactions: Stability Analysis and Period-Doubling Bifurcation in a Modified Nicholson-Bailey Model

Abstract

A Nicholas-Bailey model was initially created with the purpose of examining the dynamics of the population between a parasite and its host. In 1935, Nicholson and Bailey proposed a model for predicting the interactions between Encarsia Formosa parasites and Trialeurodes vaporariorum hosts that focused on the interaction between parasites and hosts1. In the study of biological systems, these types of models, such as discrete-time equations, can be considered invaluable tools for studying the interactions between two species.

This dissertation presents a refined iteration of the Nicholson-Bailey discrete host-parasite model in the first chapter 1. The research unfolds in several chapters. The initial chapter provides a comprehensive background and reviews pertinent literature. Subsequently, fundamental definitions of ordinary differential equations are expounded upon, elucidating key concepts in dynamical systems such as stability analysis, manifold theory and bifurcations. Moreover, essential results and theorems pertinent to the study are delineated.

In the second chapter, an investigation scrutinizes the dynamics of the newly formulated host-parasite model, featuring three essential parameters confined to the first quadrant. A rescaling technique is employed to condense the model into a two-parameter format, capturing its dynamics. Notably, the model consistently manifests two boundary steady states, with the potential emergence of a third interior steady state under specific parameter conditions. Utilizing linearized stability analysis, thresholds for system stability are identified, distinguishing between stable and unstable regimes.

Further exploration delves into the long-term stability of steady states and center manifold theory, particularly focusing on non-hyperbolic steady states and transitions from stable to unstable regions. The analysis extends to bifurcation scenarios, encompassing one or two parameter bifurcations based on varying parameter ranges. Period-doubling bifurcations, leading to chaotic behavior, are observed as eigenvalues cross critical thresholds. Numerical simulations substantiate the theoretical findings, reinforcing the validity of the analysis conclusions.