Date of Award
2025
Degree Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Specialization
Applied Mathematics
Department
Mathematics and Applied Mathematical Sciences
First Advisor
Mustafa Kulenovic
Abstract
This dissertation investigates the local and global behavior of several classes of discrete population models involving stocking, harvesting, and cooperation. The study is divided into three major manuscripts, each offering new theoretical in sights, bifurcation results, and applications to real-world data.
In Manuscript 1, we study the Beverton-Holt population models with constant and proportional stocking and harvesting, as well as sigmoid Beverton-Holt models under constant stocking and harvesting, both in autonomous and non-autonomous settings. We analyze stability and explore global dynamics using established local and global dynamics theorems, bifurcation theory, and explicit solutions when available. Special attention is given to the impact of environmental oscillations and the Allee effect. The models are applied to real-world fishery data, revealing how interventions affect population sustainability. We compare the fitted models using simulations and compute their coefficients of determination, along with appropriate statistical analyses, to assess predictive performance.
In Manuscript 2, we extend the analysis to a broader class of nonlinear discrete models, including sigmoid models with proportional stocking and harvesting, as well as surge-type functions under both constant and proportional harvesting and stocking. We determine global attractors using global dynamics analysis and numerical methods. These models are also applied to pharmacokinetics data (Theophylline concentration), where surge models are shown to capture absorption and decay patterns more accurately than classical models. A unified stability result for systems with oscillatory parameters is presented, demonstrating convergence to the equilibrium of the limiting autonomous dynamics.
In Manuscript 3, we study the global dynamics of a cooperative toggle-switch system in the plane, inspired by biological models of gene networks. We investigate both the continuous (ODE) version and its discrete Euler approximation. For the discrete case, we characterize the basins of attraction of equilibria and period two solutions using recent results on planar cooperative systems. We provide conditions under which multiple attractors coexist and describe how the state space is partitioned into distinct basins. Simulations support the theoretical findings and highlight the biological relevance of the model, particularly in representing mutualistic species or interacting genes under bistable or oscillatory dynamics.
Together, these manuscripts offer a comprehensive analysis of discrete systems in population biology and gene regulation, with implications for sustainable resource management, pharmacodynamics, and evolutionary dynamics.
Recommended Citation
Habach, Sam, "STABILITY AND GLOBAL DYNAMICS OF AUTONOMOUS AND NONAUTONOMOUS DISCRETE POPULATION MODELS WITH STOCKING, HARVESTING, AND COOPERATION" (2025). Open Access Dissertations. Paper 4514.
https://digitalcommons.uri.edu/oa_diss/4514