Date of Award
2026
Degree Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics and Applied Mathematical Sciences
First Advisor
Mustafa R. S. Kulenović
Abstract
This thesis will be presented in manuscript format. The first chapter will introduce all preliminary definitions and theorems in differential equations and difference equations that will be referenced in following chapters.
The second chapter will investigate and prove global behavior of some non-autonomous logistic and food-limited logistic differential equations. Consider the differential equations
[Mathematical formulas cannot be viewed here, refer to PDF.]
where N(t) ³ 0 and C > 0, growth rate, r, carrying capacity, K, and stocking or harvesting type, h, are positive w-periodic functions of time, r(t), K(t), and h(t) respectively. We give conditions for the global asymptotic stability of the positive w-periodic solution of this equation. We illustrate our results with simulations that include equations from mathematical biology.
The third chapter will investigate the global dynamics of Darwinian Evolution applied to a food-limited population model. The evolutionary version of a food-limited population model is given as
[Mathematical formulas cannot be viewed here, refer to PDF.]
where r, K, C are positive numbers. We assume that growth rate, r, and the carrying capacity, K, are functions of some trait that evolve over time and affect the global behavior of solutions. We expand on other author’s work and give an analytical biological interpretation for solutions found. We illustrate these findings and give biological relevance for simulations based on mathematical biology.
The fourth chapter will investigate a two gene competitive regulatory network in both the context of ordinary differential equations and difference equations. We investigate the asymptotic behavior of a proposed ordinary differential equation (ODE) model for Genetic Toggle switches using:
[Mathematical formulas cannot be viewed here, refer to PDF.]
where a, b, m, n > 0 and x(t), y(t) ≥ 0. We also investigate the asymptotic behavior of the Euler discretization of this system:
[Mathematical formulas cannot be viewed here, refer to PDF.]
where 1 – h = a1, 1 – k = a2, ah = b1, and bk = b2, which has not been considered before. Then let a1, a2 > (0, 1) and h, k > 0 be steps of discretization. Here x and y represent protein concentrations or genetic activity at a particular time in both genes and a, b, m, n > 0 and p, q > 0 respectively above. We apply the theory of competitive maps to find the basins of attractions of different equilibrium points and period-two solutions of the discretized system.
Recommended Citation
Ferrara, Carmen R., "GLOBAL DYNAMICS OF SOME AUTONOMOUS AND NONAUTONOMOUS CONTINUOUS AND DISCRETE MODELS IN MATH BIOLOGY" (2026). Open Access Dissertations. Paper 4575.
https://digitalcommons.uri.edu/oa_diss/4575