Date of Award
Doctor of Philosophy in Mathematics
This dissertation investigates the dynamics of some second-order difference equations and systems of difference equations whose defining functions satisfy certain monotonicity properties. In each study we utilize the theory for specific classes of monotone difference equations to establish local and global dynamics.
Manuscript 1 is an introduction that provides fundamental definitions and important results for difference equations that are used throughout the rest of the thesis.
Manuscript 2 presents some potential global dynamic scenarios for competitive systems of difference equations in the plane. These results are extended to apply to the class of second-order difference equations whose transition functions are decreasing in the first variable and increasing in the second. In particular, these results are applied to investigate the following equation as a case study:
[Mathematical equations cannot be displayed here, refer to PDF] (1)
where the initial conditions χ-1 and χ0 are arbitrary nonnegative numbers such that the solution is defined and the parameters satisfy C, E, a, d, f ≥ 0, C+E > 0, a + C > 0, and a + d > 0.
A rich collection of additional dynamical behaviors for Equation (1) are established to provide a nearly complete characterization of its global dynamics with the basins of attraction of equilibria and periodic solutions.
[Mathematical equations cannot be displayed here, refer to PDF] (2)
Here f is a continuous function nondecreasing in both arguments, the parameter a is a positive real number, and the initial conditions χ-1 and χ0 are arbitrary nonnegative numbers such that the solution is defined. Local and global dynamics of Equation (2) are presented in the event f is chosen to be a certain type of linear or quadratic polynomial. Particular consideration is given to the existence problem of period-two solutions.
Manuscript 4 presents an order-k generalization of Equation (2),
[Mathematical equations cannot be displayed here, refer to PDF] (3)
where f remains a function nondecreasing in all of its arguments, a > 0, and χ0, χ1,: : : , χ1-k ≥ 0. We examine several interesting examples in which f is a transcendental function. This manuscript establishes conditions under which Equation (3) possesses a unique positive equilibrium that is a global attractor of all solutions with positive initial conditions. In particular, results are presented for the special case in which f(x, : : : , x) is chosen to be a concave function.
Bertrand, Elliott J., "Global Dynamics of Some Competitive and Cooperative Discrete Dynamical Systems" (2018). Open Access Dissertations. Paper 726.