# Global Dynamics of Some Competitive and Cooperative Discrete Dynamical Systems

#### Abstract

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: ^ *x*_{n+1} = *Cx*_{n-1}^{ 2} + *Ex*_{n-1} / *ax*_{n }^{2} + *dx*_{n} + *f*, * n* = 0,1,…, where the initial conditions *x*_{ -1} and *x*_{0} 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. ^ Manuscript 3 considers the following second-order generalization of the classical Beverton-Holt equation: *x*_{n+1} = * af*(*x _{n}*,

*x*

_{n-1})/1+

*f*(

*x*,

_{n}*x*

_{n-1}),

*n*= 0,1,…. Here

*f*is a continuous function nondecreasing in both arguments, the parameter

*a*is a positive real number, and the initial conditions

*x*

_{-1}and

*x*

_{0}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),

*x*

_{n+1}=

*af*(

*x*,

_{ n}*x*

_{n-1},…,

*x*

_{ n+1-k})/1 +

*f*(

*x*,

_{n}*x*

_{n-1},…,

*x*

_{n+1-k},

*n*= 0,1,…,

*k*≥ 1, where

*f*remains a function nondecreasing in all of its arguments,

*a*> 0, and x

_{0},

*x*

_{-1},…,

*x*

_{1-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.^

#### Subject Area

Mathematics

#### Recommended Citation

Elliott J Bertrand,
"Global Dynamics of Some Competitive and Cooperative Discrete Dynamical Systems"
(2018).
*Dissertations and Master's Theses (Campus Access).*
Paper AAI10784985.

https://digitalcommons.uri.edu/dissertations/AAI10784985