Date of Award

2016

Degree Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics

Department

Mathematics

First Advisor

Mustafa Kulenovic

Abstract

Consider the difference equation

χn+1 = α + Σki=0 ai χn-I + Σki=0 Σki=0 aij χn-iχn-j / Β + Σki=0 biχn-I+ Σki=0 Σkj bij χn-i χn-j , n = 0, 1, ...

α, β, ai, bi, aij, bij, i, j, = 0, 1, …, k and the initial conditions χi, i ε {-k, …, 0} are nonnegative. We investigate the asymptotic behavior of the solutions of the considered equation. We give simple explicit conditions for the global stability and global asymptotic stability of the zero or positive equilibrium of this equation.

We investigate the global dynamics of several anti-competitive systems of rational difference equations which are special cases of general linear fractional system of the form

χn+1 = χ2n-1 / aχ2n + cχ2n-1 + ƒ , n = 0, 1, 2, …,

where the parameters a, c and f are nonnegative numbers with condition a + e + f > 0 and the initial conditions χ-1, χ0 are arbritary nonnegative numbers such that χ-1 + χ0 > 0.

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