Study of the stability of a 3 x 3 system of difference equations using Centre Manifold Theory

We study the stability of the zero equilibrium of the following system of difference equations, which is a natural extension of an one-dimensional biological model: Xn+1 = a(1)X(n) + b(1)y(n)e(-x)n, Yn+1 = a(2)y(n) + b(2)Z(n)e(-y)n, z(n+1) = a(3)z(n) + b(3)x(n)e(-z)n where a(1), a(2), a(3), b(1), b(2), b(3) are real constants and the initial values conditions x(0), y(0) and z(0) are real numbers. The stability of those systems in the special case when one of the eigenvalues has absolute value equal to 1 and the other two eigenvalues have absolute value less than 1, using centre manifold theory, is investigated. (C) 2016 Elsevier Ltd. All rights reserved.