Study of the stability of a 3×3 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 biologicalmodel: xn+1=a1xn+b1yne−xn,yn+1=a2yn+b2zne−yn,zn+1=a3zn+b3xne−znwhere a1, a2, a3, b1, b2, b3 are real constants and the initial values conditions x0, y0 and z0 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.