  We study the dynamics of the positive solutions of a second-order,
Ricker-type exponential difference equation with periodic parameters. We find
that qualitatively different dynamics occur depending on whether the period p
of the main parameter is odd or even. If p is odd then periodic and
non-periodic solutions may coexist (with different initial values) if the
amplitude of the periodic parameter is allowed to vary over a sufficiently
large range. But if p is even then all solutions converge to an asymptotically
stable limit cycle of period p if either all the odd-indexed or all the
even-indexed terms of the periodic parameter are less than 2, and the sum of
the even terms does not equal the sum of the odd terms. The key idea in our
analysis is a semiconjugate factorization of the above equation into a
triangular system of two first-order equations.
