Because these questions are so open-ended there are many different
possible answers. The logistic models
p(n + 1) = a (1 - 0.001 p(n)) p(n)
with various different values for the constant a are a rich
source of examples. If a <= 3.0 then there are no cycles.
When a > 3.0 there are always 2-cycles. As a grows
more cycles appear. However, a must always be less than 4.0.
Otherwise these models can predict negative population, which doesn't
make sense for population models. You may also look at other models, for
example, pack hunter models for additional examples.