Many biological populations breed seasonally and have nonoverlapping generations, so that their dynamics are described by 1st-order difference equations, Nt+1 = F(Nt). In many cases, F(N) as a function of N will have a hump. As such a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes". A detailed account is given of the underlying mathematics of this process and other situations reviewed (in 2- and higher dimensional systems, or in differential equation systems) where apparently random dynamics can arise from bifurcation processes. This complicated behavior, in simple deterministic models, can have disturbing implications for the analysis and interpretation of biological data.