  Combined with all density-dependent factors, the per capita growth rate of a
species may be non-monotonic. One important consequence is that species may
suffer from weak Allee effects or strong Allee effects. In this paper, we study
the permanence of a discrete-time two-species-interaction model with
non-monotonic per capita growth rates for the first time. By using the average
Lyapunov functions and extending the ecological concept of the relative
nonlinearity, we find a simple sufficient condition for guaranteeing the
permanence of systems that can model complicated two-species interactions. The
extended relative nonlinearity allows us to fully characterize the effects of
nonlinearities in the per capita growth functions with non-monotonicity. These
results are illustrated with specific two species competition and predator-prey
models of generic forms with non-monotone per capita growth rates.
