On a piecewise-smooth map arising in ecology

In this paper, we study a two-dimensional piecewise smooth map arising in ecology. Such map, containing two parameters d and β, is derived from a model describing how masting of a mature forest happens and synchronizes. Here d is the energy depletion quantity and β   is the coupling strength. Our main results are the following. First, we obtain a “weak” Sharkovskii ordering for the map on its nondiagonal invariant region for a certain set of parameters. In particular, we show that its Sharkovskii ordering is the natural number (resp., the positive even number) for β>1$\beta >1$ (resp., 0<β<1$0<\beta <1$). Second, we obtain a region of parameter space for which its corresponding global dynamics can be completely characterized.