Wada basins and qualitative unpredictability in ecological models: a graphical interpretation

The capability of ecological models to make precise predictions was questioned with the discovery of chaos. Here it is shown that an alternative form of unpredictability is associated with some nonlinear models. The notion of a Wada basin, in which three or more basins share complexly interdigitated boundaries, represents this new form of unpredictability. It signifies that a single point seemingly balanced between two basins of attraction may in fact just as easily travel to a third, seemingly unconnected, basin. A circle map approximation to coupled predator prey pairs is used to demonstrate certain qualitative properties associated with the equilibria within the basins. Most important is the demonstration that the ultimate basin in which a trajectory comes to lie may not be confidently predicted from detailed knowledge of the point of initiation. Furthermore, when Wada basin boundaries become large, a small amount of stochastic forcing may create chaos-like behavior (though not formally chaotic) in a system that has, mathematically, only stable equilbria. The generation of Wada basins is discussed, using piecewise linear maps.