A Simple Population Model with Qualitatively Uncertain Dynamics

Basins of attraction of nonlinear systems can be “riddled”; arbitrarily close to any point in a riddled basin are neighboring points which go to a different attractor. I present two chaotically forced single-species population models with riddled basins of attraction. As a result of this complex basin structure, the ultimate survival of these populations is effectively unpredictable. Riddled basins produce a level of unpredictability qualitatively greater than the familiar sensitive dependence on initial conditions within a single chaotic attractor, or the unpredictability caused by multiple attractors with fractal basin boundaries.