Period ‘bubbling’ in simple ecological models: Pattern and chaos formation in a quartic model

There has been much interest of late in the control of chaos. Particularly it has been recently shown how chaos can be averted in simple ecological models through reverse period doubling, or bubble formation. However, if the assumption that the descending tail of the 1-D map must be strictly descending is relaxed, a new form of chaos may emerge. These observations may be especially important in higher degree models, such as quartic maps, which arise in a variety of ecological applications. The combination of potential period doubling and reverse period doubling lead to cascading bubbles, and the formation of chaos across a broad range of parameter values.