Paradox of enrichment: A fractional differential approach with memory.

The paradox of enrichment (PoE) proposed by Rosenzweig [M. Rosenzweig, The paradox of enrichment, Science 171 (1971) 385-387] is still a fundamental problem in ecology. Most of the solutions have been proposed at an individual species level of organization and solutions at community level are lacking. Knowledge of how learning and memory modify behavioral responses to species is a key factor in making a crucial link between species and community levels. PoE resolution via these two organizational levels can be interpreted as a microscopic- and macroscopic-level solution. Fractional derivatives provide an excellent tool for describing this memory and the hereditary properties of various materials and processes. The derivatives can be physically interpreted via two time scales that are considered simultaneously: the ideal, equably flowing homogeneous local time, and the cosmic (inhomogeneous) non-local time. Several mechanisms and theories have been proposed to resolve the PoE problem, but a universally accepted theory is still lacking because most studies have focused on local effects and ignored non-local effects, which capture memory. Here we formulate the fractional counterpart of the Rosenzweig model and analyze the stability behavior of a system. We conclude that there is a threshold for the memory effect parameter beyond which the Rosenzweig model is stable and may be used as a potential agent to resolve PoE from a new perspective via fractional differential equations.