Search dynamics at the edge of extinction: Anomalous diffusion as a critical survival state

¹  Instituto de Física, Universidade Federal de Alagoas - 57072-970, Maceió-AL, Brazil
²  Departamento de Física Teórica e Experimental, Universidade Federal do Rio Grande do Norte, 59072-970, Natal-RN, Brazil
³  Departamento de Física, Universidade Federal do Paraná - 81531-990, Curitiba-PR, Brazil
⁴  Laboratório de Física Teórica e Computacional, Departamento de Física, Universidade Federal de Pernambuco, 50670-901, Recife-PE, Brazil

Received: 14 July 2006
Accepted: 28 November 2006

Abstract

We investigate the general problem of autonomous random walkers whose sole source of energy are search targets that are themselves diffusing random walkers. We study how the energy accumulated by the searcher varies with the target density via numerical simulations and compare the results with an analytical model for fixed targets. We report that superdiffusion of either searcher or target confers substantial energetic advantages to the former. While superdiffusion may not play a crucial role for high target densities, in contrast it confers a vital advantage in the limit of low densities at the edge of extinction: diffusive searchers rapidly die but superdiffusive searchers can survive for long periods without entering into the extinction state. The validity and relevance of our findings in broader contexts are also discussed.