The boundary of “life” for a self-organized critical evolution: the role of the interaction range

The dynamics of a tree-like evolution is investigated as a function of the range k of the interactions between competing entities which are located at the extremities of the branches. Speciation (branching) events are supposed to be driven by extremal dynamics. Extinction events are allowed and controlled by a parameter r. A transition between self-organized critical and frozen evolution occurs at some well-defined critical value rc(k). Surprisingly, the critical rc value behaves as a power of the range k (rc k−δ) with an exponent δ = −0.46±0.03. Moreover, the asymptotic case k = +∞ is herein exactly solved. The dynamics for k = +∞ is not critical and does not present any transition in contrast with finite k cases.