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 ii 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 r(c)(k). Surprisingly, the critical r(c) value behaves as a power of the range k(r(c) similar to k(-delta)) with an exponent S = -0.46+/-0.03. Moreover, the asymptotic case k = +infinity is herein exactly solved. The dynamics for k = +infinity is not critical and does not present any transition in contrast with finite k cases.