  In many non-linear systems, such as plasma oscillation, boson condensation,
chemical reaction, and even predatory-prey oscillation, the coarse-grained
dynamics are governed by an equation containing anti-symmetric transitions,
known as the anti-symmetric Lotka-Volterra (ALV) equations. In this work, we
prove the existence of a novel bifurcation mechanism for the ALV equations,
where the equilibrium state can be drastically changed by flipping the
stability of a pair of fixed points. As an application, we focus on the
implications of the bifurcation mechanism for evolutionary networks; we found
that the bifurcation point can be determined quantitatively by the microscopic
quantum entanglement. The equilibrium state can be critically changed from one
type of global demographic condensation to another state that supports global
cooperation for homogeneous networks. In other words, our results indicate that
there exist a class of many-body systems where the macroscopic properties are
invariant with a certain amount of microscopic entanglement, but they can be
changed abruptly once the entanglement exceeds a critical value. Furthermore,
we provide numerical evidence showing that the emergence of bifurcation is
robust against the change of the network topologies, and the critical values
are in good agreement with our theoretical prediction. These results show that
the bifurcation mechanism could be ubiquitous in many physical systems, in
addition to evolutionary networks.
