  We explore the concept of emergent quantum-like theory in complex adaptive
systems, and examine in particular the concrete example of such an emergent (or
"mock") quantum theory in the Lotka-Volterra system. In general, we investigate
the possibility of implementing the mathematical formalism of quantum mechanics
on classical systems, and what would be the conditions for using such an
approach. We start from a standard description of a classical system via
Hamilton-Jacobi (HJ) equation and reduce it to an effective Schr\"odinger-type
equation, with a (mock) Planck constant $\mockbar$, which is system-dependent.
The condition for this is that the so-called quantum potential VQ, which is
state-dependent, is cancelled out by some additional term in the HJ equation.
We consider this additional term to provide for the coupling of the classical
system under consideration to the "environment." We assume that a classical
system could cancel out the VQ term (at least approximately) by fine tuning to
the environment. This might provide a mechanism for establishing a stable,
stationary states in (complex) adaptive systems, such as biological systems. In
this context we emphasize the state dependent nature of the mock quantum
dynamics and we also introduce the new concept of the mock quantum, state
dependent, statistical field theory. We also discuss some universal features of
the quantum-to-classical as well as the mock-quantum-to-classical transition
found in the turbulent phase of the hydrodynamic formulation of our proposal.
In this way we reframe the concept of decoherence into the concept of "quantum
turbulence," i.e. that the transition between quantum and classical could be
defined in analogy to the transition from laminar to turbulent flow in
hydrodynamics.
