  Non-equilibrium and instability features of prey-predator-like systems
associated to topological quantum domains emerging from a quantum phase-space
description are investigated in the framework of the Weyl-Wigner quantum
mechanics. Reporting about the generalized Wigner flow for one-dimensional
Hamiltonian systems, $\mathcal{H}(x,\,k)$, constrained by $\partial^2
\mathcal{H} / \partial x \, \partial k = 0$, the prey-predator dynamics driven
by Lotka-Volterra (LV) equations is mapped onto the Heisenberg-Weyl
non-commutative algebra, $[x,\,k] = i$, where the canonical variables $x$ and
$k$ are related to the two-dimensional LV parameters, $y = e^{-x}$ and $z =
e^{-k}$. From the non-Liouvillian pattern driven by the associated Wigner
currents, hyperbolic equilibrium and stability parameters for the
prey-predator-like dynamics are then shown to be affected by quantum
distortions over the classical background, in correspondence with
non-stationarity and non-Liouvillianity properties quantified in terms of
Wigner currents and Gaussian ensemble parameters. As an extension, considering
the hypothesis of discretizing the time parameter, non-hyperbolic bifurcation
regimes are identified and quantified in terms of $z-y$ anisotropy and Gaussian
parameters. The bifurcation diagrams exhibit, for quantum regimes, chaotic
patterns highly dependent on Gaussian localization. Besides exemplifying a
broad range of applications of the generalized Wigner information flow
framework, our results extend, from the continuous (hyperbolic regime) to
discrete (chaotic regime) domains, the procedure for quantifying the influence
of quantum fluctuations over equilibrium and stability scenarios of LV driven
systems.
