Noncommutative phase-space Lotka-Volterra dynamics: The quantum analog

The Lotka-Volterra (LV) dynamics is investigated in the framework of the Weyl-Wigner (WW) quantum mechanics extended to one-dimensional Hamiltonian systems, $\mathcal{H}(x,k)$ constrained by the ${\partial }^2\mathcal{H}/\partial x\partial k=0$ condition. Supported by the Heisenberg-Weyl noncommutative algebra, where $[x,k]=i$, the canonical variables $x$ and $k$ are interpreted in terms of the LV variables, $y=e^{-x}$ and $z=e^{-k}$, eventually associated with the number of individuals in a closed competitive dynamics: the so-called prey-predator system. The WW framework provides the ground for identifying how classical and quantum evolution coexist at different scales and for quantifying quantum analog effects. Through the results from the associated Wigner currents, (non-)Liouvillian and stationary properties are described for thermodynamic and Gaussian quantum ensembles in order to account for the corrections due to quantum features over the classical phase-space pattern yielded by the Hamiltonian description of the LV dynamics. In particular, for Gaussian statistical ensembles, the Wigner flow framework provides the exact profile for the quantum modifications over the classical LV phase-space trajectories so that Gaussian quantum ensembles can be interpreted as an adequate Hilbert space state configuration for comparing quantum and classical regimes. The generality of the framework developed here extends the boundaries of the understanding of quantumlike effects on competitive microscopical biosystems.

Figure 1

Classical portrait of Lotka-Volterra associated Hamiltonians. Phase-space $x-k$ trajectories (black solid lines) for $\epsilon =6,5,4,3,2.5,2.2,2.1$, and 2.05, and corresponding species distributions (red dashed lines) $z$ and $y$, identified by $y=e^{-x}$ and $z=e^{-k}$.

Figure 2

(a) Internal energy $\mathcal{E}\left(\beta \right)$ and (b) heat capacity $\mathcal{C}\left(\beta \right)$ as function of $\beta$ for classical (dark black lines) and quantum $\left[O\left({\hslash }^2\right)\right]$ (light red lines) stationary regimes. Results are for $a=1/2$ (dotted lines), 1 (dashed lines), 2 (thin lines), and 4 (thick lines).

Figure 3

First column: Classical (dark black contours) and QM corrected (light yellow contours) Wigner function profiles ${\mathcal{W}}_0(x,k;\beta )$ and ${\mathcal{W}}_{ST}^{\left(2\right)}(x,k;\beta )$. The background depicts the quantum modified Wigner function ${\mathcal{W}}_{ST}^{\left(2\right)}(x,k;\beta )$. Second column: Features of the Wigner flow for the TD ensemble on the $x\text{−}k$ plane. Dark blue contour lines are for $J_x^{\left(2\right)}(x,k;\beta )=0$ and light orange contour lines are for $J_k^{\left(2\right)}(x,k;\beta )=0$. The green-yellow (dark-light) vector arrow color scheme shows the [quantum $O\left({\hslash }^2\right)]$ Wigner currents ${\mathcal{J}}^{\left(2\right)}(x,k;\beta )$ with the domains of quantum fluctuations bound by blue (dark) and orange (light) lines. Again, the background depicts the quantum modified Wigner function ${\mathcal{W}}_{ST}^{\left(2\right)}(x,k;\beta )$. Third column: Liouvillian quantifier (in the background) ${\mathbf{\nabla }}_{\xi }·\mathbf{w}$, superposed by the normalized quantum velocity field representation (red arrows) $\mathbf{w}/\left|\mathbf{w}\right|$. The background color scheme (from darker ${\mathbf{\nabla }}_{\xi }·\mathbf{w}\sim 0$ to lighter regions, ${\mathbf{\nabla }}_{\xi }·\mathbf{w}>0$) reinforces the approximated Liouvillian behavior for the axis $x=k$. The results are for $a=1$ (isotropic $x-k$ distribution) and $\beta =0.1$ (first row), 1 (second row), and 5 (third row) from which one notes that the quantum distortions become relevant only for lower temperatures (increasing $\beta$ values).

Figure 4

First column: Wigner currents (vector scheme) ${\mathcal{J}}^{\alpha }$ and stationarity quantifier (background scheme) ${\mathbf{\nabla }}_{\xi }·{\mathcal{J}}^{\alpha }$ for the Gaussian ensemble on the $x\text{−}k$ plane. At $\tau =0$, Gaussian ensembles do not exhibit neither vortices nor stagnation points in a kind of camouflage of the quantum distortions. The magnitude of ${\mathcal{J}}^{\alpha }$ is given by the green-yellow (dark-light) color scheme. The stationarity quantifier ${\mathbf{\nabla }}_{\xi }·\mathcal{J}$, and the modulus of $\mathcal{J}$ are described according to the dark-light blue background and blue-green-yellow (dark-light) vector arrow color schemes, respectively. The results are for the increasing spreading pattern of the Gaussian function from $\alpha =1/\sqrt{2}$ (first row), 1 (second row), and $\sqrt{2}$ (third row). Peaked Gaussians ($\alpha =\sqrt{2}$) localize the quantum distortions which result in nonstationarity. Second column: Normalized quantum velocities (red arrows) $\mathbf{w}$ and the Liouvillian quantifier ${\mathbf{\nabla }}_{\xi }·\mathbf{w}$, depicted through the background color scheme from darker ${\mathbf{\nabla }}_{\xi }·\mathbf{w}\sim 0$ to lighter regions ${\mathbf{\nabla }}_{\xi }·\mathbf{w}>0$. The classical pattern is shown as a collection of black lines.

Figure 5

Phase-space oscillating population portraits impacted by quantum corrections modulated by Gaussian ensembles. The green-yellow (dark-light) color scheme describes the phase-space trajectory evolution $\left[y\right(\tau ),z(\tau \left)\right]$ from $\tau =0$ (dark green tone) to $\tau \gg 0$ (light yellow tone). Plots are for $a=0.9,1$, 5, and $\alpha =0.8$.