Toda-like Hamiltonian as a probe for quantized prey-predator dynamics

Abstract

Phase-space features of a reduced version of the Toda-like Hamiltonian, $\mathcal{H}(x,\,k)$, written in a form constrained by the condition $\partial^2 \mathcal{H} / \partial x \partial k = 0$, with $x$ and $k$ as canonically conjugate variables, are analyzed in terms of Wigner currents. For Wigner currents convoluted with either thermodynamic or Gaussian ensembles, the underlying Hamiltonian dynamics admits analytic corrections due to quantum distortions over the classical phase-space pattern, computed and interpreted through quantifiers of quantumness and stationarity. Notably, while emulating the Lotka-Volterra (LV) dynamics that describe ecological competition systems, the Toda-like classical dynamics allows for analytical solutions with computable periods corresponding to closed phase-space orbits of isotropic prey-predator population distributions. The essential conditions for understanding how classical and quantum evolution can coexist are provided at different scales of quantumness, driven by the associated convoluting ensemble parameter. In the case of Gaussian statistical ensembles, the exact profile of the quantum distortions over classical prey-predator phase-space trajectories is obtained non-perturbatively. Our results indicate that, besides the classical stability admitted by LV models, the Toda-like patterns also exhibit quantum stability. Therefore, this can be regarded as the first step as a predictive theoretical framework towards more robust descriptions of quantum patterns in competitive microscopic biosystems.

Figures (6)

• Figure 1: (Color online) Classical portrait of LV (solid lines) and Toda-like (dashed lines) associated Hamiltonians. The phase-space $x-k$ trajectories (black thin lines) are for $\epsilon= 2.5,\, 2.2,\, 2.1,\, 2.05$ and $2.01$, and the corresponding species distributions (red thick lines), $z$ and $y$, are identified by $y=e^{-x}$ and $z=e^{-k}$. The values of $\epsilon$ are chosen to denote the consistence with Eq. \ref{['HamBm']}.
• Figure 2: (Color online) Time dependence of the parameter $\mathcal{T}(\tau)$ for the classical Toda-like dynamical system. Results are are for $\epsilon=6,\, 4,\, 2.5$ and $2.1$.
• Figure 3: (Color online) Internal energy (first plot), $\mathcal{E}(\beta)$, and heat capacity (second plot), $\mathcal{C}(\beta)$, as function of $\beta$, for classical (black lines) and quantum ( $\mathcal{O}(\hbar^2)$) (red lines) stationary regimes. The results are for $a=1/2$ (dotted lines), $1$ (dashed lines), ${2}$ (thin lines) and $4$ (thick lines).
• Figure 4: (Color online) First column: Phase-space picture of the Wigner currents, in a vector scheme, and of the stationarity quantifier, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \hbox{\boldmath $\mathcal{J}^{\alpha}$}$, in a background color scheme, for Gaussian ensembles. The magnitude of $\hbox{\boldmath $\mathcal{J}^{\alpha}$}$ is given by the greenyellow color scheme. The results are for the increasing values of $\alpha$, from $\alpha =1/\sqrt{2}$ (first row), $1$ (second row) and $\sqrt{2}$ (third row). Peaked Gaussians ($\alpha =\sqrt{2}$) localize and discretize the quantum distortions by delimiting well-defined domain walls, which a centered non-stationarity and non-liovillianity pattern. Spread Gaussians ($\alpha =1/\sqrt{2}$), in some sense, destroy the quantum pattern as they approach the classical limit. Second column: The normalized velocities (red arrows), $\mathbf{w}$, and the Liouvillian quantifier (background color scheme), $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w}$. The Liouvillian pattern runs from darker regions, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w} \sim 0$, to lighter regions, $\hbox{\boldmath $\nabla$}_{\xi} \cdot \mathbf{w} > 0$. The classical pattern is shown as a collection of black lines. The quantum results are just for the departing configurations at $\tau=0$.
• Figure 5: (Color online) Region plot scheme for the phase-space evolution of quantum critical points corresponding to classical equivalent (blue regions) equilibrium points and the subsets of locally stable (white to red regions) equilibrium points driven by the Gaussian spreading parameter, $\alpha$, from $0$ (blue tones) to $2.7$ (red tones) and with $a=4$. Results are for flux surrounding envelops with boundaries identified by $\vert\mathbf{w}\vert < 0.08$. Quantum effects emerge thorough a compensation rate where either two vortices of opposite winding numbers match each other or saddle points that mutually annihilates one each other. The spreading behavior of the Gaussian ensemble, from red "bubble" (unstable) islands to the blue ( quasi-stable) envelop, corresponding to decreasing values of $\alpha$, diffusively recovers the classical-like pattern for which the effective quantum imprints are described in Fig. \ref{['Bio03-DM']} just as an equivalent prey-predator dephasing effect.