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

Area of Science:

• Theoretical Physics
• Quantum Mechanics
• Mathematical Biology

Background:

• Phase-space dynamics are crucial for understanding classical and quantum systems.
• The Toda-like Hamiltonian and Lotka-Volterra (LV) models offer frameworks for studying complex dynamics.
• Wigner currents provide a tool to analyze quantum features in phase space.

Purpose of the Study:

• To analyze phase-space features of a reduced Toda-like Hamiltonian using Wigner currents.
• To investigate quantum distortions and their effect on classical dynamics.
• To explore the coexistence of classical and quantum evolution in ecological competition models.

Main Methods:

• Analysis of a reduced Toda-like Hamiltonian (H(x,k)) with ∂^{2}H/∂x∂k=0.
• Convolution of Wigner currents with thermodynamic or Gaussian ensembles.
• Computation and interpretation of quantum distortions using quantifiers of quantumness and stationarity.
• Nonperturbative derivation of quantum distortions for Gaussian ensembles.

Main Results:

• Analytic corrections to Hamiltonian dynamics due to quantum distortions were identified.
• The Toda-like dynamics emulate Lotka-Volterra (LV) dynamics, yielding analytical solutions for closed phase-space orbits.
• Both classical and quantum stability were observed in the Toda-like patterns, extending classical LV model stability.
• Conditions for the coexistence of classical and quantum evolution were established.

Conclusions:

• The Toda-like Hamiltonian, when emulating LV dynamics, exhibits both classical and quantum stability.
• This work presents a predictive theoretical framework for quantum patterns in competitive microscopic biosystems.
• The study bridges classical ecological models with quantum mechanics, offering new insights into complex system behavior.