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Quantum Prey–Predator Dynamics: A Gaussian Ensemble Analysis

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Abstract

Quantum frameworks for modeling competitive ecological systems and self-organizing structures have been investigated under multiple perspectives yielded by quantum mechanics. These comprise the description of the phase-space prey–predator competition dynamics in the framework of the Weyl–Wigner quantum mechanics. In this case, from the classical dynamics described by the Lotka–Volterra (LV) Hamiltonian, quantum states convoluted by statistical gaussian ensembles can be analytically evaluated. Quantum modifications on the patterns of equilibrium and stability of the prey–predator dynamics can then be identified. These include quantum distortions over the equilibrium point drivers of the LV dynamics which are quantified through the Wigner current fluxes obtained from an onset Hamiltonian background. In addition, for gaussian ensembles highly localized around the equilibrium point, stability properties are shown to be affected by emergent topological quantum domains which, in some cases, could lead either to extinction and revival scenarios or to the perpetual coexistence of both prey and predator agents identified as quantum observables in microscopic systems. Conclusively, quantum and gaussian statistical driving parameters are shown to affect the stability criteria and the time evolution pattern for such microbiological-like communities.

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Notes

  1. Akin to the statistical QM, the WW phase-space framework recovers the QM probabilistic interpretation through its marginal distributions that correspond to position and momentum probability densities given by

    $$\begin{aligned} \vert \psi _x(x)\vert ^2 = \int ^{+\infty }_{-\infty } dx\,{\mathcal {W}}(x,\, k) \leftrightarrow \vert \psi _k(k)\vert ^2 = \int ^{+\infty }_{-\infty } dk\,{\mathcal {W}}(x,\, k), \quad \text{ with }\quad \psi _x(x) = \frac{1}{2\pi }\int ^{+\infty }_{-\infty } dk\,\exp {\left[ -i \, k \,x\right] }\, \psi _k(k), \end{aligned}$$
    (4)

    which, at 1-dim, are straightforwardly encompassed by the Heisenberg–Weyl algebra in the form of a position-momentum non-commutative relation, \([x,\,k] = i\).

  2. Related to the powers of \(\hbar\), \(\hbar ^{2\eta }\), suppressed from the dimensionless notation.

  3. As previously pointed out, a similar analysis can be carried out for the thermodynamic statistical ensemble interpretation up to \({\mathcal {O}}(\hbar ^2)\) in the series expansion, Eqs. (7) and (8).

  4. After noticing that

    $$\begin{aligned} \sum _{\eta =0}^{\infty }{{h}}_{2\eta +1} (\alpha \chi )\frac{s^{2\eta +1}}{(2\eta +1)!} = \sinh (2s\,\alpha \chi ) \exp \big [-s^2\big ]. \end{aligned}$$
    (15)

  5. In terms of gaussian error functions, \({{Erf}}[\dots ]\).

  6. In fact, as it can be numerically verified for any gaussian configuration with \(\alpha < 1\).

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Acknowledgements

The work of AEB is supported by the Brazilian Agencies FAPESP (Grant No. 2020/01976-5 and Grant No. 2023/00392-8, São Paulo Research Foundation (FAPESP)) and CNPq (Grant No. 301485/2022-4).

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Authors and Affiliations

  1. Departamento de Física, Universidade Federal de São Carlos, PO Box 676, 13565-905, São Carlos, SP, Brazil

    A. E. Bernardini

  2. Departamento de Física e Astronomia, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    O. Bertolami

  3. Centro de Física do Porto, Rua do Campo Alegre 687, 4169-007, Porto, Portugal

    O. Bertolami

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Bernardini, A.E., Bertolami, O. Quantum Prey–Predator Dynamics: A Gaussian Ensemble Analysis. Found Phys 53, 63 (2023). https://doi.org/10.1007/s10701-023-00703-z

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