Population ecology is mainly based on nonlinear equations of the Lotka-Volterra type, which provide mathematical models for describing the dynamics of interacting species. However, for many interacting populations, these equations entail complex dynamical behavior and unpredictability, generating such difficulties and problematical situations as illustrated by the 'paradox of the plankton' and the 'paradox of enrichment', for instance. A careful analysis shows that an ecosystem is a fundamentally contextual system, hence any formalism describing such systems should incorporate contextuality from the very beginning. But existing approaches are based on classical physics and probability theory, and introduce contextuality as an external effect, so that they cannot generally explain the main peculiarities of ecosystems. Basing ourselves on a contextual formalism elaborated to study microscopic systems in quantum mechanics and including appropriate nonlinear equations, we construct a generalization of the Lotka-Volterra equations for contextual systems, apply these equations to discuss some paradoxical situations encountered in ecology, and propound alternative solutions to those currently existing in the literature.
Population ecology is mainly based on nonlinear equations of the Lotka-Volterra type, which provide mathematical models for describing the dynamics of interacting species. However, for many interacting populations, these equations entail complex dynamical behavior and unpredictability, generating such difficulties and problematical situations as illustrated by the 'paradox of the plankton' and the 'paradox of enrichment', for instance. A careful analysis shows that an ecosystem is a fundamentally contextual system, hence any formalism describing such systems should incorporate contextuality from the very beginning. But existing approaches are based on classical physics and probability theory, and introduce contextuality as an external effect, so that they cannot generally explain the main peculiarities of ecosystems. Basing ourselves on a contextual formalism elaborated to study microscopic systems in quantum mechanics and including appropriate nonlinear equations, we construct a generalization of the Lotka-Volterra equations for contextual systems, apply these equations to discuss some paradoxical situations encountered in ecology, and propound alternative solutions to those currently existing in the literature.
The laws of physics enter the doctrine of population dynamics through various interactions whereby energy and matter, or more broadly, information are exchanged between different species and between species and their natural environment. The dynamics then emerge as a continual process involving many such causal interactions. Quantum physics, on the other hand, shows that the coordination between different processes must obey certain rules. We hypothetically ask: Can this game of species benefit from unique features of quantum mechanics such as entanglement and nonlocality? Here we extend quantum game theory to analyze this question and demonstrate that in certain models describing ecological systems where several predators feed on the same prey, the strength of quantum entanglement between the various species has a profound effect on the asymptotic behavior of the system. For example, if there are sufficiently many predator species who are all equally correlated with their prey, they are all driven to extinction. These results were derived in two ways: by analyzing the asymptotic dynamics of the system, and also by modeling the system as a quantum correlation network, which enabled us to apply methods from network theory in the above scenarios. Several generalizations and applications are discussed.
Physical systems that display competitive non-linear dynamics have played a key role in the development of mathematical models of Nature. Important examples include predator-prey models in ecology, biology, consumer-resource models in economics, and reaction-diffusion equations in chemical reactions. However, as real world systems are embedded in complex environments, where it is difficult or even impossible to control external parameters, quantitative comparison between measurements and simple models remains challenging. This motivates the search for competitive dynamics in isolated physical systems, with precise control. An ideal candidate is laser excitation in dilute atomic ensembles. For example, atoms in highly-excited Rydberg states display rich many-body dynamics including ergodicity breaking, synchronisation and time crystals. Here, we demonstrate predator-prey dynamics by laser excitation and ionisation of Rydberg atoms in a room temperature vapour cell. Ionisation of excited atoms produce electric fields that suppress further excitation. This starves the ionisation process of resource, giving rise to predator-prey dynamics. By comparing our results to the Lotka-Volterra model, we demonstrate that as well applications in non-linear dynamics, our experiment has applications in metrology, and remote sensing of localised plasmas.
Phase-space features of a reduced version of the Toda-like Hamiltonian, H(x,k), written in a form constrained by the condition ∂2H/∂x∂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.
The Lotka-Volterra (LV) dynamics is investigated in the framework of the Weyl-Wigner (WW) quantum mechanics extended to one-dimensional Hamiltonian systems, H(x,k) constrained by the ∂^{2}H/∂x∂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.
The work presented in this thesis considers statistical models of quantum statistics. It sets up a framework to analyse experimental data, which is independent of any particular theory, being careful to eliminate any bias towards the theory of quantum mechanics. This framework allows us to separate the discussion of quantum statistics, which are the predictions of quantum theory, from the theory of quantum mechanics enabling an evaluation of alternative models in reproducing the same statistics. The framework allows one to evaluate the non-classicality of a process or phenomenon, without relying on their definitions in quantum mechanics. The two main non-classical phenomenon analysed in this thesis are contextuality and the weak value. We provide completely statistical definitions for both, which is completely independent of quantum mechanics. The two are defined as statistical properties of experimental data. The presence of contextuality in the data for a set of experiments is shown to pose certain obstructions to the statistical modelling of the data, specifically it is shown to require the state of the system to be updated in a non-Markovian way. A statistical model of this type is presented as proof of existence of statistical models for contextual data. The construction of this model is used to o_er insight into the mechanisms required to construct statistical models for contextual data. Additionally an explicit connection is established between contextuality and classical simulability for any sub-theory of quantum mechanics. This connection is used to evaluate the contextuality found in the qubit-stabilizer sub-theory to calculate a lower bound for the minimum number of classical bits a model would require to simulate the statistics. The weak value was previously thought of an a quintessentially quantum phenomenon. We analyse the dynamics of the process that leads to the weak value using classical Hamiltonian mechanics. A statistical analysis of the experiment explains the physical objects associated with the real and imaginary parts of the weak value and its emergence as a result of post-selection. The weak value calculated using classical Hamiltonian dynamics matches the quantum predictions.
Phase-space features of a reduced version of the Toda-like Hamiltonian H(x,k), written in a form constrained by the condition ∂^{2}H/∂x∂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 nonperturbatively. 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 toward more robust descriptions of quantum patterns in competitive microscopic biosystems.