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Hong Qian - One of the best experts on this subject based on the ideXlab platform.
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a thermodynamic theory of ecology helmholtz theorem for lotka volterra Equation extended conservation law and stochastic predator prey dynamics
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2015Co-Authors: Hong QianAbstract:We carry out mathematical analyses, a la Helmholtz’s and Boltzmann’s 1884 studies of monocyclic Newtonian dynamics, for the Lotka–Volterra (LV) Equation exhibiting predator–prey oscillations. In doing so, a novel ‘thermodynamic theory’ of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic Equation (e.g. the LV Equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter α and in initial conditions: Helmholtz’s theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyse the relationships among mean ecological activeness θ , quantities characterizing dynamic ranges of populations A and α , and the ecological force F α . The analyses identify an entire orbit as a stationary ecology, and establish the notion of an ‘Equation of ecological states’. Studies of the stochastic dynamics with finite populations show the LV Equation as the robust, fast cyclic underlying behaviour. The mathematical narrative provides a novel way of capturing long-term dynamical behaviours with an emergent conservative ecology .
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a thermodynamic theory of ecology helmholtz theorem for lotka volterra Equation extended conservation law and stochastic predator prey dynamics
arXiv: Mathematical Physics, 2014Co-Authors: Hong QianAbstract:We carry out mathematical analyses, {\em \`{a} la} Helmholtz's and Boltzmann's 1884 studies of monocyclic Newtonian dynamics, for the Lotka-Volterra (LV) Equation exhibiting predator-prey oscillations. In doing so a novel "thermodynamic theory" of ecology is introduced. An important feature, absent in the classical mechanics, of ecological systems is a natural stochastic population dynamic formulation of which the deterministic Equation (e.g., the LV Equation studied) is the infinite population limit. Invariant density for the stochastic dynamics plays a central role in the deterministic LV dynamics. We show how the conservation law along a single trajectory extends to incorporate both variations in a model parameter $\alpha$ and in initial conditions: Helmholtz's theorem establishes a broadly valid conservation law in a class of ecological dynamics. We analyze the relationships among mean ecological activeness $\theta$, quantities characterizing dynamic ranges of populations $\mathcal{A}$ and $\alpha$, and the ecological force $F_{\alpha}$. The analyses identify an entire orbit as a stationary ecology, and establish the notion of "Equation of ecological states". Studies of the stochastic dynamics with finite populations show the LV Equation as the robust, fast cyclic underlying behavior. The mathematical narrative provides a novel way of capturing long-term dynamical behaviors with an emergent {\em conservative ecology}.
Erwin Frey - One of the best experts on this subject based on the ideXlab platform.
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Topologically robust zero-sum games and Pfaffian orientation: How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra Equation
Physical Review E, 2018Co-Authors: Philipp Geiger, Johannes Knebel, Erwin FreyAbstract:To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra Equation (ALVE). The ALVE is the replicator Equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio phi = 1.6180 ..., cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.
Philipp Geiger - One of the best experts on this subject based on the ideXlab platform.
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Topologically robust zero-sum games and Pfaffian orientation: How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra Equation
Physical Review E, 2018Co-Authors: Philipp Geiger, Johannes Knebel, Erwin FreyAbstract:To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra Equation (ALVE). The ALVE is the replicator Equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio phi = 1.6180 ..., cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.
Johannes Knebel - One of the best experts on this subject based on the ideXlab platform.
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Topologically robust zero-sum games and Pfaffian orientation: How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra Equation
Physical Review E, 2018Co-Authors: Philipp Geiger, Johannes Knebel, Erwin FreyAbstract:To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra Equation (ALVE). The ALVE is the replicator Equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio phi = 1.6180 ..., cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.
Frey Erwin - One of the best experts on this subject based on the ideXlab platform.
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Topological phase transition in coupled rock-paper-scissor cycles
'American Physical Society (APS)', 2020Co-Authors: Knebel Johannes, Geiger, Philipp M., Frey ErwinAbstract:A hallmark of topological phases is the occurrence of topologically protected modes at the system`s boundary. Here we find topological phases in the antisymmetric Lotka-Volterra Equation (ALVE). The ALVE is a nonlinear dynamical system and describes, e.g., the evolutionary dynamics of a rock-paper-scissors cycle. On a one-dimensional chain of rock-paper-scissor cycles, topological phases become manifest as robust polarization states. At the transition point between left and right polarization, solitonic waves are observed. This topological phase transition lies in symmetry class $D$ within the "ten-fold way" classification as also realized by 1D topological superconductors
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Topologically robust zero-sum games and Pfaffian orientation -- How network topology determines the long-time dynamics of the antisymmetric Lotka-Volterra Equation
'American Physical Society (APS)', 2018Co-Authors: Geiger, Philipp M., Knebel Johannes, Frey ErwinAbstract:To explore how the topology of interaction networks determines the robustness of dynamical systems, we study the antisymmetric Lotka-Volterra Equation (ALVE). The ALVE is the replicator Equation of zero-sum games in evolutionary game theory, in which the strengths of pairwise interactions between strategies are defined by an antisymmetric matrix such that typically some strategies go extinct over time. Here we show that there also exist topologically robust zero-sum games, such as the rock-paper-scissors game, for which all strategies coexist for all choices of interaction strengths. We refer to such zero-sum games as coexistence networks and construct coexistence networks with an arbitrary number of strategies. By mapping the long-time dynamics of the ALVE to the algebra of antisymmetric matrices, we identify simple graph-theoretical rules by which coexistence networks are constructed. Examples are triangulations of cycles characterized by the golden ratio $\varphi = 1.6180...$, cycles with complete subnetworks, and non-Hamiltonian networks. In graph-theoretical terms, we extend the concept of a Pfaffian orientation from even-sized to odd-sized networks. Our results show that the topology of interaction networks alone can determine the long-time behavior of nonlinear dynamical systems, and may help to identify robust network motifs arising, for example, in ecology.Comment: 43 pages, 12 figure
