A Stochastic Differential Equation Model for Predator-Avoidance Fish Schooling

This paper presents a system of stochastic differential equations (SDEs) as mathematical model to describe the spatial-temporal dynamics of predator-prey system in an artificial aquatic environment with schooling behavior imposed upon the associated prey. The proposed model follows the particle-like approach where interactions among the associated units are manifested through combination of attractive and repulsive forces analogous to the ones occurred in molecular physics. Two hunting tactics of the predator are proposed and integrated into the general model, namely the center-attacking and the nearest-attacking strategy. Emphasis is placed upon demonstrating the capacity of the proposed model in: (i) discovering the predator-avoidance patterns of the schooling prey, and (ii) showing the benefit of constituting large prey school in better escaping the predator's attack. Based on numerical simulations upon the proposed model, four predator-avoidance patterns of the schooling prey are discovered, namely Split and Reunion, Split and Separate into Two Groups, Scattered, and Maintain Formation and Distance. The proposed model also successfully demonstrates the benefit of constituting large group of schooling prey in mitigating predation risk. Such findings are in agreement with real-life observations of the natural aquatic ecosystem, hence confirming the validity and exactitude of the proposed model

Spacing distribution for quantum Rabi models

The asymmetric quantum Rabi model (AQRM) is a fundamental model in quantum optics describing the interaction of light and matter. Besides its immediate physical interest, the AQRM possesses an intriguing mathematical structure which is far from being completely understood. In this paper, we focus on the distribution of the level spacing, the difference between consecutive eigenvalues of the AQRM in the limit of high energies, i.e. large quantum numbers. In the symmetric case, that is the quantum Rabi model (QRM), the spacing distribution for each parity (given by the $\mathbb{Z}_2$-symmetry) is fully clarified by an asymptotic expression derived by de Monvel and Zielinski, though some questions remain for the full spectrum spacing. However, in the general AQRM case, there is neither a parity decomposition or an asymptotic expression for the eigenvalues. In connection with numerically exact studies for the first 40,000 eigenstates we describe the spacing distribution for the AQRM which is characterized by a new type of periodicity and symmetric behavior of the distribution with respect to the bias parameter. The results reflects the hidden symmetry of the AQRM known to appear for half-integer bias. In addition, we observe in the AQRM the excited state quantum phase transition for large values of the bias parameter, analogous to the QRM with large qubit energy, and an internal symmetry of the level spacing distribution for fixed bias. This novel symmetry is independent from the symmetry for half-integer bias and not explained by current theoretical knowledge.Comment: 29 pages. 15 figures. Improved presentation and Remark 4.3 was adde