This paper presents a mathematical model based on stochastic differential equations (SDEs) to depict the dynamics of a predator-prey system in an aquatic environment characterized by schooling behavior among the prey. The model employs a particle-like approach, incorporating attractive and repulsive forces, akin to phenomena observed in molecular physics, to capture the interactions among the constituent units. Two hunting tactics of the predator, center-attacking and nearest-attacking strategies, are integrated into the model. Numerical simulations of this model unveil four distinct predator-avoidance patterns exhibited by schooling prey: Split and Reunion, Split and Separate into Two Groups, Scattered, and Maintain Formation and Distance. Our results also confirm the effectiveness of large groups of schooling prey in mitigating predation risk, consistent with real-life observations in natural aquatic ecosystems. These findings validate the accuracy and applicability of our model.

This paper presents a bounded confidence gossip algorithm for describing the process of opinion formation over a communication network. Each agent in the network keeps a time-varying opinion vector (or state) which represents its opinion about a set of matters. A common confidence threshold is set for all of the agents. The states of agents in the network will be updated time by time according to an iterative procedure: At each time, (i) one agent is chosen randomly, then it chooses one of its neighbors on the communication graph to contact with; (ii) they exchange their states; and (iii) if they have different states and the distance between t eir states is strictly smaller than the confidence threshold, they update their states as the average of the two. This algorithm converges almost surely to some equilibrium point such that any two adjacent agents either have the same state or have distinct states whose distance is no less than the confidence threshold. This is called the constant confidence threshold algorithm. An increasing confidence threshold algorithm, which repeats the constant confidence threshold algorithm several times with increasing confidence threshold, is also proposed. The algorithm is also convergent almost surely to some equilibrium point. Applicability of the method to clustering problems is shown through numerical examples.

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 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 no parity decomposition for the eigenvalues. In connection with numerically exact studies and recent theoretical results, 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, reflecting the hidden symmetry of the AQRM. 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.

A stochastic forest model of young and old age class trees is studied. First, we prove existence, uniqueness and boundedness of global nonnegative solutions. Second, we investigate asymptotic behavior of solutions by giving a sufficient condition for sustainability of the forest. Under this condition, we show existence of a Borel invariant measure. Third, we present several sufficient conditions for decline of the forest. Finally, we give some numerical examples.