Narrow Results

In this paper, we study the stationary and dynamical properties of three-dimensional trapped Bose–Einstein condensates with attractive interactions subjected to a random potential. To this end, a variational method is applied to solve the underlying Gross–Pitaevskii equation. We derive analytical predictions for the energy, the equilibrium width, and evolution laws of the condensate parameter. The breathing mode oscillations’ frequency of the condensate has been also calculated in terms of the gas and disorder parameters. We analyze in addition the dynamics of collapse from the Gaussian approximation. Surprisingly, we find that the intriguing interplay of the attractive interaction and disorder effects leads to prevent collapse of the condensate.

In this paper, a class of three-species multi-delay Lotka–Volterra ratio-dependent predator–prey model with feedback controls and shelter for the prey is considered. A set of easily verifiable sufficient conditions which guarantees the permanence of the system and the global attractivity of positive solution for the predator–prey system are established by developing some new analysis methods and using the theory of differential inequalities as well as constructing a suitable Lyapunov function. Furthermore, some conditions for the existence, uniqueness and stability of positive periodic solution for the corresponding periodic system are obtained by using the fixed point theory and some new analysis techniques. In addition, some numerical solutions of the equations describing the system are given to show that the obtained criteria are new, general, and easily verifiable. Finally, we still solve numerically the corresponding stochastic predator–prey models with multiplicative noise sources, and obtain some new interesting dynamical behaviors of the system. At the same time, the influence of the delays and shelters on the dynamics behavior of the system is also considered by solving numerically the predator–prey models.