A second-order stochastic model describing a large scale crowd is formulated, and an efficient evacuation strategy for agents in complex surroundings is proposed and solved numerically. The method consists in reshaping the crowd contour by making use of the crowd density feedback that is commonly available from geolocation technologies, and Kantorovich distance is used to transport the current shape into the desired one. The availability of the crowd density enables to solve the otherwise challenging forward-backward problem. Using this approach, we demonstrate via numerical results that the crowd migrates through the complex environment as designed.

The term “Habitat Complexity” is used to measure the coupling level between different species in an ecosystem. It plays an important role to an ecosystem in which two or more species are interacting with each other. In the present paper, effects of the habitat complexity on predator–prey ecosystems with or without noise disturbances are investigated. Stochastic analysis methods of the Ito differential equations, the stochastic averaging method, and the Monte Carlo simulation are applied to obtain the stationary probability densities of the predator and prey population densities. It is found that the effects of the habitat complexity on the system dynamical behaviors are quite different for the deterministic case and the stochastic case. In absence of the noises, there are two bifurcation points dividing the level of habitat complexity into three regions: weak, moderate and strong. In the three regions, limit cycles, co-existing equilibrium states, and predator-extinct equilibrium states are present, respectively. In the stochastic case, the stable invariant measures in terms of probability distributions replace both the limit cycles and coexisting equilibriums in the regions of weak and moderate habitat complexity. For the case of strong habitat complexity, although the predator remains extinct, the prey is not in an equilibrium state, but possesses a probability distribution.