Abstract. We consider a collective behavior model in which individuals try to imitate each others' velocity and have a preferred speed. We show that a phase change phenomenon takes place as diffusion decreases, bringing the system from a "disordered" to an "ordered" state. This effect is related to recently noticed phenomena for the diffusive Vicsek model. We also carry out numerical simulations of the system and give further details on the phase transition.
In this paper, we study simulations of the schooling and swarming behavior of a mathematical model for the motion of pelagic fish. We use a derivative of a discrete model of interacting particles originated by Vicsek, Czirók et al. [6] [5] [23] [24]. Recently, a system of ODEs was derived from this model [2], and using these ODEs, we find transitory and long-term behavior of the discrete system. In particular, we numerically find stationary, migratory, and circling behavior in both the discrete and the ODE model and two types of swarming behavior in the discrete model. The migratory solutions are numerically stable and the circling solutions are metastable. We find a stable circulating ring solution of the discrete system where the fish travel in opposite directions within an annulus. We also find the origin of noise-driven swarming when repulsion and attraction are absent and the fish interact solely via orientation.
We present an agent-based model to simulate gang territorial development motivated by graffiti marking on a two-dimensional discrete lattice. For simplicity, we assume that there are two rival gangs present, and they compete for territory. In this model, agents represent gang members and move according to a biased random walk, adding graffiti with some probability as they move and preferentially avoiding the other gang's graffiti. All agent interactions are indirect, with the interactions occurring through the graffiti field. We show numerically that as parameters vary, a phase transition occurs between a well-mixed state and a well-segregated state. The numerical results show that system mass, decay rate and graffiti rate influence the critical parameter. From the discrete model, we derive a continuum system of convection-diffusion equations for territorial development. Using the continuum equations, we perform a linear stability analysis to determine the stability of the equilibrium solutions and we find that we can determine the precise location of the phase transition in parameter space as a function of the system mass and the graffiti creation and decay rates.
We present a shape-based approach to three-dimensional image reconstruction from diffuse optical data. Our approach differs from others in the literature in that we jointly reconstruct object and background characterization and localization simultaneously, rather than sequentially process for optical properties and postprocess for edges. The key to the efficiency and robustness of our algorithm is in the model we propose for the optical properties of the background and anomaly: We use a low-order parameterization of the background and another for the interior of the anomaly, and we use an ellipsoid to describe the boundary of the anomaly. This model has the effect of regularizing the inversion problem and provides a natural means of including additional physical properties if they are known a priori. A Gauss-Newton-type algorithm with line search is implemented to solve the underlying nonlinear least-squares problem and thereby determine the coefficients of the parameterizations and the descriptors of the ellipsoid. Numerical results show the effectiveness of this method.
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