Center for Complex and Nonlinear Science

We develop the off-lattice model to simulate the life cycle of Myxococcus xanthus. When the food is abundant, they grow as swarms that spread away from the colony. In this stage, their movement and coordination are determined by their A-motility and S-motility engines. However, when they are in starvation, C-signaling between cells takes place and changes their cell-cell coordination. This allows them to form an aggregate which eventually develops into a fruiting body. Cells inside the fruiting body differentiate into round nonmotile spores which are resistant to adverse condition. In this paper, the Dynamic Energy Budget model is used as a trigger mechanism for cell growth and cell division, and then for switching from the swarming stage to the stage of fruiting body formation. Moreover, the logistic equation is implemented to count the number of C-signal molecules on each cell surface, which is then used as a switch for transitions between the stages of fruiting body formation.

Kolmogorov's statistical theory of turbulence is based on the existence of the invariant measure of the Navier-Stokes flow. Recently the existence of the invariant measure was established in the three-dimensional case. It was established earlier by the author for uni-directional flow and for rivers. We discuss how one can try to go about approximating the invariant measure in three dimensions.

The viscous Moore-Greitzer equation modeling the airflow through the compression system in turbomachines, such as a jet engine, is derived using a scaled Navier-Stokes equation. The method utilizes a separation of scales argument, based on the different spatial scales in the engine and the different time scales in the flow. The pitch and size of the rotor-stator pair of blades provides a small parameter, which is the size of the local cell. The motion of the stator and rotor blades in the compressor produces a very turbulent flow on a fast time scale. The leading order equation, for the fast-time and local scale, describes this turbulent flow. The next order equations, produce an axi-symmetric swirl and a flow-pattern analogous to Rayleigh-B´enard convection rolls in Rayleigh-B´enard convection. On a much larger spatial scale and a slower time scale, there exist modulations of the flow including instabilities called surge and stall. A higher order equation, in the small parameter, describes these global flow modulations, when averaged over the small (local) spatial scales, the fast time scale and the time scale of the vortex rotations. Thus a more general system of spatially global, slow-time equations is obtained. This system can be solved numerically without any approximations. The viscous Moore- Greitzer equation is obtained when small inertial terms are dropped from these slow-time, spatially global equations, averaged once more in the axial direction. The new equations are simulated with two different simplifying assumptions and the results compared with simulations of the viscous Moore-Greitzer equations.

We study how an ODE description of schools of fish by Birnir (2007) changes in the presence of a random acceleration. The model can be reduced to one ODE for the direction of the velocity of a generic fish and another one for its speed. These equations contain the mean speed v and a Kuramoto order parameter for the phases of the fish velocities, r. We show that their stationary solutions consist of an incoherent unstable solution with r=v=0 and a globally stable solution with r=1 and a constant v > 0. In the latter solution, all fishes move uniformly in the same direction with v and the direction of motion determined by the initial configuration of the school.In the second part, the directional headings of the particles are perturbed, in two distinct ways, and the speeds accelerated in order to obtain two distinct classes of non-stationary, complex solutions. We show that the system has similar behavior as the unperturbed one, and derive the resulting constant value of the average speed, verified numerically. Finally, we show that the system exhibits a similar bifurcation to that in Cirok and Vishek et al. 1995, between phases of synchronization and disorder. In one case, the variance of the angular noise, which is Brownian, is varied, and in the other case, varying the turning rate causes a similar phase transition.

The existence of solutions describing the turbulent flow in rivers is proven. The existence of an associated invariant measure describing the statistical properties of this one dimensional turbulence is established. The turbulent solutions are not smooth but H\"older continuous with exponent $3/4$. The scaling of the solutions' second structure (or width) function gives rise to Hack's law \cite{H57}; stating that the length of the main river, in mature river basins, scales with the area of the basin $l \sim A^{h}$, $h = 0.568$ being Hack's exponent.

We construct the 1962 Kolmogorov-Obukhov statistical theory of turbulence from the stochastic Navier-Stokes equations driven by generic noise. The intermittency corrections to the scaling exponents of the structure functions of turbulence are given by the She-Leveque intermittency corrections. We show how they are produced by She-Waymire log-Poisson processes, that are generated by the Feynmann-Kac formula from the stochastic Navier-Stokes equation. We find the Kolmogorov-Hopf equations and  compute the invariant measures of turbulence for 1-point and 2-point statistics. Then projecting these measures we find the formulas for the probability distribution functions (PDFs) of the velocity differences in the structure functions. In the limit of zero intermittency, these PDFs reduce to the Generalized Hyperbolic Distributions of Barndorff-Nilsen.

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, Czir´ok 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.

The existence and uniqueness of solutions of the Navier-Stokes equation driven with additive noise in three dimensions is proven, in the presence of a strong uni-directional mean flow with some rotation. The physical relevance of this solution and its relation to the classical solution, whose existence and uniqueness is also proven, is explained. The existence of a unique invariant measure is established and the properties of this measure are described. The invariant measure is used to prove Kolmogorov's scaling in 3-dimensional turbulence including the celebrated -5/3 power law for the decay of the power spectrum of a turbulent 3-dimensional flow.

We model the spawning migration of the Icelandic capelin stock using an interacting particle model with added environmental fields. Without artificial forcing terms or a homing instinct, we qualitatively reproduce several observed spawning migrations using available temperature data and approximated currents. The simulations include orders of magnitude more particles than many similar models, affecting the global behavior of the system. Without environmental fields, we analyze how various parameters scale with the number of particles. In particular we present scaling behavior between the size of the time step, radii of the sensory zones and the number of particles in the system. We then discuss incorporating environmental data into the model.