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 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.

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.

A system of ordinary differential equations (ODEs) is derived from a discrete system of Vicsek, Czir ok et al. 1995, describing the motion of a school of fish. Classes of linear and stationary solutions of the ODEs are found and their stability explored using equivariant bifurcation theory. The existence of periodic and toroidal solutions is also proven under deterministic perturbations and structurally stable heteroclinic connections are found. Applications of the model to the migration of the capelin, a pelagic fish that undertakes an extensive migration in the North Atlantic, are dissussed and simulation of the ODEs presented.

Recent advances in the theory of turbulent solutions of the Navier-Stokes equations are discussed and the existence of their associated invariant measures. The statistical theory given by the invariant measures is described and associated with historically-known scaling laws. These are Hack's law in one dimension, the Bachelor-Kraichnan law in two dimensions and the Kolmogorov's scaling law in three dimensions. Applications to problems in turbulence are discussed and applications to Reynolds Averaged Navier Stokes (RANS) and Large Eddy Simulation (LES) models in computational turbulence.

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.

We will consider the 2-dimensional Navier-Stokes equation for an incompressible fluid with periodic boundary condition, and with a random perturbation that is in the form of white noise in time and a deterministic perturbation due to the large deviation principle. Our ultimate goal is to find appropriate conditions on the initial data and the forcing terms so that global existence and uniqueness of a mild solution is guaranteed. We will use the Picard's iteration method to prove existence of local mild solution and then prove the existence of a maximal solution which then leads to global existence. The result is applied to the backward Kolmogorov-Obukhov energy cascade and the forward Kraichnan enstrophy cascade in 2D turbulence.