The mechanisms and universality class underlying the remarkable phenomena at the transition to turbulence remain a puzzle 130 years after their discovery . The statistical behaviour is thought to be related to directed percolation (DP; refs 6,11-13). Attempts to understand transitional turbulence dynamically invoke periodic orbits and streamwise vortices [14][15][16][17][18][19] , the dynamics of long-lived chaotic transients 20, and model equations based on analogies to excitable media 21. Here we report direct numerical simulations of transitional pipe flow, showing that a zonal flow emerges at large scales, activated by anisotropic turbulent fluctuations; in turn, the zonal flow suppresses the small-scale turbulence leading to stochastic predator-prey dynamics. We show that this ecological model of transitional turbulence, which is asymptotically equivalent to DP at the transition 22, reproduces the lifetime statistics and phenomenology of pipe flow experiments. Our work demonstrates that a fluid on the edge of turbulence exhibits the same transitional scaling behaviour as a predator-prey ecosystem on the edge of extinction, and establishes a precise connection with the DP universality class.Turbulent fluids are ubiquitous in nature, arising for sufficiently large characteristic speeds U , depending on the kinematic viscosity ν and a characteristic system scale, such as the diameter of a pipe D. Turbulent flows are complex, stochastic, and unpredictable in detail, but transition at lower velocities to a laminar flow, which is simple, deterministic and predictable. This transition is controlled by the dimensionless parameter known as the Reynolds number, which in the pipe geometry of interest here is given by Re ≡ UD/ν, and occurs in the range 1,700 Re 2,300. The laminar-turbulence transition has presented a challenge to experiment and theory since Osborne Reynolds' original observation of intermittent 'flashes' of turbulence 1 . To explore this transitional regime, we have performed direct numerical simulations of the Navier-Stokes equations in a pipe of length L = 10D, using the open-source code 'Open Pipe Flow' 23 , as described in Supplementary Methods. The Reynolds number at which transitional turbulence occurs is higher for short pipes 23 , and the simulations reported here for L = 10D were performed at a nominal value Re = 2,600, which we estimate to be equivalent to Re 2,200 in long pipe data 7 based on estimates of when puff decay transitions to puff splitting. We confirmed that our results did not qualitatively change for a longer pipe with L = 20D. We denote the time-dependent velocity deviation from the Hagen-Poiseuille flow by u = (u z , u θ , u r ). Because we were interested in transitional behaviour, we looked for a decomposition 2,6,24,25 of large-scale modes that would indicate some form of collective behaviour, as well as small-scale modes that would be representative of turbulent dynamics. In particular, we report here the behaviour of the velocity field (u z , u θ , u r ), where the bar de...

The transitional and well-developed regimes of turbulent shear flows exhibit a variety of remarkable scaling laws that are only now beginning to be systematically studied and understood. In the first part of this article, we summarize recent progress in understanding the friction factor of turbulent flows in rough pipes and quasi-two-dimensional soap films, showing how the data obey a two-parameter scaling law known as roughness-induced criticality, and exhibit power-law scaling of friction factor with Reynolds number that depends on the precise form of the nature of the turbulent cascade. These results hint at a nonequilibrium fluctuation-dissipation relation that applies to turbulent flows. The second part of this article concerns the lifetime statistics in smooth pipes around the transition, showing how the remarkable super-exponential scaling with Reynolds number reflects deep connections between large deviation theory, extreme value statistics, directed percolation and the onset of coexistence in predator-prey ecosystems. Both these phenomena reflect the way in which turbulence can be fruitfully approached as a problem in non-equilibrium statistical mechanics.

Genetic variation in a population can sometimes arise so fast as to modify ecosystem dynamics. Such phenomena have been observed in natural predator-prey systems and characterized in the laboratory as showing unusual phase relationships in population dynamics, including a π phase shift between predator and prey (evolutionary cycles) and even undetectable prey oscillations compared to those of the predator (cryptic cycles). Here we present a generic individual-level stochastic model of interacting populations that includes a subpopulation of low nutritional value to the predator. Using a master equation formalism and by mapping to a coherent state path integral solved by a system-size expansion, we show that evolutionary and cryptic quasicycles can emerge generically from the combination of intrinsic demographic fluctuations and clonal mutations alone, without additional biological mechanisms.

Biological organisms experience constantly changing environments, from sudden changes in physiology brought about by feeding, to the regular rising and setting of the Sun, to ecological changes over evolutionary timescales. Living organisms have evolved to thrive in this changing world but the general principles by which organisms shape and are shaped by time varying environments remain elusive. Our understanding is particularly poor in the intermediate regime with no separation of timescales, where the environment changes on the same timescale as the physiological or evolutionary response. Experiments to systematically characterize the response to dynamic environments are challenging since such environments are inherently high dimensional. This roadmap deals with the unique role played by time varying environments in biological phenomena across scales, from physiology to evolution, seeking to emphasize the commonalities and the challenges faced in this emerging area of research.