On the geometry of solutions of the quasi-geostrophic and Euler equations

We study solutions of the two-dimensional quasi-geostrophic thermal active scalar equation involving simple hyperbolic saddles. There is a naturally associated notion of simple hyperbolic saddle breakdown. It is proved that such breakdown cannot occur in finite time. At large time, these solutions may grow at most at a quadruple-exponential rate. Analogous results hold for the incompressible three-dimensional Euler equation.

A simple explanation for taxon abundance patterns

For taxonomic levels higher than species, the abundance distributions of the number of subtaxa per taxon tend to approximate power laws but often show strong deviations from such laws. Previously, these deviations were attributed to finite-time effects in a continuous-time branching process at the generic level. Instead, we describe herein a simple discrete branching process that generates the observed distributions and find that the distribution's deviation from power law form is not caused by disequilibration, but rather that it is time independent and determined by the evolutionary properties of the taxa of interest. Our model predicts—with no free parameters—the rank-frequency distribution of the number of families in fossil marine animal orders obtained from the fossil record. We find that near power law distributions are statistically almost inevitable for taxa higher than species. The branching model also sheds light on species-abundance patterns, as well as on links between evolutionary processes, self-organized criticality, and fractals.

“Coarse” stability and bifurcation analysis using time-steppers: A reaction-diffusion example

Evolutionary, pattern forming partial differential equations (PDEs) are often derived as limiting descriptions of microscopic, kinetic theory-based models of molecular processes (e.g., reaction and diffusion). The PDE dynamic behavior can be probed through direct simulation (time integration) or, more systematically, through stability/bifurcation calculations; time-stepper-based approaches, like the Recursive Projection Method [Shroff, G. M. & Keller, H. B. (1993) SIAM J. Numer. Anal. 30, 1099–1120] provide an attractive framework for the latter. We demonstrate an adaptation of this approach that allows for a direct, effective (“coarse”) bifurcation analysis of microscopic, kinetic-based models; this is illustrated through a comparative study of the FitzHugh-Nagumo PDE and of a corresponding Lattice–Boltzmann model.

Diversity, functionality, and stability of the T cell repertoire derived in vivo from a single human T cell precursor

In this report, we have analyzed the human T cell repertoire derived in vivo from a single T cell precursor. A unique case of X-linked severe combined immunodeficiency in which a reverse mutation occurred in an early T cell precursor was analyzed to this end. It was determined that at least 1,000 T cell clones with unique T cell receptor-β sequences were generated from this precursor. This diversity seems to be stable over time and provides protection from infections in vivo. A similar estimation was obtained in an in vitro murine model of T cell generation from a single cell precursor. Overall, our results document the large diversity potential of T cell precursors and provide a rationale for gene therapy of the block of T cell development.

The cutoff phenomenon in finite Markov chains.

Natural mixing processes modeled by Markov chains often show a sharp cutoff in their convergence to long-time behavior. This paper presents problems where the cutoff can be proved (card shuffling, the Ehrenfests' urn). It shows that chains with polynomial growth (drunkard's walk) do not show cutoffs. The best general understanding of such cutoffs (high multiplicity of second eigenvalues due to symmetry) is explored. Examples are given where the symmetry is broken but the cutoff phenomenon persists.

The challenge of paleoecological stasis: reassessing sources of evolutionary stability.

The paleontological record of the lower and middle Paleozoic Appalachian foreland basin demonstrates an unprecedented level of ecological and morphological stability on geological time scales. Some 70-80% of fossil morphospecies within assemblages persist in similar relative abundances in coordinated packages lasting as long as 7 million years despite evidence for environmental change and biotic disturbances. These intervals of stability are separated by much shorter periods of ecological and evolutionary change. This pattern appears widespread in the fossil record. Existing concepts of the evolutionary process are unable to explain this uniquely paleontological observation of faunawide coordinated stasis. A principle of evolutionary stability that arises from the ecosystem is explored here. We propose that hierarchical ecosystem theory, when extended to geological time scales, can explain long-term paleoecological stability as the result of ecosystem organization in response to high-frequency disturbance. The accompanying stability of fossil morphologies results from "ecological locking," in which selection is seen as a high-rate response of populations that is hierarchically constrained by lower-rate ecological processes. When disturbance exceeds the capacity of the system, ecological crashes remove these higher-level constraints, and evolution is free to proceed at high rates of directional selection during the organization of a new stable ecological hierarchy.