Coevolution Maintains Diversity in the Stochastic "Kill the Winner" Model

The "Kill the Winner" hypothesis is an attempt to address the problem of diversity in biology. It argues that host-specific predators control the population of each prey, preventing a winner from emerging and thus maintaining the coexistence of all species in the system. We develop a stochastic model for the "Kill the Winner" paradigm and show that the stable coexistence state of the deterministic "Kill the Winner" model is destroyed by demographic stochasticity, through a cascade of extinction events. We formulate an individual-level stochastic model in which predator-prey coevolution promotes high diversity of the ecosystem by generating a persistent population flux of species

Numerical Renormalization Group Calculations For Similarity Solutions and Travelling Waves

We present a numerical implementation of the renormalization group (RG) for partial differential equations, constructing similarity solutions and travelling waves. We show that for a large class of well-localized initial conditions, successive iterations of an appropriately defined discrete RG transformation in space and time will drive the system towards a fixed point. This corresponds to a scale-invariant solution, such as a similarity or travelling-wave solution, which governs the long-time asymptotic behavior. We demonstrate that the numerical RG method is computationally very efficient.Comment: 14 pages, Postscript file of paper and 3 figures distributed as self-unpacking uuencoded compressed tar file.Also available by anonymous ftp to gijoe.mrl.uiuc.edu (128.174.119.153), file /pub/numrg.u

Biology's next revolution

The interpretation of recent environmental genomics data exposes the far-reaching influence of horizontal gene transfer, and is changing our basic concepts of organism, species and evolution itself.Comment: Slightly expanded version of invited essay published in Nature. The most important addition is a complete set of references that could not be included in the published version due to space limitations and acknowledgment of the grant that supported our wor

Block Spins for Partial Differential Equations

We investigate the use of renormalisation group methods to solve partial differential equations (PDEs) numerically. Our approach focuses on coarse-graining the underlying continuum process as opposed to the conventional numerical analysis method of sampling it. We calculate exactly the coarse-grained or `perfect' Laplacian operator and investigate the numerical effectiveness of the technique on a series of 1+1-dimensional PDEs with varying levels of smoothness in the dynamics: the diffusion equation, the time-dependent Ginzburg-Landau equation, the Swift-Hohenberg equation and the damped Kuramoto-Sivashinsky equation. We find that the renormalisation group is superior to conventional sampling-based discretisations in representing faithfully the dynamics with a large grid spacing, introducing no detectable lattice artifacts as long as there is a natural ultra-violet cut off in the problem. We discuss limitations and open problems of this approach.Comment: 8 pages, RevTeX, 8 figures, contribution to L.P. Kadanoff festschrift (J. Stat. Phys

Experimental Constraints on the Pairing State of the Cuprate Superconductors: an Emerging Consensus

We present a critical discussion of recent experimental probes of the pairing state of the high temperature superconductors, focusing primarily, but not exclusively, on \Yba, where the best data currently exist. Penetration depth measurements near \Tc\ give no indication of an extra transition, indicating that the pairing state is a one-dimensional representation of the crystal symmetry. Penetration depth measurements at low temperatures provide strong evidence for a change in sign of the gap function over the Fermi surface. Quantum mechanical phase interference experiments generally confirm this and in addition show that the nodal positions are consistent with a \dx2y2\ pairing state. This pairing state is consistent with photoemission measurements of the gap function, Raman scattering, the effect on \Tc\ of impurities, and many other data (reviewed by two of us previously) which indicate the presence of low lying excitations in the superconducting state. We also discuss evidence that apparently does not fit in with a \dx2y2\ pairing state, and we describe possible alternative scenarios.Comment: 81 pages, macro package (modified version of uiucmac.tex) included in submission, figures NOT available (but not essential