For abstracts, see below.
Erik M. Rauch, Mark M. Millonas and Dante R. Chialvo. Pattern
Formation and Functionality in Swarm Models. Physics Letters A
207 185-193 (1995). [html]
[postscript]
[pdf]
Erik M. Rauch and Mark M. Millonas. The Role of
Trans-membrane
Signal Transduction in Turing-type Cellular Pattern Formation. Journal of
Theoretical
Biology 226: 401-407 (2004). [pdf]
J. Asikainen, A. Aharony, B.B. Mandelbrot, E.M. Rauch, and J.-P. Hovi. Fractal geometry of critical Potts clusters. European Physical Journal B 34: 479 (2003). [pdf]
Erik M. Rauch. Discrete, Amorphous Physical Models. International Journal of Theoretical Physics 42: 329-348 (2003).
M. A.M. de Aguiar, H. Sayama, E. M. Rauch, Y. Bar-Yam, and M. Baranger. Stability and Instability of Polymorphic Populations and the Role of Multiple Breeding Seasons in Phase III of Wright's Shifting Balance Theory. Physical Review E 65: 031909 (2002). [pdf]
Erik M. Rauch, Hiroki Sayama and Yaneer Bar-Yam. The
relationship
between measures of fitness and time scale in evolution. Physical Review Letters 88:
228101
(2002). [pdf]
Erik M. Rauch, Hiroki Sayama and Yaneer Bar-Yam. Dynamics and
genealogy
of strains in spatially extended host-pathogen models. Journal of
Theoretical
Biology 221: 655-664 (2003). [pdf]
M.A.M. de Aguiar, E. M. Rauch, and Y. Bar-Yam. Mean-field approximation
to a
spatial host-pathogen model. Physical
Review E 67: 047102 (2003). [pdf]
M.A.M. de Aguiar, E. M. Rauch, and Y. Bar-Yam. Invasion and Extinction
in the
Mean Field Approximation for a Spatial Host-Pathogen Model. Journal of
Statistical
Physics 114: 1417-1451 (2004). [pdf]
Erik M. Rauch and Yaneer Bar-Yam. Theory
predicts the uneven distribution of genetic diversity within species.
Nature 431, 449-452 (2004).
Erik M. Rauch, Hiroki Sayama and Yaneer
Bar-Yam. The
relationship
between measures of fitness and time scale in evolution. Physical Review Letters 88:
228101
(2002). [pdf]
M. A.M. de Aguiar, H. Sayama, E. M. Rauch, Y. Bar-Yam, and M. Baranger. Stability and Instability of Polymorphic Populations and the Role of Multiple Breeding Seasons in Phase III of Wright's Shifting Balance Theory. Physical Review E 65: 031909 (2002). [pdf]
It is generally difficult for a large population at a fitness peak to acquire the genotypes of a higher peak, because the intermediates produced by allelic recombination between types at different peaks are of lower fitness. In his shifting-balance theory, Wright proposed that fitter genotypes could, however, become fixed in small isolated demes by means of random genetic fluctuations. These demes would then try to spread their genome to nearby demes by migration of their individuals. The resulting polymorphism, the coexistence of individuals with different genotypes, would give the invaded demes a chance to move up to a higher fitness peak. This last step of the process, namely, the invasion of lower fitness demes by higher fitness genotypes, is known as phase III of Wright's theory. Here we study the invasion process from the point of view of the stability of polymorphic populations. Invasion occurs when the polymorphic equilibrium, established at low migration rates, becomes unstable. We show that the instability threshold depends sensitively on the average number of breeding seasons of individuals. Iteroparous species (with many breeding seasons! have lower thresholds than semelparous species (with a single breeding season). By studying a particular simple model, we are able to provide analytical estimates of the migration threshold as a function of the number of breeding seasons. Once the threshold is crossed and polymorphism becomes unstable, any imbalance between the different demes is sufficient for invasion to occur. The outcome of the invasion, however, depends on many parameters, not only on fitness. Differences in fitness, site capacities, relative migration rates, and initial conditions, all contribute to determine which genotype invades successfully. Contrary to the original perspective of Wright's theory for continuous fitness improvement, our results show that both upgrading to higher fitness peaks and downgrading to lower peaks are possible.Erik M. Rauch, Mark M. Millonas and Dante R. Chialvo. Pattern Formation and Functionality in Swarm Models. Physics Letters A 207 185-193 (1995). [html] [postscript] [pdf]
We explore a simplified class of models we call swarms, which are inspired by the collective behavior of social insects. We perform a mean-field stability analysis and numerical simulations of the model. Several interesting types of behavior emerge in the vicinity of a second-order phase transition in the model, including the formation of stable lines of traffic flow, and memory reconstitution and bootstrapping. Tn addition to providing an understanding of certain classes of biological behavior, these models bear a generic resemblance to a number of pattern formation processes in the physical sciences.Erik M. Rauch and Mark M. Millonas. The Role of Trans-membrane Signal Transduction in Turing-type Cellular Pattern Formation. Journal of Theoretical Biology 226: 401-407 (2004). [pdf]
The Turing mechanism for the production of a broken spatial symmetry in an initially homogeneous system of reacting and diffusing substances has attracted much interest as a potential model for certain aspects of morphogenesis such as pre-patterning in the embryo. The two features necessary for the formation of Turing patterns are short-range autocatalysis and long-range inhibition which usually only occur when the diffusion rate of the inhibitor is significantly greater than that of the activator. This observation has sometimes been used to cast doubt on applicability of the Turing mechanism to cellular patterning since many messenger molecules that diffuse between cells do so at more-or-less similar rates. Here we show that Turing-type patterns will be able to robustly form under a wide variety of realistic physiological conditions though plausible mechanisms of intra-cellular chemical communication without relying on differences in diffusion rates. In the mechanism we propose, reactions occur within cells. Signal transduction leads to the production of messenger molecules, which diffuse between cells at approximately equal rates, coupling the reactions occurring in different cells. These mechanisms also suggest how this process can be controlled in a rather precise way by the genetic machinery of the cell.J. Asikainen, A. Aharony, B.B. Mandelbrot, E.M. Rauch, and J.-P. Hovi. Fractal geometry of critical Potts clusters. European Physical Journal B 34: 479 (2003). [pdf]
Numerical simulations on the total mass, the numbers of bonds on the hull, external perimeter, singly connected bonds and gates into large fjords of the Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at criticality are presented. The data are found consistent with the recently derived corrections-to-scaling theory. However, the approach to the asymptotic region is slow, and the present range of the data does not allow a unique identification of the exact correction exponents.Erik M. Rauch. Discrete, Amorphous Physical Models. International Journal of Theoretical Physics 42: 329-348 (2003).
Discrete models of physical phenomena are an attractive alternative to continuous models such as partial differential equations. In discrete models, such as cellular automata, space is treated as having finitely many locations per unit volume, and physical processes are modelled by rules that depend on a small number of nearby locations. Such models depend critically on a regular (crystalline) lattice, as well as the global synchronization of all sites. We should ask, on the grounds of minimalism, whether the global synchronization and crystalline lattice are inherent in any discrete formulation. Is it possible to do without these conditions and still have a useful physical model? Or are they somehow fundamental? We will answer this question by presenting a class of models that are "extremely local" in the sense that the update rule does not depend on synchronization with the other sites, or on knowledge of the lattice geometry. All interactions involve only a single pair of sites. The models have the further advantage that they exactly conserved the analog of quantities such as momentum and energy which are conserved in physics. An example model of waves is given, and evidence is given that it agrees well qualitatively and quantitatively with continuous differential equations.