Susan R. McKay

Chaotic dynamics in a multiple species fishery: a model of community predation

Abstract This paper summarizes a dynamic model of a multiple species fishery that incorporates a community level phenomenon — community predation or the tendency of big fish of almost any species to eat little fish of almost any species. The model is used to explore the conditions under which the dynamic behavior of a multiple species complex characterized by community predation qualitatively conforms with the observed dynamic processes of real fisheries systems. The specifications necessary to replicate qualitatively the dynamic behavior of such system lead to the chaotic behavior of individual populations but stability of the entire system. The presence of chaos may explain why the dynamic path of individual populations in real fisheries has been so resistant to prediction, but more important, indicates that predictable order, the basis of management control of these kinds of multiple species systems, may be found at the system rather than at the individual population level.

Chaotic spin glasses: An upper critical dimension (invited)

The chaotic renormalization‐group trajectories exhibited by frustrated hierarchical Ising models have been interpreted as signaling a spin‐glass phase, since, as the system is probed at successive length scales, strong and weak correlations are encountered in a chaotic sequence. Cluster‐hierarchical models have been introduced, with susceptibilities behaving as in Bravais lattices. Frustrated cluster‐hierarchical models again show an ordered phase characterized by chaotic rescaling and a smooth specific heat at the transition (α<−5). Scans in dimensionality reveal an upper critical dimension for the chaotic spin‐glass phase, via a boundary crisis mechanism. Beyond this dimension, the system has no long‐range order at any temperature. Nevertheless, a low‐temperature regime can be distinctly identified, exhibiting intermediate‐range chaotic spin‐glass order.

Amorphously packed, frustrated hierarchical models: Chaotic rescaling and spin‐glass behavior

Hierarchical Ising models have been constructed with competing ferromagnetic and antiferromagnetic interactions, and solved exactly. As frustration is increased, a low‐temperature phase is encountered, characterized by chaotic renormalization‐group trajectories. This spin‐glass phase is microscopically described in terms of subsets of spins which are strongly correlated, yet noncontiguous. Further, the amorphous packing of real systems is approximated in these models by a random distribution of shapes at various hierarchical levels, which eliminates unphysical limit cycle behavior in favor of chaos.

Equimagnetization lines in the hybrid‐order phase diagram of the d=3 random‐field Ising model (invited)

The three‐dimensional random‐field Ising model has been approximately solved by pursuing the global renormalization‐group trajectories of the full, coupled probability distributions of local fields and bonds. This study, which in effect is the global analysis of the renormalization‐group flows of 220 physical quantities (based on a generalization of the Migdal–Kadanoff approximation), yields several results that were previously unsuspected. The boundary between the ferromagnetic and paramagnetic phases is hybrid order, in the sense that the magnetization is discontinuous (as in a first‐order transition) and the specific heat has a power‐law singularity (as in a second‐order transition). The magnetic properties of the d=3 random‐field Ising model are presented in the form of equimagnetization lines. The study has been repeated for a variety of dimensions d. The lower critical dimension dl=2 is correctly obtained. Moreover, this study indicates the existence of an intermediate‐critical dimension dt (between...

Managing unpredictable resources: Traditional policies applied to chaotic populations

Abstract Conventional theory for the management of living ocean resources assumes a predictable link between current management actions and the future state of managed populations. As a practical matter, however, it is very hard to establish this kind of predictable relationship. It is possible that the dynamics of these populations exhibit chaotic variation. This paper addresses the question of appropriate management policies in a regime characterized by chaotic population dynamics. The problem is approached through a bioeconomic simulator that has chaotic properties. With light fishing, policies that alter the conditions of fishing perform better than policies dependent upon population predictions.

Hierarchical Models and Chaotic Spin Glasses

Renormalization-group studies in position space have led to the discovery of hierarchical models which are exactly solvable, exhibiting nonclassical critical behavior at finite temperature. Position-space renormalization-group approximations that had been widely and successfully used are in fact alternatively applicable as exact solutions of hierarchical models, this realizability guaranteeing important physical requirements. For example, a hierarchized version of the Sierpiriski gasket is presented, corresponding to a renormalization-group approximation which has quantitatively yielded the multicritical phase diagrams of submonolayers on graphite. Hierarchical models are now being studied directly as a testing ground for new concepts. For example, with the introduction of frustration, chaotic renormalization-group trajectories were obtained for the first time. Thus, strong and weak correlations are randomly intermingled at successive length scales, and a new microscopic picture and mechanism for a spin glass emerges. An upper critical dimension occurs via a boundary crisis mechanism in cluster-hierarchical variants developed to have well-behaved susceptibilities.

Local entropy and structure in a two-dimensional frustrated system

We calculate the local contributions to the Shannon entropy and excess entropy and use these information theoretic measures as quantitative probes of the order arising from quenched disorder in the diluted Ising antiferromagnet on a triangular lattice. When one sublattice is sufficiently diluted, the system undergoes a temperature-driven phase transition, with the other two sublattices developing magnetizations of equal magnitude and opposite sign as the system is cooled.(1) The diluted sublattice has no net magnetization but exhibits spin glass ordering. The distribution of local entropies shows a dramatic broadening at low temperatures; this indicates that the systems total entropy is not shared equally across the lattice. The entropy contributions from some regions exhibit local reentrance, although the entropy of the system decreases monotonically as expected. The average excess entropy shows a sharp peak at the critical temperature, showing that the excess entropy is sensitive to the structural changes that occur as a result of the spin glass ordering.

Ising spin‐glass critical and multicritical fixed distributions from a renormalization‐group calculation with quenched randomness

The Ising model with a quenched random distribution of ferromagnetic and antiferromagnetic interactions has been investigated by tracking the probability distribution of interactions under rescaling. As the initial probability of a ferromagnetic interaction is increased, the phase diagram in three dimensions shows three ordered phases: antiferromagnetic, spin glass and ferromagnetic. We find that, within the spin‐glass region, effective interactions between pairs of spins at successive length scales occur in a chaotic sequence, while the average magnitudes of the ferromagnetic and the antiferromagnetic interactions both flow to infinity under iteration. Our calculated spin‐glass multicritical points are on the Nishimori line in cases of both bimodal (±J) and diluted (trimodal) distributions, and these points flow under rescaling to a common fixed distribution with a specific heat exponent α of approximately −3.7.