Eric Foxall

Bio: Eric Foxall is an academic researcher from Arizona State University. The author has contributed to research in topics: Population & Random walk. The author has an hindex of 6, co-authored 39 publications receiving 152 citations. Previous affiliations of Eric Foxall include University of Victoria & University of British Columbia.

A scaling law for random walks on networks

Abstract: The dynamics of many natural and artificial systems are well described as random walks on a network: the stochastic behaviour of molecules, traffic patterns on the internet, fluctuations in stock prices and so on. The vast literature on random walks provides many tools for computing properties such as steady-state probabilities or expected hitting times. Previously, however, there has been no general theory describing the distribution of possible paths followed by a random walk. Here, we show that for any random walk on a finite network, there are precisely three mutually exclusive possibilities for the form of the path distribution: finite, stretched exponential and power law. The form of the distribution depends only on the structure of the network, while the stepping probabilities control the parameters of the distribution. We use our theory to explain path distributions in domains such as sports, music, nonlinear dynamics and stochastic chemical kinetics.

Explicit construction of chaotic attractors in Glass networks

Abstract: Chaotic dynamics have been observed in example piecewise-affine models of gene regulatory networks. Here we show how the underlying Poincare maps can be explicitly constructed. To do this, we proceed in two steps. First, we consider a limit case, where some parameters tend to ∞, and then consider the case with finite parameters as a perturbation of the previous one. We provide a detailed example of this construction, in 3-d, with several thresholds per variable. This construction is essentially a topological horseshoe map. We show that the limit situation is conjugate to the golden mean shift, and is thus chaotic. Then, we show that chaos is preserved for large parameters, relying on the structural stability of the return map in the limit case. We also describe a method to embed systems with several thresholds into binary systems, of higher dimensions. This shows that all results found for systems having several thresholds remain valid in the binary case.

Tragedy of the commons in the chemostat.

Abstract: We present a proof of principle for the phenomenon of the tragedy of the commons that is at the center of many theories on the evolution of cooperation. Whereas the tragedy is commonly set in a game theoretical context, and attributed to an underlying Prisoner's Dilemma, we take an alternative approach based on basic mechanistic principles of species growth that does not rely on the specification of payoffs which may be difficult to determine in practice. We establish the tragedy in the context of a general chemostat model with two species, the cooperator and the cheater. Both species have the same growth rate function and yield constant, but the cooperator allocates a portion of the nutrient uptake towards the production of a public good -the "Commons" in the Tragedy- which is needed to digest the externally supplied nutrient. The cheater on the other hand does not produce this enzyme, and allocates all nutrient uptake towards its own growth. We prove that when the cheater is present initially, both the cooperator and the cheater will eventually go extinct, hereby confirming the occurrence of the tragedy. We also show that without the cheater, the cooperator can survive indefinitely, provided that at least a low level of public good or processed nutrient is available initially. Our results provide a predictive framework for the analysis of cooperator-cheater dynamics in a powerful model system of experimental evolution.

New Results for the Two-Stage Contact Process

Abstract: In this paper, we continue the work started by Steve Krone on the two-stage contact process. We give a simplified proof of the duality relation and answer most of the open questions posed in Krone (1999). We also fill in the details of an incomplete proof.

Scaling properties of paths on graphs

Abstract: Let G be a directed graph on finitely many vertices and edges, and assign a positive weight to each edge on G. Fix vertices u and v and consider the set of paths that start at u and end at v, self-intersecting in any number of places along the way. For each path, sum the weights of its edges, and then list the path weights in increasing order. The asymptotic behaviour of this sequence is described, in terms of the structure and type of strongly connected components on the graph. As a special case, for a Markov chain the asymptotic probability of paths obeys either a power law scaling or a weaker type of scaling, depending on the structure of the transition matrix. This generalizes previous work by Mandelbrot and others, who established asymptotic power law scaling for special classes of Markov chains.

Differential Equations and Dynamical Systems

Abstract: Series Preface * Preface to the Third Edition * 1 Linear Systems * 2 Nonlinear Systems: Local Theory * 3 Nonlinear Systems: Global Theory * 4 Nonlinear Systems: Bifurcation Theory * References * Index

The protein folding network

Abstract: The conformation space of a 20 residue antiparallel β-sheet peptide, sampled by molecular dynamics simulations, is mapped to a network. Snapshots saved along the trajectory are grouped according to secondary structure into nodes of the network and the transitions between them are links. The conformation space network describes the significant free energy minima and their dynamic connectivity without requiring arbitrarily chosen reaction coordinates. As previously found for the Internet and the World-Wide Web as well as for social and biological networks, the conformation space network is scale-free and contains highly connected hubs like the native state which is the most populated free energy basin. Furthermore, the native basin exhibits a hierarchical organization, which is not found for a random heteropolymer lacking a predominant free-energy minimum. The network topology is used to identify conformations in the folding transition state (TS) ensemble, and provides a basis for understanding the heterogeneity of the TS and denatured state ensemble as well as the existence of multiple pathways.

A Complete Bibliography of the Journal of Statistical Physics: 2000{2009

Abstract: (2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].