State Aggregation and Population Dynamics in Linear Systems

We consider complex systems that are composed of many interacting elements, evolving under some dynamics. We are interested in characterizing the ways in which these elements may be grouped into higher-level, macroscopic states in a way that is compatible with those dynamics. Such groupings may then be thought of as naturally emergent properties of the system. We formalize this idea and, in the case that the dynamics are linear, prove necessary and sufficient conditions for this to happen. In cases where there is an underlying symmetry among the components of the system, group theory may be used to provide a strong sufficient condition. These observations are illustrated with some artificial life examples

A non-linear degenerate equation for direct aggregation and traveling wave dynamics

The gregarious behavior of individuals of populations is an important factor in avoiding predators or for reproduction. Here, by using a random biased walk approach, we build a model which, after a transformation, takes the general form [u_{t}=[D(u)u_{x}]_{x}+g(u)] . The model involves a density-dependent non-linear diffusion coefficient [D] whose sign changes as the population density [u] increases. For negative values of [D] aggregation occurs, while dispersion occurs for positive values of [D] . We deal with a family of degenerate negative diffusion equations with logistic-like growth rate [g] . We study the one-dimensional traveling wave dynamics for these equations and illustrate our results with a couple of examples. A discussion of the ill-posedness of the partial differential equation problem is included

Kinetics of self-induced aggregation in Brownian particles

We study a model of interacting random walkers that proposes a simple mechanism for the emergence of cooperation in group of individuals. Each individual, represented by a Brownian particle, experiences an interaction produced by the local unbalance in the spatial distribution of the other individuals. This interaction results in a nonlinear velocity driving the particle trajectories in the direction of the nearest more crowded regions; the competition among different aggregating centers generates nontrivial dynamical regimes. Our simulations show that for sufficiently low randomness, the system evolves through a coalescence behavior characterized by clusters of particles growing with a power law in time. In addition, the typical scaling properties of the general theory of stochastic aggregation processes are verified.Comment: RevTeX, 9 pages, 9 eps-figure

Aggregation and Control of Populations of Thermostatically Controlled Loads by Formal Abstractions

This work discusses a two-step procedure, based on formal abstractions, to generate a finite-space stochastic dynamical model as an aggregation of the continuous temperature dynamics of a homogeneous population of Thermostatically Controlled Loads (TCL). The temperature of a single TCL is described by a stochastic difference equation and the TCL status (ON, OFF) by a deterministic switching mechanism. The procedure is formal as it allows the exact quantification of the error introduced by the abstraction -- as such it builds and improves on a known, earlier approximation technique in the literature. Further, the contribution discusses the extension to the case of a heterogeneous population of TCL by means of two approaches resulting in the notion of approximate abstractions. It moreover investigates the problem of global (population-level) regulation and load balancing for the case of TCL that are dependent on a control input. The procedure is tested on a case study and benchmarked against the mentioned alternative approach in the literature.Comment: 40 pages, 21 figures; the paper generalizes the result of conference publication: S. Esmaeil Zadeh Soudjani and A. Abate, "Aggregation of Thermostatically Controlled Loads by Formal Abstractions," Proceedings of the European Control Conference 2013, pp. 4232-4237. version 2: added references for section

Patchiness and Demographic Noise in Three Ecological Examples

Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only -at most- local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals. Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the "square-root" intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.Comment: 20 pages, 5 figures. To appear in J. Stat. Phy

Decentralized Convergence to Nash Equilibria in Constrained Deterministic Mean Field Control

This paper considers decentralized control and optimization methodologies for large populations of systems, consisting of several agents with different individual behaviors, constraints and interests, and affected by the aggregate behavior of the overall population. For such large-scale systems, the theory of aggregative and mean field games has been established and successfully applied in various scientific disciplines. While the existing literature addresses the case of unconstrained agents, we formulate deterministic mean field control problems in the presence of heterogeneous convex constraints for the individual agents, for instance arising from agents with linear dynamics subject to convex state and control constraints. We propose several model-free feedback iterations to compute in a decentralized fashion a mean field Nash equilibrium in the limit of infinite population size. We apply our methods to the constrained linear quadratic deterministic mean field control problem and to the constrained mean field charging control problem for large populations of plug-in electric vehicles.Comment: IEEE Trans. on Automatic Control (cond. accepted

Allee Effects May Slow the Spread of Parasites in a Coastal Marine Ecosystem

Allee effects are thought to mediate the dynamics of population colonization, particularly for invasive species. However, Allee effects acting on parasites have rarely been considered in the analogous process of infectious disease establishment and spread. We studied the colonization of uninfected wild juvenile Pacific salmon populations by ectoparasitic salmon lice (Lepeophtheirus salmonis) over a 4-year period. In a data set of 68,376 fish, we observed 85 occurrences of precopular pair formation among 1,259 preadult female and 613 adult male lice. The probability of pair formation was dependent on the local abundance of lice, but this mate limitation is likely offset somewhat by mate-searching dispersal of males among host fish. A mathematical model of macroparasite population dynamics that incorporates the empirical results suggests a high likelihood of a demographic Allee effect, which can cause the colonizing parasite populations to die out. These results may provide the first empirical evidence for Allee effects in a macroparasite. Furthermore, the data give a rare detailed view of Allee effects in colonization dynamics and suggest that Allee effects may dampen the spread of parasites in a coastal marine ecosystem