Emergence of coherent motion in aggregates of motile coupled maps

In this paper we study the emergence of coherence in collective motion described by a system of interacting motiles endowed with an inner, adaptative, steering mechanism. By means of a nonlinear parametric coupling, the system elements are able to swing along the route to chaos. Thereby, each motile can display different types of behavior, i.e. from ordered to fully erratic motion, accordingly with its surrounding conditions. The appearance of patterns of collective motion is shown to be related to the emergence of interparticle synchronization and the degree of coherence of motion is quantified by means of a graph representation. The effects related to the density of particles and to interparticle distances are explored. It is shown that the higher degrees of coherence and group cohesion are attained when the system elements display a combination of ordered and chaotic behaviors, which emerges from a collective self-organization process.Comment: 33 pages, 12 figures, accepted for publication at Chaos, Solitons and Fractal

Deriving mesoscopic models of collective behaviour for finite populations

Animal groups exhibit emergent properties that are a consequence of local interactions. Linking individual-level behaviour to coarse-grained descriptions of animal groups has been a question of fundamental interest. Here, we present two complementary approaches to deriving coarse-grained descriptions of collective behaviour at so-called mesoscopic scales, which account for the stochasticity arising from the finite sizes of animal groups. We construct stochastic differential equations (SDEs) for a coarse-grained variable that describes the order/consensus within a group. The first method of construction is based on van Kampen's system-size expansion of transition rates. The second method employs Gillespie's chemical Langevin equations. We apply these two methods to two microscopic models from the literature, in which organisms stochastically interact and choose between two directions/choices of foraging. These `binary-choice' models differ only in the types of interactions between individuals, with one assuming simple pair-wise interactions, and the other incorporating higher-order effects. In both cases, the derived mesoscopic SDEs have multiplicative, or state-dependent, noise. However, the different models demonstrate the contrasting effects of noise: increasing order in the pair-wise interaction model, whilst reducing order in the higher-order interaction model. Although both methods yield identical SDEs for such binary-choice, or one-dimensional, systems, the relative tractability of the chemical Langevin approach is beneficial in generalizations to higher-dimensions. In summary, this book chapter provides a pedagogical review of two complementary methods to construct mesoscopic descriptions from microscopic rules and demonstrates how resultant multiplicative noise can have counter-intuitive effects on shaping collective behaviour.Comment: Second version, 4 figures, 2 appendice

Equation-Free Multiscale Computational Analysis of Individual-Based Epidemic Dynamics on Networks

The surveillance, analysis and ultimately the efficient long-term prediction and control of epidemic dynamics appear to be one of the major challenges nowadays. Detailed atomistic mathematical models play an important role towards this aim. In this work it is shown how one can exploit the Equation Free approach and optimization methods such as Simulated Annealing to bridge detailed individual-based epidemic simulation with coarse-grained, systems-level, analysis. The methodology provides a systematic approach for analyzing the parametric behavior of complex/ multi-scale epidemic simulators much more efficiently than simply simulating forward in time. It is shown how steady state and (if required) time-dependent computations, stability computations, as well as continuation and numerical bifurcation analysis can be performed in a straightforward manner. The approach is illustrated through a simple individual-based epidemic model deploying on a random regular connected graph. Using the individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of Simulated Annealing I compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level under a pairwise representation perspective

Mathematical magic

This graduate project investigates magic tricks that are fundamentally mathematical in nature. They have an important place in the curriculum because curiosity about their working leads to an understanding of important mathematics, particularly at the early college level. Their use, however, is not restricted to collegiate undergraduates. The tricks can be performed by magicians of all ages for people of all ages. They require that a person be able to add, subtract, multiply and divide whole numbers and fractions, and to think logically. Rigorous proof of the mathematics involved in each trick is the unique contribution of this graduate project. The Three Pile Card Trick uses analysis, number theory, and numerical analysis. Digit Mind Reading requires number theory. The Four Fibonacci Tricks use algebra and number theory. Three Object Divination requires logic. Random Number Prediction looks to limits, arithmetic series, probability, and mathematical expectation. Even and Odd, I Know Your Number and Stover's Prediction require algebra. Many books contain mathematical magic. Usually the books include detailed information about how to perform each trick. Some also offer mathematical explanations of why the tricks work. The tricks for this project were not accompanied by mathematical proofs. They are original as is one of the tricks, Random Number Prediction.California State University, Northridge. Department of Mathematics.Includes bibliographical references (leaves 55-58

A theory of motivation in mathematics

California State University, Northridge. Department of Mathematics.Includes bibliographical references (pages 42-46

Multiple behaviors for turning performance of Pacific bluefin tuna (Thunnus orientalis)

Tuna are known for exceptional swimming speeds, which are possible because of their thunniform lift-based propulsion, large muscle mass and rigid fusiform body. A rigid body should restrict maneuverability with regard to turn radius and turn rate. To test if turning maneuvers by the Pacific bluefin tuna (Thunnus orientalis) are constrained by rigidity, captive animals were videorecorded overhead as the animals routinely swam around a large circular tank or during feeding bouts. Turning performance was classified into three different types: (1) glide turns, where the tuna uses the caudal fin as a rudder; (2) powered turns, where the animal uses continuous near symmetrical strokes of the caudal fin through the turn; and (3) ratchet turns, where the overall global turn is completed by a series of small local turns by asymmetrical stokes of the caudal fin. Individual points of the rostrum, peduncle and tip of the caudal fin were tracked and analyzed. Frame-by-frame analysis showed that the ratchet turn had the fastest turn rate for all points with a maximum of 302 deg s(-1). During the ratchet turn, the rostrum exhibited a minimum global 0.38 body length turn radius. The local turn radii were only 18.6% of the global ratchet turn. The minimum turn radii ranged from 0.4 to 1.7 body lengths. Compared with the performance of other swimmers, the increased flexion of the peduncle and tail and the mechanics of turning behaviors used by tuna overcomes any constraints to turning performance from the rigidity of the anterior body morphology

The evolution of distributed sensing and collective computation in animal populations

Many animal groups exhibit rapid, coordinated collective motion. Yet, the evolutionary forces that cause such collective responses to evolve are poorly understood. Here, we develop analytical methods and evolutionary simulations based on experimental data from schooling fish. We use these methods to investigate how populations evolve within unpredictable, time-varying resource environments. We show that populations evolve toward a distinctive regime in behavioral phenotype space, where small responses of individuals to local environmental cues cause spontaneous changes in the collective state of groups. These changes resemble phase transitions in physical systems. Through these transitions, individuals evolve the emergent capacity to sense and respond to resource gradients (i.e. individuals perceive gradients via social interactions, rather than sensing gradients directly), and to allocate themselves among distinct, distant resource patches. Our results yield new insight into how natural selection, acting on selfish individuals, results in the highly effective collective responses evident in nature.National Science Foundation (NSF)Office of Naval ResearchArmy Research OfficeHuman Frontier Science ProgramNSFJames S McDonnell Foundatio

Kinematics of swimming of the manta ray: three-dimensional analysis of open water maneuverability

For aquatic animals, turning maneuvers represent a locomotor activity that may not be confined to a single coordinate plane, making analysis difficult particularly in the field. To measure turning performance in a three-dimensional space for the manta ray (Mobula birostris), a large open-water swimmer, scaled stereo video recordings were collected. Movements of the cephalic lobes, eye and tail base were tracked to obtain three-dimensional coordinates. A mathematical analysis was performed on the coordinate data to calculate the turning rate and curvature (1/turning radius) as a function of time by numerically estimating the derivative of manta trajectories through three-dimensional space. Principal component analysis (PCA) was used to project the three-dimensional trajectory onto the two-dimensional turn. Smoothing splines were applied to these turns. These are flexible models that minimize a cost function with a parameter controlling the balance between data fidelity and regularity of the derivative. Data for 30 sequences of rays performing slow, steady turns showed the highest 20% of values for the turning rate and smallest 20% of turn radii were 42.65+16.66 deg s-1 and 2.05+1.26 m, respectively. Such turning maneuvers fall within the range of performance exhibited by swimmers with rigid bodies