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

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

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

Effects of Demographic Stochasticity on Population Persistence in Advective Media

Many populations live and disperse in advective media. A fundamental question, known as the “drift paradox” in stream ecology, is how a closed population can survive when it is constantly being transported downstream by the flow. Recent population-level models have focused on the role of diffusive movement in balancing the effects of advection, predicting critical conditions for persistence. Here, we formulate an individual-based stochastic analog of the model described in (Lutscher et al., SIAM Rev. 47(4):749–772, 2005) to quantify the effects of demographic stochasticity on persistence. Population dynamics are modeled as a logistic growth process and dispersal as a position-jump process on a finite domain divided into patches. When there is no correlation in the interpatch movement of residents, stochasticity simply smooths the persistence-extinction boundary. However, when individuals disperse in “packets” from one patch to another and the flow field is memoryless on the timescale of packet transport, the probability of persistence is greatly enhanced. The latter transport mechanism may be characteristic of larval dispersal in the coastal ocean or wind-dispersed seed pods

Modeling the influence of flow on invertebrate drift across spatial scales using a 2D hydraulic model and a 1D population model

Methods for creating explicit links in environmental flow assessments between changes in physical habitat and the availability and delivery rate of macroinvertebrates that comprise fish diets are generally lacking. Here, we present a hybrid modelling approach to simulate the spatial dynamics of macroinvertebrates in a section of the Merced River in central California, re-engineered to improve the viability of Chinook salmon. Our efforts focused on quantifying the influence of the hydrodynamic environment on invertebrate drift dispersal, which is a key input to salmon bioenergetics models. We developed a two-dimensional hydrodynamic model that represented flow dynamics well at baseflow and 75% bankfull discharges. Hydraulic predictions from the 2D model were coupled with a particle tracking algorithm to compute drift dispersal, where the settling rates of simulated macroinvertebrates were parameterized from the literature. Using the cross-sectional averaged velocities from the 2D model, we then developed a simpler 1D representation of how dispersal distributions respond to flow variability. These distributions were included in 1D invertebrate population models that represent variability in drift densities over reach scales. Dispersal distributions in the 2D simulation and 1D representation responded strongly to spatial changes in flow. When included in the 1D population model, dispersal responses to flow 'scaled-up' to yield distributions of drifting macroinvertebrates that showed a strong inverse relationship with flow velocity. The strength of the inverse relationship was influenced by model parameters, including the rate at which dispersers settle to the benthos. Finally, we explore how the scale of riffle/pool variability relative to characteristic length scales calculated from the 1D population model can be used to understand drift responses for different settling rates and at different discharges. We show that, under the range of parameter values explored, changes in velocity associated with transitions between riffles and pools produce local changes in drift density of proportional magnitude. This simple result suggests a means for confronting model predictions against field data. © 2013 Elsevier B.V

Modeling the influence of flow on invertebrate drift across spatial scales using a 2D hydraulic model and a 1D population model

Methods for creating explicit links in environmental flow assessments between changes in physical habitat and the availability and delivery rate of macroinvertebrates that comprise fish diets are generally lacking. Here, we present a hybrid modelling approach to simulate the spatial dynamics of macroinvertebrates in a section of the Merced River in central California, re-engineered to improve the viability of Chinook salmon. Our efforts focused on quantifying the influence of the hydrodynamic environment on invertebrate drift dispersal, which is a key input to salmon bioenergetics models. We developed a two-dimensional hydrodynamic model that represented flow dynamics well at baseflow and 75% bankfull discharges. Hydraulic predictions from the 2D model were coupled with a particle tracking algorithm to compute drift dispersal, where the settling rates of simulated macroinvertebrates were parameterized from the literature. Using the cross-sectional averaged velocities from the 2D model, we then developed a simpler 1D representation of how dispersal distributions respond to flow variability. These distributions were included in 1D invertebrate population models that represent variability in drift densities over reach scales. Dispersal distributions in the 2D simulation and 1D representation responded strongly to spatial changes in flow. When included in the 1D population model, dispersal responses to flow 'scaled-up' to yield distributions of drifting macroinvertebrates that showed a strong inverse relationship with flow velocity. The strength of the inverse relationship was influenced by model parameters, including the rate at which dispersers settle to the benthos. Finally, we explore how the scale of riffle/pool variability relative to characteristic length scales calculated from the 1D population model can be used to understand drift responses for different settling rates and at different discharges. We show that, under the range of parameter values explored, changes in velocity associated with transitions between riffles and pools produce local changes in drift density of proportional magnitude. This simple result suggests a means for confronting model predictions against field data. © 2013 Elsevier B.V

Parallel Simulation for a Fish Schooling Model on a General-Purpose Graphics Processing Unit

We consider an individual-based model for fish schooling, which incorporates a tendency for each fish to align its position and orientation with an appropriate average of its neighbors' positions and orientations, in addition to a tendency for each fish to avoid collisions. To accurately determine the statistical properties of the collective motion of fish whose dynamics are described by such a model, many realizations are typically required. This carries a very high computational cost. The current generation of graphics processing units is well suited to this task. We describe our implementation and present computational experiments illustrating the power of this technology for this important and challenging class of problems