Daniel Grünbaum

Self-Organized Fish Schools: An Examination of Emergent Properties

Heterogeneous, “aggregated” patterns in the spatial distributions of individuals are almost universal across living organisms, from bacteria to higher vertebrates. Whereas specific features of aggregations are often visually striking to human eyes, a heuristic analysis based on human vision is usually not sufficient to answer fundamental questions about how and why organisms aggregate. What are the individual-level behavioral traits that give rise to these features? When qualitatively similar spatial patterns arise from purely physical mechanisms, are these patterns in organisms biologically significant, or are they simply epiphenomena that are likely characteristics of any set of interacting autonomous individuals? If specific features of spatial aggregations do confer advantages or disadvantages in the fitness of group members, how has evolution operated to shape individual behavior in balancing costs and benefits at the individual and group levels? Mathematical models of social behaviors such as schooling in fishes provide a promising avenue to address some of these questions. However, the literature on schooling models has lacked a common framework to objectively and quantitatively characterize relationships between individual-level behaviors and group-level patterns. In this paper, we briefly survey similarities and differences in behavioral algorithms and aggregation statistics among existing schooling models. We present preliminary results of our efforts to develop a modeling framework that synthesizes much of this previous work, and to identify relationships between behavioral parameters and group-level statistics.

Schooling as a strategy for taxis in a noisy environment

Many aquatic animals face a fundamental problem during foraging and migratory movements: while their resources commonly vary at large spatial scales, they can only sample and assess their environment at relatively small, local spatial scales. Thus, they are unable to choose movement directions by directly sampling distant parts of their environment. A common strategy to overcome this problem is taxis, a behaviour in which an animal performs a biased random walk by changing direction more rapidly when local conditions are getting worse. Such an animal spends more time moving in right directions than wrong ones, and eventually gets to a favourable area. Taxis is inefficient, however, when environmental gradients are weak or overlain by ‘noisy’ small-scale fluctuations. In this paper, I show that schooling behaviour can improve the ability of animals performing taxis to climb gradients, even under conditions when asocial taxis would be ineffective. Schooling is a social behaviour incorporating tendencies to remain close to and align with fellow members of a group. It enhances taxis because the alignment tendency produces tight angular distributions within groups, and dampens the stochastic effects of individual sampling errors. As a result, more school members orient up-gradient than in the comparable asocial case. However, overly strong schooling behaviour makes the school slow in responding to changing gradient directions. This trade-off suggests an optimal level of schooling behaviour for given spatio-temporal scales of environmental variations. Social taxis may enhance the selective value of schooling in pelagic grazers such as herrings, anchovies and Antarctic krill. Furthermore, the degree of aggregation in a population of schooling animals may affect directly the rate and direction of migration and foraging movements.

Oscillator Models and Collective Motion

This article describes PCOD, a cooperative control framework for stabilizing relative equilibria in a model of self-propelled, steered particles moving in the plane at unit speed. Relative equilibria correspond either to motion of all of the particles in the same direction or to motion of all of the particles around the same circle. Although the framework applies to time-varying and directed interaction between individuals, we focus here on time-invariant and undirected interaction, using the Laplacian matrix of the interaction graph to design a set of decentralized control laws applicable to mobile sensor networks. Since the direction of motion of each particle is represented in the framework by a point on the unit circle, the closed-loop model has coupled-phase oscillator dynamics.

MODELLING SOCIAL ANIMAL AGGREGATIONS

It is hard to find animals in nature that do not aggregate for one reason or another. The details of such aggregations are important because they influence numerous fundamental processes like mate-finding, prey-detection, predator avoidance, and disease transmission. Yet, despite the near universality of aggregation and its profound consequences, biologists have only recently begun to probe its underlying mechanisms. In this chapter we review theoretical approaches to animal aggregation, concentrating on aggregations which are caused by social interactions. We emphasize methods and limitations, and suggest what we think are the most promising avenues for future research. For an earlier review article on dynamical aspects of animal grouping consult Okubo (1986).

BLACK‐BROWED ALBATROSSES FORAGING ON ANTARCTIC KRILL: DENSITY‐DEPENDENCE THROUGH LOCAL ENHANCEMENT?

Many Antarctic seabirds depend on prey that are patchy, cryptic, ephemeral, and unpredictable in location. These predators typically employ two alternative behavioral strategies for locating resource patches: direct visual or olfactory detection, and indirect detection (local enhancement) by sighting other predators that are already exploiting a patch. We developed a model of direct detection and local enhancement in seabirds that predicts how foraging success varies with behavioral strategy, seabird densities, and prey swarm density and detectability. Application of the model to Black-browed Albatrosses foraging for Antarctic krill near South Georgia suggests that local enhancement is generally a highly effective foraging strategy, and that the fraction of time albatrosses spend in feeding flocks should show strong interactions between prey and conspecific densities. To test these predictions, we analyzed survey data collected near South Georgia in January-March 1986. Our analysis suggests a strong Allee-type density dependence in foraging success that was qualitatively and quantitatively consistent with model predictions. This density dependence suggests a potential for destabilizing patterns of resource utilization and reproductive suc- cess in Black-browed Albatrosses that may have important implications for conservation of albatrosses and other Antarctic species.

Good eaters, poor swimmers: compromises in larval form.

Compromises between swimming and feeding affect larval form and behavior. Two hypotheses, with supporting examples, illustrate these feeding-swimming trade-offs. (1) Extension of ciliated bands into long loops increases maximum clearance rates in feeding but can decrease stability of swimming in shear flows. A hydromechanical model of swimming by ciliated bands on arms indicates that morphologies with high performance in swimming speed and weight-carrying ability in still water differ from morphologies conferring high stability to external disturbances such as shear flows. Instability includes movement across flow lines from upwelling to downwelling water in vertical shear. Thus a hypothesis for the high arm elevation angles of sea urchin larvae, which reduce speed in still water, is that they reduce a downward bias imposed by the vertical shear in turbulence. Observations of sea urchin larvae in vertical shear and comparisons among brittle star larvae are consistent with the performance trade-offs predicted by the model. (2) Structures and behaviors that reduce swimming speed can enhance filtering for feeding. In the opposed-band feeding mechanisms of veligers and many trochophores, cilia push water to swim but movement of cilia relative to the water occurs when cilia overtake and capture particles. Features that may increase clearance rates at the expense of speed and weight capacity include structures that increase drag or body weight and a ciliary band that beats in opposition to the feeding-swimming current. Larval feeding mechanisms inherited from distant ancestors result in different swimming-feeding trade-offs. The different trade-offs further diversify larval form and behavior.

Form, performance and trade-offs in swimming and stability of armed larvae

Diverse larval forms swim and feed with ciliary bands on arms or analogous structures. Armed morphologies are varied: numbers, lengths, and orientations of arms differ among species, change through development, and can be plastic in response to physiological or environmental conditions. A hydromechanical model of idealized equal-armed larvae was used to examine functional consequences of these varied arm arrangements for larval swimming performance. With effects of overall size, ciliary tip speed, and viscosity factored out, the model suggested trade-offs between morphological traits conferring high swimming speed and weight-carrying ability in still water (generally few arms and low arm elevations), and morphologies conferring high stability to external disturbances such as shear flows (generally many arms and high arm elevations). In vertical shear, larvae that were passively stabilized by a center of buoyancy anterior to the center of gravity tilted toward and consequently swam into downwelling flows. Thus, paradoxically, upward swimming by passively stable swimmers in vertical shear resulted in enhanced downward transport. This shear-dependent vertical transport could affect diverse passively stable swimmers, not just armed larvae. Published descriptions of larvae and metamorphosis of 13 ophiuroids suggest that most ophioplutei fall into two groups: those approximating modeled forms with two arms at low elevations, predicted to enhance speed and weight capacity, and those approximating modeled forms with more numerous arms of equal length at high elevations, predicted to enhance stability in shear.

The Dynamics of Animal Grouping

Though the term “grouping” is used very ambiguously in this chapter, we mean it to be a phenomenon such as insect swarming or fish schooling in which a number of animals are involved in movement as a group. In this chapter an attempt is made to describe the motion of swarming individuals, and to comprehend the grouping from the standpoint of advection—diusion processes.

The logic of ecological patchiness

Most ecological interactions occur in environments that are spatially and temporally heterogeneous—‘patchy’—across a wide range of scales. In contrast, most theoretical models of ecological interactions, especially large-scale models applied to societal issues such as climate change, resource management and human health, are based on ‘mean field’ approaches in which the underlying patchiness of interacting consumers and resources is intentionally averaged out. Mean field ecological models typically have the advantages of tractability, few parameters and clear interpretation; more technically complex spatially explicit models, which resolve ecological patchiness at some (or all relevant) scales, generally lack these advantages. This report presents a heuristic analysis that incorporates important elements of consumer–resource patchiness with minimal technical complexity. The analysis uses scaling arguments to establish conditions under which key mechanisms—movement, reproduction and consumption—strongly affect consumer–resource interactions in patchy environments. By very general arguments, the relative magnitudes of these three mechanisms are quantified by three non-dimensional ecological indices: the Frost, Strathmann and Lessard numbers. Qualitative analysis based on these ecological indices provides a basis for conjectures concerning the expected characteristics of organisms, species interactions and ecosystems in patchy environments.

From individual behaviour to population models: a case study using swimming algae.

Trajectories of swimming algae are analysed, and two random-walk models developed to link the individual-level behaviour of cells to population-level advection-diffusion models for the spatial-temporal distribution of cells. The models are both of the advection-diffusion form but are based on two different sets of assumptions about the underlying random-walk behaviours, a velocity jump behaviour and a velocity diffusion behaviour. The mathematical models were developed to allow for an arbitrary (non-weak) bias in the random walk and a variable swimming speed in order to represent the experimental data. For the algal species considered, Heterosigma akashiwo, the mean upward swimming speed was computed as 40 microm s(-1), and the calculated diffusion constants ranged from 2 x 10(3) to 4 x 10(4) microm(2) s(-1) depending on the details of the models. That two widely used modelling approaches yield substantially different population-level predictions when applied to the same empirical data suggests that better theoretical tools are needed for identifying adequate approximations for behavioural characteristics.