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We employ discrete mathematical models to explore how two different kinds of structures arise in fish groups. The first structure is the ellipsoidal spatial arrangement of many fish schools, observed both in the laboratory and in the wild. Our model is the first to account for the imperfect packing observed in actual schools, and to predict the shape of the ellipsoid.
The second structure is the acyclic set of pairwise dominance relationships observed in small groups (joint work with I. Chase). Recent experiments demonstrate that this social structure is self-organizing. How could the individual behaviors that generate such structure evolve? We hypothesize a behavioral trait, reactivity, which is consistent with laboratory experiment. For a simple model of hierarchy formation, the presence of this trait increases the probability of acyclicity. Moreover, there is positive selection pressure for reactivity, under realistic assumptions about the relation between fitness and hierarchy rank.
As mentioned in the abstract, Dr. Tovey addresses two different classes of fish groupings- those of larger fish schools versus smaller groupings.
This analysis is approached from the point of view of a fish school as a multi-agent, self-organizing dynamic structure, where control is distributed and each agent has limited information and computational capacity. In general, empirical observations indicate effective group coordination, with an absence of individual leaders ("alpha" fish). All members of a school tend to maintain the same speed and orientation. School behavior is often altered in the presence of a predator; for example, increased orientation in schools has been observed in such situations. In the talk, three categories of fish schools are cited, but the analysis focuses on that of obligate schooling (observed in herring & pilchard), which is characterized by a consistent form, maintained polarization, and a strong tendency of member fish to remain in the school (any fish that get seperated from the group become agitated and less organized).
Obligate schools tend to be ellipsoidal in shape when observed in their natural habitat, and occasionaly so when observed in tanks. It has been observed that the following factors affect school structure: hunger, light, time of day, background noise, presence of predators, blindness, and deafness (yes, researchers actually managed to plug the fishes' ears!). Contrary to this, the removal of the forebrains of member fishes had less effect in obligate schools. Also, fishes in obligate schools exhibit distance and angle preferences.
To predict obligate school shapes, Dr. Tovey applies a lattice-type model, which is suggested by the distance and angle preferences of member fish. The model begins with one fish in a lattice position, and then individually adds fish into available spaces with either uniform or weighted probability. The results of this model indicate elliptical formations with low eccentricity (within a few percent), and "holes" within the lattice structure. Not surprisingly, the formation dimensions as well as fish density is sensitive to the lattice structure used. Analysis of the algorithm shows that the expected diameter of a school given n fish is Theta[n^(1/3)]. Dr. Tovey indicates a desire to test these outcomes empirically, but describes the current state of this endeavor as "dead fish".
One weakness in this model is its incompleteness in representing all dynamical phenomena, such as the merging of fish schools. Also, an exact lattice structure is not adhered to by actual fish schools. Dr. Tovey characterizes the main strength of the model as its ability to explain the resulting dense ellipsoidal shape using distributed control.
When fish meets fish, chasing, nipping, lunging & retreating, and liplocks are often observed, indicating a dominance of one fish over another. In groups, all pairs interact, and the dominance relations formed tend to be transitive ("cycles" are rare). Empirical observations of fish dominance hierarchies do not provide a clear characterization of the principles of formation behind these hierarchies; it is concluded that this is a self-organizing phenomenon. This not only raises the question of how these hierarchies emerge, but also of the evolutionary selection pressure that brought about this behavior, which must be investigated on an individual fish basis.
Here a "reactivity" model of hierarchy formation is presented. In it, each fish is assigned a fixed reactivity parameter B_i, where B_i=0 indicates fish i is not reactive. With this, the parameter mu_i = I B_i is defined (I is defined in a moment). Pairs of fish are randomly chosen to have an encounter, and the outcome between fish i and fish j is determined by comparing randomly selected values from normal distributions with mean mu_i: N(mu_i,1) vs. N(mu_j,1). The variable I in the formula for mu_i is then set as follows: I=0 if fish i has had no encounters, I=1 if fish i has won its most recent encounter, I=-1 if fish i has lost its most recent encounter.
Clearly, if a fish loses a dominance encounter, it is most likely to lose the next (and vice-versa). Also, although reactivity implies longer strings of wins, it does not strongly affect the expected number of wins. This results in hierarchy ranks being pushed to extremes (to the upside for higher reactivity); this, coupled with the fact that higher ranking fish have more progeny, imply that fish with high reactivity have higher evolutionary fitness than those with lower reactivity.
A theorem based on this model is presented, which states that for n<=4, the probability of transitivity in the dominance hierarchy increases in B (the reactivity). Unfortunately, even as B -> Infinity, the probability of transitivity remains strictly less than 1, implying that reactivity alone does not suffice to explain the observed high frequency of transitivity. The model can be modified to include a "prior attribute" parameter a_i, which is a measure of weight, strength, and other attributes that can affect the outcomes of dominance encounters. After redefining mu_i as mu_i= a_i + I B_i, it is shown that for n=3, transitivity remains more likely with increased reactivity, but not for n=4. Thus more work needs to be done.