. The Biological bulletin. Biology; Zoology; Biology; Marine Biology. 1 0 -1 (9) • -1 • 1 0 -1 (h). 1/3 D 2/3 D 1/3 D 2/3 D Figure 3. Caricatures of the attraction-repulsion functions used by various authors to simulate schooling in fish (see also Table 1). Repulsion is always negative on the y-axis, whereas attraction is positive. Zones of no repulsion or attraction (, parallel orientation) are at zero. The .v-axis is distance from fish of interest, displayed in arbitrary units, (a) Aoki (1982). (b) Warburton and Lazarus (1991). (c) Huth and Wissel (1990, 1
. The Biological bulletin. Biology; Zoology; Biology; Marine Biology. 1 0 -1 (9) • -1 • 1 0 -1 (h). 1/3 D 2/3 D 1/3 D 2/3 D Figure 3. Caricatures of the attraction-repulsion functions used by various authors to simulate schooling in fish (see also Table 1). Repulsion is always negative on the y-axis, whereas attraction is positive. Zones of no repulsion or attraction (, parallel orientation) are at zero. The .v-axis is distance from fish of interest, displayed in arbitrary units, (a) Aoki (1982). (b) Warburton and Lazarus (1991). (c) Huth and Wissel (1990, 1992. 1994). (d) Curves showing the relative strengths of repulsion (left), parallel orientation (center), and attraction (right) in Reuter and Breckling (1994). Note ihat total social force always sums to (e) Romey (1996). (f) The attraction strength used by Vabe and Nettestad (1997). (g) Stocker (1999). (h) Our simulation. these aspects (among others), direct comparisons of their results are not possible. Finally, although most authors quantified school charac- teristics in their simulations with one or more index of aggregation, there is little or no consensus on which indices are most biologically relevant. Consequently, different stud- ies tend to cite different statistics, again making compari- sons difficult. For example, it is difficult to determine whether Romey's (1996) model provides the same amount of polarization as Huth and Wissel's (1992), because the former did not report that particular emergent property. A related but more subtle problem is that no single paper has simultaneously presented indices at the individual, group, and population levels (Table 1). In part this is because the distinction between group and population is meaningless in some models (, all individuals always interact and there- fore the entire population is always part of the same group. , Aoki, 1982; Reuter and Breckling, 1994). Even when this is not the case, most studies u
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