Schooling as a strategy for taxis in a noisy environment

Journal: Evolutionary Ecology: Evolutionary Ecology is a conceptually oriented journal of basic biology at the interface of ecology and evolution. The journal publishes original research, reviews and discussion papers dealing with evolutionary ecology, including evolutionary aspects of behavioral and population ecology. The objective is to promote the conceptual, theoretical and empirical development of ecology and evolutionary biology; the scope extends to all organisms and systems. Research papers present the results of empirical and theoretical investigations, testing current theories in evolutionary ecology.

Author: Daniel Grunbaum: My research program seeks to establish quantitative relationships between short-term, small-scale processes, such as individual movement behaviors, and their long-term, large-scale population level effects, such as population fluxes and distributions.

• The Dynamics of Animal Grouping
• Mathematical Treatment of Biological Diffusion
• Oscillator Models and Collective Motion

Abstract

• 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.
  • Consider voters switching from Bush->Obama->Trump
• Such an animal spends more time moving in right directions than wrong ones, and eventually gets to a favourable area. Taxis is ineffcient, 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.
  • This has implications for everything from human social interaction to ANN design.

Notes

• Because limiting resources typically have `patchy’ distributions in which concentrations may vary by orders of magnitude, success or failure in finding favourable areas often has an enormous impact on growth rates and reproductive success. To locate resource concentrations, many aquatic organisms display tactic behaviours, in which they orient with respect to local variations in chemical stimuli or other environmental properties. (pp 503)
• Here, I propose that schooling behaviours improve the tactic capabilities of school members, and enable them to climb faint and noisy gradients which they would otherwise be unable to follow. (pp 504)
• Schooling is thought to result from two principal behavioural components: (1) tendencies to move towards neighbours when isolated, and away from them when too close, so that the group retains a characteristic level of compactness; and (2) tendencies to align orientation with those of neighbours, so that nearby animals have similar directions of travel and the group as a whole exhibits a directional polarity. (pp 504)
  • My models indicate that attraction isn’t required, as long as there is a distance-graded awareness. In other words, you align most strongly with those agents that are closest.
• I focus in this paper on schooling in aquatic animals, and particularly on phytoplankton as a distributed resource. However, although I do not examine them specifically, the modelling approaches and the basic results apply more generally to other environmental properties (such as temperature), to other causes of population movement (such as migration) and to other socially aggregating species which form polarized groups (such as flocks, herds and swarms). (pp 504)
• Under these circumstances, the search of a nektonic filter-feeder for large-scale concentrations of phytoplankton is analogous to the behaviour of a bacterium performing chemotaxis. The essence of the analogy is that, while higher animals have much more sophisticated sensory and cognitive capacities, the scale at which they sample their environment is too small to identify accurately the true gradient. (pp 505)
  • And, I would contend for determining optimal social interactions in large groups.
• Bacteria using chemotaxis usually do not directly sense the direction of the gradient. Instead, they perform random walks in which they change direction more often or by a greater amount if conditions are deteriorating than if they are improving (Keller and Segel, 1971; Alt, 1980; Tranquillo, 1990). Thus, on average, individuals spend more time moving in favourable directions than in unfavourable ones. (pp 505)
• A bacterial analogy has been applied to a variety of behaviours in more complex organisms, such as spatially varying di€usion rates due to foraging behaviours or food-handling in copepods and larval ®sh (Davis et al., 1991), migration patterns in tuna (Mullen, 1989) and restricted area searching in ladybugs (Kareiva and Odell, 1987) and seabirds (Veit et al., 1993, 1995). The analogy provides for these higher animals a quantitative prediction of distribution patterns and abilities to locate resources at large space and time scales, based on measurable characteristics of small-scale movements. (pp 505)
• I do not consider more sophisticated (and possibly more effective) social tactic algorithms, in which explicit information about the environment at remote points is actively or passively transmitted between individuals, or in which individual algorithms (such as slowing down when in relatively high concentrations) cause the group to function as a single sensing unit (Kils, 1986, described in Pitcher and Parrish, 1993). (pp 506)
  • This is something that could be easily added to the model. There could be a multiplier for each data cell that acts as a velocity scalar of the flock. That should have significant effects! This could also be applied to gradient descent. The flock of Gradient Descent Agents (GDAs) could have a higher speed across the fitness landscape, but slow and change direction when a better value is found by one of the GDAs. It occurs to me that this would work with a step function, as long as the baseline of the flock is sufficiently broad.
• When the noise predominates (d <= 1), the angular distribution of individuals is nearly uniform, and the up-gradient velocity is near zero. In a range of intermediate values of d(0.3 <= d <= 3), there is measurable but slow movement up-gradient. The question I will address in the next two sections is: Can individuals in this intermediate signal-to-noise range with slow gradient-climbing rates improve their tactic ability by adopting a social behaviour (i.e. schooling)? (pp 508)
• The key attributes of these models are: (1) a decreasing probability of detection or responsiveness to neighbours at large separation distances; (2) a social response that includes some sort of switch from attractive to repulsive interactions with neighbours, mediated by either separation distance or local density of animals*; and (3) a tendency to align with neighbours (Inagaki et al., 1976; Matuda and Sannomiya, 1980, 1985; Aoki, 1982; Huth and Wissel, 1990, 1992; Warburton and Lazarus, 1991; Grunbaum, 1994). (pp 508)
  • * Though not true of belief behavior (multiple individuals can share the same belief), for a Gradient Descent Agent (GDA), the idea of attraction/repulsion may be important.
• If the number of neighbours is within an acceptable range, then the individual does not respond to them. On the other hand, if the number is outside that range, the individual turns by a small amount, Δθ<sub>3</sub>, to the left or right according to whether it has too many or too few of them and which side has more neighbours. In addition, at each time step, each individual randomly chooses one of its visible neighbours and turns by a small amount, Δθ<sub>4</sub>, towards that neighbour’s heading. (pp 508)
• The results of simulations based on these rules show that schooling individuals, on average, move more directly in an up-gradient direction than asocial searchers with the same tactic parameters. Figure 4 shows the distribution of individuals in simulations of asocial and social taxis in a periodic domain (i.e. animals crossing the right boundary re-enter the left boundary, etc.). (pp 509)
• 
• As predicted by Equation (5), asocial taxis results in a broad distribution of orientations, with a peak in the up-gradient (positive x-axis) direction but with a large fraction of individuals moving the wrong way at any given time (Fig. 5a,b). By comparison, schooling individuals tend to align with one another, forming a group with a tightened angular distribution. There is stochasticity in the average velocity of both asocial and social searchers (Fig. 5c). On average, however, schooling individuals move up-gradient faster and more directly than asocial ones. These simulation results demonstrate that it is theoretically possible to devise tactic search strategies utilizing social behaviours that are superior to asocial algorithms. That is, one of the advantages of schooling is that, potentially, it allows more successful search strategies under `noisy’ environmental conditions, where variations on the micro-scales at which animals sense their environment obscure the macro-scale gradients between ecologically favourable and unfavourable regions. (pp 510)
• School-size effects must depend to some extent on the tactic and schooling algorithms, and the choices of parameters. However, underlying social taxis are the statistics of pooling outcomes of independent decisions, so the numerical dependence on school size may operate in a similar manner for many comparable behavioural schemes. For example, it seems reasonable to expect that, in many alternative schooling and tactic algorithms, decisions made collectively by less than 10 individuals would show some improvement over the asocial case but also retain much of the variability. Similarly, in most scenarios, group statistics probably vary only slowly with group size once it reaches sizes of 50-100. (pp 514)
• when group size becomes large, the behaviour of model schools changes in character. With numerous individuals, stochasticity in the behaviour of each member has a relatively weaker effect on group motion. The behaviour of the group as a whole becomes more consistent and predictable, for longer time periods. (pp 514)
  • I think that this should be true in belief spaces as well. It may be difficult to track one person’s trajectory, but a group in aggregate, particularly a polarized group may be very detectable.
• An example of group response to changing gradient direction shows that there can be a cost to strong alignment tendency. In this example, the gradient is initially pointed in the negative y-direction (Fig. 9). After an initial period of 5 time units, during which the gradient orients perpendicularly to the x-axis, the gradient reverts to the usual x-direction orientation. The school must then adjust to its new surroundings by shifting to climb the new gradient. This example shows that alignment works against course adjustment: the stronger the tendency to align, the slower is the group’s reorientation to the new gradient direction. This is apparently due to a non-linear interaction between alignment and taxis: asymmetries in the angular distribution during the transition create a net alignment flux away from the gradient direction. Thus, individuals that pay too much attention to neighbours, and allow alignment to overwhelm their tactic tendencies, may travel rapidly and persistently in the wrong direction. (pp 516)
  • So, if alignment (and velocity matching) are strong enough, the conditions for a stampede (group behavior with negative outcomes – in this case, less food) emerge
• The models also suggest that there is a trade-off in strengthening tendencies to align with neighbours: strong alignment produces tight angular distributions, but increases the time needed to adjust course when the direction of the gradient changes. A reasonable balance seems to be achieved when individuals take roughly the same time to coalesce into a polarized group as they do to orient to the gradient in asocial taxis. (pp 518)
  • There is something about the relationship between explore and exploit in this statement that I really need to think about.
• Social taxis is potentially effective in animals whose resources vary substantially over large length scales and for whom movements over these scales are possible. (pp 518)
  • Surviving as a social animal requires staying in the group. Since belief can cover wide ranges (e.g. religion), does there need to be a mechanism where individuals can harmonize their beliefs? From Social Norms and Other Minds The Evolutionary Roots of Higher Cognition :  Field research on primate societies in the wild and in captivity clearly shows that the capacity for (at least) implicit appreciation of permission, prohibition, and obligation social norms is directly related to survival rates and reproductive success. Without at least a rudimentary capacity to recognize and respond appropriately to these structures, remaining within a social group characterized by a dominance hierarchy would be all but impossible.
• Interestingly, krill have been reported to school until a food patch has been discovered, whereupon they disperse to feed, consistent with a searching function for schooling. The apparent effectiveness of schooling as a strategy for taxis suggests that these schooling animals may be better able to climb obscure large-scale gradients than they would were they asocial. Interactive effects of taxis and sociality may affect the evolutionary value of larger groups both directly, by improving foraging ability with group size, and indirectly, by constraining alignment rates. (pp 518)
• An example where sociality directly affects foraging strategy is forage area copying, in which unsuccessful fish move to the vicinity of neighbours that are observed to be foraging successfully (Pitcher et al., 1982; Ranta and Kaitala, 1991; Pitcher and Parrish, 1993). Pitcher and House (1987) interpreted area copying in goldfish as the result of a two-stage decision process: (1) a decision to stay put or move depending on whether feeding rate is high or low; and (2) a decision to join neighbours or not based upon whether or not further solitary searching is successful. Similar group dynamics have been observed in foraging seabirds (Porter and Seally, 1982; Haney et al., 1992).
  • This is an interesting refinement of the multi-armed bandit problem. It might be modeled in this: Multi-armed bandits in the presence of side observations in social networks
• Synchrokinesis depends upon the school having a relatively large spatial extent: part of a migrating school encounters an especially favourable or unfavourable area. The response of that section of the school is propagated throughout the school by alignment and grouping behaviours, with the result that the school as a whole is more effective at route-finding than isolated individuals. Forage area copying and synchrokinesis are distinct from social taxis in that an individual discovers and reacts to an environmental feature or resource, and fellow group members exploit that discovery. In social taxis, no individual need ever have greater knowledge about the environment than any other — social taxis is essentially bound up in the statistics of pooling the outcomes of many unreliable decisions. Synchrokinesis and social taxis are complementary mechanisms and may be expected to co-occur in migrating and gradient-climbing schools. (pp 519)
• For example, in the comparisons of taxis among groups of various sizes, the most successful individuals were in the asocial simulation, even though as a fraction of the entire population they were vanishingly small. (pp 519)
  • Explorers have the highest payoff for the highest risks