Slime molds, ants, people on a lawn and termites coordinate through their surroundings rather than with each other: they leave marks that fade and spread, and steer toward the strongest.
The self-avoiding walk steers away from its own trail. The agents here steer toward trails, and in groups they build networks, pick the shorter of two routes and pile up pellets.
In the 1950s Pierre-Paul Grassé proposed that termites rebuilding a nest coordinate through the work itself: each addition changes what the next termite responds to. He called this stigmergy, from the Greek for mark and work.
The version that recurs across biology has three parts: agents write into a shared field as they move, the field fades and spreads, and agents steer toward more of it. For a concentration written by agents at ,
with deposit , decay rate and diffusion . Where agents pass at rate , the field settles at order . If agents tell concentrations apart on a scale , then at every direction looks equally marked, and at marks are too faint to follow. A mark can be read from about away, which sets how close two trails must be to merge.
Physarum polycephalum is a single cell that can spread across a dish many centimetres wide as a network of tubes; tubes that carry more protoplasmic flow thicken and the others decline (Tero and colleagues 2010). Jeff Jones (2010) reproduced these networks with particles on a grid. Each particle samples the field at three points ahead and turns toward the strongest; it steps forward and deposits if the cell ahead is empty, and otherwise picks a random heading. Each step the field is blurred over 3×3 cells and multiplied by a decay factor. Without the one-particle-per-cell rule the particles collapse into one or two dense bands.
Figure 1. Jones's particle model of Physarum. Particles start scattered over 6% of cells; brightness is the field. Sliders take effect immediately. The deposit of 5 and the decay factor of 0.9 are the values Jones and colleagues give for this model (Strano, Adamatzky and Jones 2012); the 45° angles and sensor distance of 9 are one setting that makes networks, chosen for this page.
The particles condense into a web of strands that keeps coarsening. A particle reacts to strands within about one sensor distance: at 20 the holes come out two to three times larger in area than at 9 after the same number of steps. At decay 0.99 a mark lasts about a hundred steps and strands grow thick; at 0.8 it fades within five and the web turns faint.
The real cell also prunes. Nakagaki, Yamada and Tóth (2000) filled a maze with a plasmodium and put food at two exits; the dead-end tubes withdrew, leaving one along the shortest route. Tero, Kobayashi and Nakagaki (2007) modelled tubes carrying a fixed flow between the food sources. Flow divides as current does in a circuit, for a tube of length and conductivity , and each tube adapts to its flow, which plays the part of the mark:
Figure 2. The tube model in a seeded 13×8 maze with ten loops. One unit of flow runs from the top-left food source to the bottom-right; every corridor starts at and line width shows . Dashed: the shortest route. The chart plots the total length of tubes with against time (Euler steps of 0.2).
Dead ends carry no flow and fade within a few time units; loops take longer, because their two arms split the flow until one pulls ahead. At the network ends on the shortest route, as Bonifaci, Mehlhorn and Varma (2012) proved for any network whose shortest route is unique; 49 of 50 seeded mazes had settled by , the last (two near-equal routes) by 1214. Above 1 the network commits several times faster (median 8 time units at , 50 at 1), sometimes wrongly: at 2.5, 4 of 50 mazes kept a route up to 0.7% longer. At 0.5 it keeps about twice the shortest length, and only 30% of its tubes would cut the food sources apart if lost, against all of them on a single route.
Tero and colleagues (2010) found the same trade in the organism. Grown between food placed at 36 cities around Tokyo, its networks matched the rail network on average distance between cities and were slightly shorter, but the rail network was more robust: one broken link isolated part of it 4% of the time, against 14% for slime-mold networks grown with light standing in for mountains and lakes.
Argentine ants (Linepithema humile) lay trail pheromone in both directions. Simon Goss, Jean-Louis Deneubourg and colleagues (1989) connected a nest to food by a bridge that split in two, and within minutes one branch was clearly preferred. With equal branches either could win (the left won 12 of 26 trials); with one branch 1.4 times as long, the short one won 15 of 18, and at twice as long, 14 of 14. Their rule, from earlier experiments: an ant at a fork takes branch over with probability
where and count the ants that have already crossed each branch. Marks much weaker than make no difference, and an exponent above 1 amplifies a small lead. Figure 3 isolates the exponent: equal branches, and ants that cross one at a time.
Figure 3. Equal branches; each ant adds one mark to the branch it takes. Each column is 400 seeded colonies of 1,000 ants at one exponent , shaded by the share of colonies ending in each band of final share on branch A (fixed scale). The bars at right show the selected column; tap a column or use the slider. Changing reruns every column.
Below choices pull back toward an even split. At exactly 1 the rule is a Pólya urn, and colonies spread around the middle (standard deviation 0.073 here at ; the urn formula gives 0.080). Above 1 the even split is unstable: at , 72% of colonies send more than 80% of their ants one way. A larger delays the break; at 100, half the colonies are still between 40% and 60% after 1,000 ants.
The rule has no term for length: on unequal branches the first ants back mark both ends of the short branch before anyone finishes the long trip.
Run 20 colonies simulates twenty independent colonies at the current settings and plots where each one ends up.
Figure 4. The double bridge. Each branch is a one-dimensional field of cells (the short one is 30); line width shows pheromone. 100 ants leave the nest over the first few hundred steps, deposit one unit per cell, and at each end choose with the rule above (k = 20, n = 2) from the three cells nearest that end. The chart tracks the short branch's share of the last 100 departures; the histogram counts final shares in a batch.
With the long branch twice the short one, about three colonies in four settle on the short branch (a batch of 20 usually gives 12 to 17), less reliably than the ants. Whichever branch leads after the first 100 departures goes on to win 9 times in 10.
When Goss and colleagues opened the short branch only after 30 minutes, 2 of 18 colonies switched; in the model without evaporation, none do. Figure 5 adds evaporation.
Figure 5. Figure 4's model, long branch twice the short, 8000 steps, short branch open from the start (blue) or from step 1500 (red), 30 seeded colonies per setting to begin. Bottom: dots are colonies' final shares on the short branch; lines, the share of colonies ending with most traffic there. Top: mean pheromone in the long branch's end cells at step 1500 (late colonies), the steady value with deposits per cell per step (dashed), and .
Before the short branch opens, the long branch's end cells sit just under . While that is several times , ants trying the new branch cannot catch up: up to no late colony switched. Switching starts as the field nears twice (16 of 30 at 0.04) and peaks at 0.044 to 0.048, with 23 and 26 of 30. Past about 0.056 trails fade faster than ants refresh them and traffic splits. Dorigo's Ant System, an optimization algorithm built on trail-following, also evaporates its trails; with the trail weighted too heavily every ant soon follows one tour, which Dorigo and colleagues called stagnation.
Worn paths across a lawn run the same loop with trampled grass as the field. In the active walker model of Dirk Helbing, Joachim Keltsch and Péter Molnár (1997), the ground recovers with durability and is worn by each footstep at a rate that saturates as the grass dies:
A walker sees the ground smoothed over a visibility radius , a field that plays the part of diffusion, and heads along the direction to its destination plus the pull of worn ground. The figure weights the pull by :
Figure 6. The active walker model on a 120×80 lawn with an entrance at each corner; each walker leaves by another corner at random, so the direct routes are the four sides and both diagonals. Colour runs from grass (G = 0) to bare earth (G = 1). Smoothing is a Gaussian with σ = 10 cells in place of Helbing's exponential kernel, and the pull κ∇V is capped at 0.9 of the destination's so a walker always gains ground (uncapped, a strong trail across its path can hold it in place). Detour is walked length over straight-line distance for the last 100 walkers; worn ground is the share with G > 0.3.
At a durability of a few hundred steps, faint wear traces the six direct routes and walkers cover 1.00 times the straight line. At the default the diagonals bow outward and braid across the middle, and walkers go about 7% farther (13% at the longest durability): what Helbing and colleagues call a compromise between direct routes and the shortest network joining the entrances, which they compared with photographs of real lawns.
Grassé's termites carry soil pellets. In Deneubourg's 1977 model, fresh material carries a pheromone that attracts termites carrying pellets. Figure 7 combines that with the rules from Deneubourg's later model of brood sorting: pellets emit pheromone that spreads and decays; a loaded termite drifts up the gradient and drops its pellet with probability (plus a base rate of 0.002); an empty one picks a pellet up with probability .
Figure 7. Pellet clustering on a 120×72 grid with wrap-around edges. About 690 pellets start scattered one per cell over 8% of the grid; 150 termites walk at random (yellow: carrying a pellet, grey: empty). Brown squares are pellets, darker for taller stacks; the amber glow is pheromone. K = 6 and diffusion D = 0.2. The readout counts piles of ten or more pellets and compares the field of a uniform scatter, ρq/λ, with K. Changing a control resets the grid.
At the default lifetime of 20, a lone pellet's field is about half of and a pile's about four times , so lone pellets are about ten times as likely to be picked up; by step 12,000 about fifteen piles hold nearly all the pellets.
By the estimate at the start, clustering should fail once the scatter's own field, , passes : here, at a lifetime of 75 steps.
Figure 8. Figure 7's model at twelve pheromone lifetimes, each run to 12,000 steps from seeded starts (two seeds to begin). The share of pellets in piles of ten or more after 1,000, 4,000 and 12,000 steps; dots are single runs, lines their mean. The dashed line marks .
The line marks a slowdown, not a wall. At a lifetime of 75, 42% to 43% of pellets are in piles at step 4,000 (95% at a lifetime of 15) and 60% to 68% by 12,000; the share falls to about 31% at 150 and 2% or less at 500. At a lifetime of 2 the marks are faint next to and no pile of ten exists at step 1,000, yet by 12,000 every pellet sits in one of 4 to 7 piles.
The cement pheromone itself has held up poorly: Green and colleagues (2017) found Macrotermes building organized around busy excavation sites, and Calovi and colleagues (2019) found soil moved according to surface curvature. Paper wasps respond to the geometry of cells already built, a cue that never decays, and lattice models on that cue alone (Theraulaz and Bonabeau 1995) grow recognizable nests.
Here the agents write a field and the field spreads. In the next essay the animals themselves spread, and that changes the boom and bust of predators and prey.