Thousands of Eciton workers leave the bivouac, lay pheromone, and come back with prey. The paper asks whether the trail structure that emerges from these simple rules is optimal: the most food brought home for the fewest ants in the field. Below is the model, running live, with the parameter values the paper's search algorithm settled on for three species.
Cumulative fitness F over time: at each step, food entering the nest divided by ants in the arena, summed.
Solé et al. 2000: three front sites, Table 1 parameters
Deneubourg et al. 1989: two front sites, Figure 2 of Franks et al. 1991
Trail parameters (change while running)
W is the number of sites across; H is the number of sites between the bivouac row and the far edge. The 2000 paper uses 200 × 200.
Food sources: fraction of sites occupied, and units per source
Each panel runs the model with its own Table 1 parameters and food layout, from the same seed, stepping in lockstep. The differences the paper describes are easiest to see here: a narrow trail with a few scattered ants for large, rare food; a broad dendritic fan for small, common food; something in between for the mix. Trail following is weak at μ ≈ 47, so the contrast is subtle; the μ override lets you try the value of 5 used in the original Deneubourg model.
Army ant colonies of the genus Eciton forage in raids involving thousands of workers. The raid is not directed by anyone. Each ant reads the trail pheromone on the ground in front of it, moves or waits, and adds pheromone of its own. From that, a colony-wide structure emerges, and the structure differs by species in a way that lines up with diet.
Deneubourg, Goss, Franks, and Pasteels had shown in 1989 that one set of individual rules, exposed to different prey distributions, produces all three patterns, and Franks and colleagues confirmed in 1991 that an E. burchelli colony given a different prey layout changes its raid to look like E. rapax. Solé and coauthors take that model as given and ask a further question: has evolution tuned these rules so that the resulting pattern is optimal? They pick a concrete measure of optimality, return on investment, and search the model's parameter space for the best performers under each diet. If the best performers look like the real raids, that is evidence the raid structure reflects a selected mechanism rather than an accident of the rules.
Ants are identical, discrete units on a square lattice of side L = 200. Each ant is in one of two states: searching (moving away from the bivouac) or returning (moving back, always carrying one food item). A pheromone field φ(i, j) ≥ 0 lives on the lattice sites.
Searching ants leave one unit of pheromone per step, unless the site already holds more than σ1. Returning ants leave q units, with a separate ceiling σ2. The field evaporates at rate δ every time step:
Movement follows two rules, applied in sequence each step. The ant looks only at the three sites directly ahead of it (left-front, front, right-front), so "ahead" means away from the bivouac for searchers and toward it for returners.
Three housekeeping rules complete it. No site may hold more than Am = 30 ants; a move into a full site is cancelled. Ants that reach the edge of the lattice are lost and removed. Each step, Nb = 10 new ants enter at the bivouac, located at the middle of the bottom row, unless the bivouac site is already full. The authors report that other sigmoid-shaped choices for the direction rule give the same results.
Solé and coauthors describe their model as following Deneubourg et al. (1989), but the rules they write down differ from the earlier model in ways that change what the patterns look like. Franks, Gomez, Goss, and Deneubourg (1991) reprint the 1989 simulations as their Figure 2, and three things are visible in it that the 2000 rules with Table 1 parameters cannot produce.
The 1989 presets in the simulator use: two front sites; μ = 5; one unit of pheromone per step outbound and q = 10 inbound; a ceiling of 1,000 units; δ = 1/30; 10 ants entering per step; and a site capacity of 10 with the retry rule. The first five follow the 1989 model as reported in the paper and in later descriptions of it. The capacity of 10 and the retry rule are inferences: they are what makes the front a flat band rather than a blob, and the 1989 text can be read either way about what a blocked ant does. With these settings, the three food layouts of Figure 2 give a swarm front on a bare principal trail (A), a trail system that splits into several small subswarms (B), and a single cohesive fan with a looping network behind it (C), at step 800 to 1,200 in a 200 × 300 arena.
Figure 1 of the 1991 paper is different in kind: it is field data, redrawn from Burton and Franks (1985), not a simulation output. The model reproduces two of its three families. Figure 2C is the E. burchelli swarm of Figure 1C, and Figure 2B, which the 1991 authors label as rapax-like, is the many-small-swarms system of Figure 1B. Neither paper shows a simulated E. hamatum raid, and in this model making prey rarer or larger than the Figure 2B setting does not give the long branching dendrites of Figure 1A; it pushes the raid back toward the no-food pattern, because the sweeping front finds nothing to return from. The 1991 field experiments made a burchelli swarm raid like rapax by concentrating its prey; they did not make it raid like hamatum.
Return on investment is measured over a run of T = 1,000 steps on the 200 × 200 lattice. At each step the amount of food entering the nest, αnest, is divided by the number of ants currently in the arena, Nants, and these ratios are summed:
Food rewards the colony; ants in the field are the cost. Five parameters are searched: q, μ, δ, σ1, and σ2. The search is the macroevolutionary algorithm of Marín and Solé rather than a genetic algorithm. It keeps a population of Ns = 16 candidate parameter sets and, each generation:
The paper uses τ = 0.2 and ε = 0.2, draws the initial candidates uniformly from 1 < q < 50, 0 < μ < 50, 10−3 < δ < 1, and 0 < σ1, σ2 < 1,000, and runs for about 40 generations. Three food landscapes are tried, each standing in for a species' diet: 1% of sites holding 200 units (large, rare), 50% of sites holding one unit (small, common), and a mix of 25% one-unit sites with 0.125% 200-unit sites.
The same algorithm runs below in a background thread. The paper's setting (200 × 200 sites, T = 1,000, 16 runs per generation) takes several minutes per 40 generations in a browser, so the defaults use a smaller arena and shorter runs. Fitness values depend on arena size and run length, so compare shapes and parameter trends with the paper rather than absolute numbers.
Food landscape
| parameter | best so far | Table 1, this diet |
|---|---|---|
| σ1 | ||
| σ2 | ||
| q | ||
| δ | ||
| μ | ||
| fitness F |
Best (solid) and mean (faint) fitness per generation. The paper's Figures 2 to 4 plot the same curve.
Best pattern in the latest generation.
In each landscape, the first generations are populated by random walkers: the raid envelope grows like the square root of distance, and almost nothing comes home. By generation 15 or so the algorithm has found short trails, and by generation 35 the pattern has settled. The three settled patterns match the three species.
| landscape | σ1 | σ2 | q | δ | μ | pattern |
|---|---|---|---|---|---|---|
| large, rare | 431 | 592 | 46 | 2.5 × 10−3 | 47.5 | single trails |
| poor, common | 51 | 390 | 47 | 2.1 × 10−3 | 47 | fractal |
| intermediate | 72 | 675 | 50 | 3.2 × 10−3 | 48.5 | few trails |
Three things stand out in the table. First, q, μ, and δ barely differ across landscapes: the best solutions all deposit as much pheromone as allowed, evaporate it as slowly as allowed, and still give empty sites strong attraction. A good strategy keeps a persistent trail while leaving room to explore. Second, the saturation ceilings σ1 and σ2 are what differ, and they are much higher for large, rare prey. Strong trails to big sources pay off; for scattered small prey, lower ceilings keep the system flexible. Third, σ1 < σ2 in every case: ants on the way out lay less than ants on the way back with prey, which is what is observed in real army ants and here falls out of maximizing efficiency rather than being assumed.
The paper draws one testable prediction from this: E. hamatum foragers should lay markedly more pheromone than the other two species.
Deneubourg and colleagues had shown that one rule set can produce all three patterns, but their comparison with real raids was by eye. This study does not quantify the shapes either, but it quantifies what the shapes do for the colony, and finds that generating the optimal pattern for each diet requires different parameter values, specifically different σ1 and σ2. Those two ceilings are the natural candidates for what evolution tuned to fit each species to its niche. Since the raid emerges from rules followed by individuals, the conclusion is that colony-level selection has shaped individual behavior so that self-organization implements an optimal foraging strategy: variation in individual genotypes shows up as colony-level pattern.
The paper describes the rules in prose and leaves a few details implicit. The choices made here, so that results can be read with the right amount of trust: