Bacterial Foraging Optimization (BFO) Explorer

A layered exploration of Passino's (2002) Bacterial Foraging Optimization (BFO) — this tab isolates the cell-to-cell signaling and chemotaxis that drive each individual bacterium. Tab ② extends the same mechanics to 2-D. Tab ③ adds reproduction and elimination-dispersal to complete the full algorithm.

Cell-to-cell communication is implemented with attractants and repellants that diffuse into the surrounding space. Each bacterium emits an attractant J_a (positive Gaussian, above axis) and a repellant J_r (negative Gaussian, below axis). Their sum J_cc creates a ring of preferred separation: short-range repulsion fading to moderate-range attraction. This emergent spacing is BFO's key ecological advantage over algorithms like PSO — bacteria maintain population diversity automatically through mutual repulsion, without a separate crowding or niching mechanism.

Drag the fixed bacteria (●) or the mobile bacterium (◆ m). Enable the nutrient gradient to add an environmental reward J_env; toggle all mobile to let every bacterium chemotax simultaneously. In all-mobile mode each bacterium navigates its own personal J_total (excluding its own self-signal), so the stable configuration is a Nash equilibrium: no individual can improve by moving given the others' positions. Enable all landscapes to see each agent's private J_total curve and observe why their peaks differ.

Bacteria on 1-D Landscape — idle —

Fixed bacteria Mobile (m) J_a (fixed) J_a (mobile) J_r (fixed) J_r (mobile) J_cc (mobile) Mobile trail

Population

Fixed N 3

Attractant

h_a 0.10

σ_a 2.24

Repellant

d_r 0.10

σ_r 0.32

Chemotaxis

Step C 0.30

Swim N_s 4

Display

Show J_a / J_r components

Nutrient gradient J_env

All mobile

All landscapes (per-agent J_total)

Chemotaxis Options

Include self in J_cc

Controls

Speed 3 /s

Mobile θ

—

J_cc(θ_m)

—

J_env(θ_m)

—

J_total(θ_m)

—

Algorithm Reference — BFO Cell-to-Cell Signaling & Chemotaxis (Passino 2002)

Cell-to-Cell Signal

given bacteria positions {θ_i}, i = 1…S h_a, σ_a ← attractant height & width (std. dev.) d_r, σ_r ← repellant depth & width (std. dev.) ▷ Per-bacterium contributions at test point θ: J_i^a(θ) ← h_a · exp(−‖θ−θ_i‖² / σ_a²) ← positive peak, above axis J_i^r(θ) ← −d_r · exp(−‖θ−θ_i‖² / σ_r²) ← negative well, below axis ▷ Total cell-to-cell reward at θ: J_cc(θ) ← Σ_i=1^S [ J_i^a(θ) + J_i^r(θ) ] J_total(θ) ← J_cc(θ) + J_env(θ) ← maximise σ_r² ≪ σ_a² → narrow repellant, wide attraction ⟹ moat of preferred separation d* exists σ_r² ≥ σ_a² → repellant as wide as attractant ⟹ moat collapses → pure repulsion everywhere

Note: Passino (2002) uses d_a (depth) and h_r (height) under his cost-minimization convention where attractant subtracts from cost and repellant adds to it. Here the sign is flipped to maximization, so the attractant is a positive peak (height → h_a) and the repellant is a negative well (depth → d_r). Passino's w_a = 1/σ_a², w_r = 1/σ_r². Sliders use σ directly; the kernel squares it internally. Self-signaling: Passino's original formulation sums over all S bacteria including self. This widget excludes self by default, which can help stabilize trajectories, but inclusion can be toggled back on.

Chemotaxis Phase (1-D)

given step size C, swim length N_s θ_mob ← current position of the mobile bacterium J_total(θ) = J_cc(θ) + J_env(θ) ← higher J_total = better position ▷ Tumble: Δ ← random direction (±1) ← unit tumble in 1-D θ_new ← clamp(θ_mob + C · Δ, domain) ▷ Swim (greedy ascent on J_total): if J_total(θ_new) > J_total(θ_mob) then θ_mob ← θ_new ← accept uphill step for s = 1 to N_s do θ_nxt ← clamp(θ_mob + C · Δ) if J_total(θ_nxt) > J_total(θ_mob) then θ_mob ← θ_nxt ← keep swimming ↑ else break ← downhill: stop end for end if ← else: tumble again next step

Same cell-to-cell signaling and chemotaxis as Tab 1, now in a 2-D spatial environment. The heatmap shows a chosen signal quantity across the plane. Bacteria are draggable. Signal parameters (h_a, σ_a, d_r, σ_r) and chemotaxis parameters (C, N_s) are shared with Tab 1.

Bacteria in 2-D Landscape — idle —

VIEW J_cc(m) J_a J_r J_env J_total(m)

bars show locally sensed J_total as % of max

Nutrient J_env

Function

Population

Fixed N 3

Attractant (shared)

h_a 0.10

σ_a 2.24

Repellant (shared)

d_r 0.10

σ_r 0.32

Chemotaxis (shared)

Step C 0.30

Swim N_s 4

Options

All mobile

Include self in J_cc

Controls

Speed 3 /s

Mobile (x, y)

—

J_cc(m)

—

J_env(m)

—

J_total(m)

—

Algorithm Reference — BFO Cell-to-Cell Signaling & Chemotaxis (Passino 2002) — 2-D

Cell-to-Cell Signal

given bacteria {θ_i} in ℝ², i = 1…S h_a, σ_a ← attractant height & width d_r, σ_r ← repellant depth & width ▷ Per-bacterium contributions at θ = (x,y): J_i^a(θ) ← h_a · exp(−‖θ−θ_i‖² / σ_a²) J_i^r(θ) ← −d_r · exp(−‖θ−θ_i‖² / σ_r²) ▷ Total cell-to-cell reward at θ: J_cc(θ) ← Σ_i=1^S [ J_i^a(θ) + J_i^r(θ) ] J_total(θ) ← J_cc(θ) + J_env(θ) ← maximise σ_r² ≪ σ_a² → moat of preferred separation σ_r² ≥ σ_a² → moat collapses

Chemotaxis Phase (2-D)

given step size C, swim length N_s θ_mob ← current position of the mobile bacterium J_total(θ) = J_cc(θ) + J_env(θ) ▷ Tumble: φ ← uniform random angle in [0, 2π) ← random direction Δ ← (cos φ, sin φ) ← unit vector θ_new ← clamp(θ_mob + C·Δ, domain) ▷ Swim (greedy ascent on J_total): if J_total(θ_new) > J_total(θ_mob) then θ_mob ← θ_new ← accept uphill step for s = 1 to N_s do θ_nxt ← clamp(θ_mob + C·Δ) if J_total(θ_nxt) > J_total(θ_mob) then θ_mob ← θ_nxt ← keep swimming ↑ else break end for end if

Full Bacterial Foraging Optimization (BFO) — all three nested loops running on a shared 2-D landscape. Signal parameters are shared with Tabs 1 & 2. All S bacteria are equal agents; none is privileged as "mobile." Toggle Reproduction and Elimination-Dispersal on/off to observe each loop's contribution layer by layer. Bacteria remain draggable during a run to manually perturb the population.

Bacteria in 2-D Landscape — idle —

Algorithm Loops

N_ed 3

N_re 4

N_c 8

Toggles

Reproduction

Elimination-Dispersal

p_ed 0.25

Nutrient J_env

Function

Population

Size S 6

Chemotaxis (shared)

Step C 0.30

Swim N_s 4

Options

Include self in J_cc

Controls

Speed 5 /s

Ready 0%

Phase

—

i_ed / N_ed

—

i_re / N_re

—

i_c / N_c

—

Algorithm Reference — Full BFO (Passino 2002) — Nested Loop Structure

Outer Loops

given S bacteria {θ_i}, N_ed, N_re, N_c, p_ed for i_ed = 1 to N_ed do ← E-D loop for i_re = 1 to N_re do ← reproduction loop for i_c = 1 to N_c do ← chemotaxis loop for i = 1 to S do ← each bacterium tumble / swim ← see Tab ① J_i^health += J_total(θ_i) end for end for ▷ Reproduction event [see right →] rank {θ_i} by J_i^health; kill bottom ½, clone top ½ reset J_i^health ← 0 for all i end for ▷ Elimination-Dispersal event [see right →] each θ_i scattered randomly with prob p_ed end for

Reproduction & Elimination-Dispersal (detail)

▷ Reproduction ← fires after inner i_c loop ends J_i^health = cumulative J_total(θ_i) over N_c steps sort i = 1…S by J_i^health (descending) for i = ⌈S/2⌉+1 to S do θ_i ← θ_S+1−i ← copy top half onto bottom half J_i^health ← 0 end for Fitter bacteria survive and reproduce; unfit bacteria are replaced by copies of their healthier counterparts. ▷ Elimination-Dispersal ← fires after i_re loop ends for i = 1 to S do r ~ Uniform(0, 1) if r < p_ed then θ_i ← random point in domain J_i^health ← 0 ← reset health on dispersal end if end for Prevents stagnation on local optima. Higher p_ed → more exploration, less exploitation.