Mathematical model describing the decision-making process of a swarm of honeybees for selecting one nest-site among many

N.F. Britton, N.R. Franks, S.C. Pratt, T.D. Seeley, Deciding on a new home: how do honeybees agree ?, Proceedings of the Royal Society of London. Series B: Biological Sciences 269 (1498), 1383-1388 (2002) [Link]

X(0)=N(475,5)Y1(0)=U(350,500)Y2(0)=U(100,150)Z1(0)=N(35,1.5)Z2(0)=N(35,1.5)X˙=β1XY1β2XY2Y1˙=β1XY1γY1+δβ1Y1Z1+αβ1Y1Z2Y2˙=β2XY2γY2+δβ2Y2Z2+αβ2Y2Z1Z1˙=γY1δβ1Y1Z1αβ2Y2Z1Z2˙=γY2δβ2Y2Z2αβ1Y1Z2β1,β2=0.001γ=0.3δ=0.5α=0.7

Description

The population of the honeybees is divided in three main groups: X are the bees that are neutral (or unbelievers) in the decision, Y is the class of the bees that believes either in one site (Y1) or in another site (Y2) and they are actively spreading their beliefs (called evangelical bees) to the other bees, Z is the class of the bees that believes either in one site (Z1) or in the other site (Z2) but they do not spread (called not-evangelical bees) their beliefs to the others.

Probabilistic Program Computing the Bees Model (Source Code)

x = Normal(475, 5)
y1 = Uniform(350, 400)
y2 = Uniform(100, 150)
z1 = Normal(35, 1.5)
z2 = Normal(35, 1.5)

#dt = 0.1
#beta = 0.001         - We assume that beta = beta1 = beta2 
#gamma = 0.3
#delta_beta = 0.0005  - delta * beta
#alpha_beta = 0.0007  - alpha * beta 

while true:
    x, y1, y2, z1, z2 = x + dt*((-beta)*x*y1 - beta*x*y2), y1 + dt*(beta*x*y1 - gamma*y1 + delta_beta *y1*z1 + alpha_beta*y1*z2), y2 + dt*(beta*x*y2 - gamma*y2 + delta_beta*y2*z2 + alpha_beta*y2*z1), z1 + dt*(gamma*y1 - delta_beta*y1*z1 - alpha_beta*y2*z1), z2 + dt*(gamma*y2 - delta_beta*y2*z2 - alpha_beta*y1*z2)
end


Program features Value Dependency Graph
if statements Yes (L) Linear
(NL) nonlinear
Dependency Graph
State space Infinite, continuous
Circular Dependency Yes
Symbolic Constants Yes
Effective Variables No
Defective Variables Yes (x, y1, y2, z1, z2)


Probabilistic Program Simulation:

Parameter Current Value Tuning
Number of program executions:
Number of loop iterations (n):
Integration step (dt):
Parameter (beta):
Parameter (gamma):
Parameter (delta):
Parameter (alpha):