Thermoregulation of HoneybeesJeremy Chiu

Supervised by: Dr. JF Williams, Dr. John Stockie

Simon Fraser University

1. Introduction

MOTIVATION: Honeybee clusters may survive temperaturesas low as -15◦ without a hive. How?•Continuum model of a honeybee swarm; based on Wat-

mough and Camazine’s Self-Organized Thermoregula-tion of Honeybee Clusters[1]• Variables:

– t: time– r(t): distance from center of swarm– T (r, t): temperature– ρ(r, t): density–R(t): radius of swarm

Figure 1: a hiveless honeybee cluster[2]

2. Assumptions

1. Bees are self-organized2. Attraction/repulsion based on warmth3. Bees metabolise when cold4. Swarm exhibits spherical symmetry5. Heat transfer based on conduction6. Conservation of bees

3. Derivation

TEMPERATURE EQUATION:

c∂T

∂t= Diffusive Term + Source Term

• Parameters: c heat capacity; λ(ρ) (non-constant) conduc-tivity; f (T ) metabolism function

•Diffusive Term = ∇ · (λ(ρ)∇T )• Source Term = ρf (T )

Figure 2: metabolism function f

DENSITY EQUATION:

∂ρ

∂t= Diffusive Term + Thermotactic Term

• Parameters: µ(ρ) motility function; χ(T ) thermotactic ve-locity•Diffusive Term = ∇ · (µ(ρ)∇ρ)

Figure 3: motility function µ

• Thermotactic Term = −∇ · (χ(T )ρ∇T )• χ: positive when bees are cold; negative when warm

RADIAL EQUATION:• In a sphere, for fixed t:

Number of Bees = 4π

∫ R(t)

0r2ρ(r, t)dr

• Assumption 6, Leibniz integral rule, chain rule,...

dR

dt= −

µ(ρ)∂ρ∂rρ

∣∣∣∣r=R

+ χ(T )∂T

∂r

∣∣∣∣r=R

4. Numerics

NORMALIZE:• r(t)→ x =

r(t)R(t)

• T (r, t)→ u(x, t); ρ(r, t)→ v(x, t)

SPATIAL DISCRETIZATION:

Figure 4: spatial discretization

SOLVING:

•Centered difference scheme on half stencil•Method of freezing coefficients:

– ~un+1 depends on Rn, ~un, ~vn

– ~vn+1 depends on Rn, ~un+1, ~vn

–Rn+1 depends on Rn, ~un+1, ~vn+1

• Finally, to solve say u, we have M a tridiagonal matrix ofknown values, ~D a vector of known values, so:

M~un+1 = ~D

=⇒ ~un+1 =M\ ~D

5. Future

• Angry Bees: Honeybees vs Hornets• 2D bees (= penguins)

References

[1] J. Watmough and S. Camazine, Self-Organized Ther-moregulation of Honeybee Clusters, J. theor. Biol.(1995) 176, 391-402

[2] Wilde, Erik. Bee Ball. 2012. Photograph. N.p.

6. Conclusion

RESULTS:

SNAPSHOTS: