ant foraging dynamics: from reaction-diffusion to ... · ant foraging dynamics: from reaction-di...
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Ant foraging dynamics: From reaction-diffusion toindividual-based models
Paulo AmorimRicardo Alonso (PUC Rio), Thierry Goudon (U. Nice, INRIA)
Instituto de Matematica - Universidade Federal do Rio de Janeiro
46th John H. Barrett Memorial Lectures 2016
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Leafcutter ant foraging trail
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
• Why study ants?
Image: www.alexanderwild.com (l), W.P. Armstrong (r).
• Besides scientific interest, some ant species like Solenopsis invicta(the Fire Ant, left) and Tapinoma melanocephalum (Ghost Ant,right) are considered invasive or pests.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Ant foraging
• One of the most interesting emergent behaviors in the animal kingdom• Individuals following simple, local, behavioral rules produce complex,“intelligent” large-scale strategies
• In general, ants have weak vision and very limited cognitive ability• How can they originate complex, organized collective behavior?
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Ant communication
• Ants use tactile, visual, chemical, and even acousticcommunication.
• One of the most important:pheromones.• Chemical compounds whichindividual ants secrete and leaveon the substrate.• Used to signal alarm, presenceof food, or colony-specific olfac-tory signatures.• The queen uses pheromonesto inhibit workers’ sexual devel-opment.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Foraging dynamics
• Goal: Efficient discovery and removal of food sources
• Key: Ants are attracted to foraging pheromone
• Pheromone laying produces a Feedback loop:
1 Ants find food through random foraging;2 Return to nest carrying food and depositing pheromone;3 Other ants detect pheromone and follow the trail to food source,
repeating the process;4 Trail is reinforced.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Modeling foraging
• We wish to find a mathematical framework where evolutionaccording to simple rules produces trail formation.
• Previous approaches: Mostly discrete models. Continuous modelsonly for isolated parts of the foraging dynamics.
• No “complete” model so far, incorporating the several stages offoraging.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Modeling animal movement
• Suppose the ant moves randomly on a 1-d territory.• The ant starts from x = 0 at t = 0.• p(x , t) = Probability of finding the ant at x at time t. Suppose in eachδt increment, it moves δx (to the left) or −δx (to the right). Then,
p(x , t) =1
2p(x − δx , t − δt) +
1
2p(x + δx , t − δt).
Or:
If the ant is ar x at time t, then either it was at x − δx or at x + δx attime t − δt, with equal probability.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
(∗) p(x , t) =1
2p(x − δx , t − δt) +
1
2p(x + δx , t − δt)
Taylor expansion:
p(x − δx , t − δt) = p(x , t)
− δx ∂p∂x− δt ∂p
∂t+ (δx)2∂
2p
∂x2+ δtδx
∂2p
∂x∂t+ (δt)2∂
2p
∂t2+ · · ·
p(x + δx , t − δt) = p(x , t)
+ δx∂p
∂x− δt ∂p
∂t+ (δx)2∂
2p
∂x2− δtδx ∂2p
∂x∂t+ (δt)2∂
2p
∂t2+ · · ·
Plugging into (∗), we get
∂p
∂t=
(δx)2
2δt
∂2p
∂x2+δt
2
∂2p
∂t2+ · · ·
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
∂p
∂t=
(δx)2
2δt
∂2p
∂x2+δt
2
∂2p
∂t2+ · · ·
• Say δt, δx → 0 in such a way that (δx)2
2δt → D ≡ constant. In the limitwe get
∂p
∂t= D
∂2p
∂x2,
which is just the diffusion equation. In n-d,
∂p
∂t= D∆p.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Reaction-diffusion-advection equation
More generally:
∂p
∂t− D∆p
↑Diffusion due to
random movement
+
Transport in the directionof a velocity vector V
↓div(pV) = F (p, x , t)
↑Source or sink
• p may be seen as the ant density
• Velocity V =??
• what if V is the concentration gradient of a chemical?
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Chemotaxis
• Foraging behavior falls into the scope of chemotaxis:
Chemotaxis system ∂tρ−∆ρ + div(χρ∇c) = 0
∂tc −∆c + τc = ρ
• Used for modeling aggregations of bacteria, slime molds, cells, etc.
• ρ is the density of bacteria
• c is the density of the chemoattractant
• The model contains: random movement of bacteria, production ofthe chemical by the bacteria, attraction to the chemical, evaporation,diffusion of the chemical.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Chemotaxis
∂tρ−∆ρ + div(ρχ∇c) = 0
∂tc −∆c + τc = ρ
• Some mathematical properties:
• Possible blow-up of solutions (depending on spatial dimension andinitial data)
• Aggregation
• Formation of spatio-temporal patterns
• Rich mathematical structure
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A chemotactical model of ant foraging
• Ant population is represented by density functions of t > 0 andx ∈ Ω ⊂ R2
• Population is divided into two groups
• Foraging ants u(t, x)
• Returning ants w(t, x)
• Pheromone concentration v(t, x)
• Food distribution c(t, x)
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A chemotactical model of ant foraging
• Simple dynamical rules:
• Foraging ants emerge from the nest and disperse randomly with atendency to follow the direction of increasing pheromone, if any;
• Once reaching food, foraging ants turn into returning ants andcollect food;
• Returning ants deposit pheromone as they return to the nest;
• Upon reaching the nest, returning ants turn into foraging ants.
• Pheromone diffuses and evaporates.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Modeling assumptions
• Returning to the nest
We prescribe a homebound vector field ∇a(x). Supported inliterature: ants use various non-pheromone methods to return to thenest (astromenotaxis, path integration, visual landmarks, CO2concentration, home range marking, etc...).
• Dynamics near nest and food
The nest is a special area where returning ants change into foragingants. We do not attempt to model other methods of recruitment nearthe nest.The food site is a special area where foraging ants turn into returningants, depleting the food.
• Experimental fact: pheromone deposition decreases as antsapproach the nest.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A chemotactical model of ant foraging
∂tu − αu∆u + div
(u βu∇v
)= −λ1uc + λ2wN(x) + M(t)N(x)
∂tw − αw∆w + div(w βw∇a
)= λ1uc − λ2wN(x)
∂tv = µP(x)w − δv + αv∆v
∂tc = −γu c .
• Foraging ants u(t, x)
• Returning ants w(t, x)
• Pheromone concentration v(t, x)
• Food distribution c(t, x)
• M,N and P describe the nest, emergence of foraging ants from the nest,and decrease in pheromone deposition approaching the nest.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A chemotactical model of ant foraging
• Nondimensional system
P.A., JTB 2015
∂tu − ∆u︸︷︷︸random dispersal
+
follow increasing pheromone︷ ︸︸ ︷div(u χu∇v
)= −uc + λwN(x) + M(t)N(x)︸ ︷︷ ︸
ant type change and emergence from nest
∂tw − Dw∆w + div(w ∇a
)︸ ︷︷ ︸follow home-bound field
= uc − λwN(x)
∂tv = P(x)w − εv + Dv∆v︸ ︷︷ ︸pheromone dynamics
∂tc = −u c︸ ︷︷ ︸food removal
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
No-flux Boundary conditions
(∇u − χuu∇v
)· n = 0(
Dw∇w − w∇a)· n = 0
∇v · n = 0, c = 0
• Conservation of total population:
∫Ωu(x) + w(x) dx = C ,
after all the ants have left the nest.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical simulation
• Numerical difficulties: Sharp fronts, narrow trails with largegradients
• We use an upwind finite difference scheme for the spatial derivativesand a fourth order Runge–Kutta scheme for the time evolution
• Numerical results exhibit the main phenomenon: trail formationto and from the nest:
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results - foraging ants
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results - returning ants
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results - pheromone concentration
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results - food
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results - food mass evolution
Hours
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results - returning ants with topography
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Numerical results with topography
Returning ants Foraging antsPaulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Parameter space exploration: conditions for trail formation
• Evaporation rate ε vs. Chemotactic sensitivity χu
• Conditions for trail formation inforaging ants. A star ? indicatessimulations where, for fixed ε, thefood removal efficiency is greatest.A white star indicates simulationswhere, for fixed χu, the food re-moval efficiency is greatest.
• Values with marked trail formationcoincide with greater food removalefficiency.
Figure 5: Conditions for trail formation in foraging ants. From left to right, " =0.01, 0.1, 0.5, 1, 2.5, 5. Bottom to top, u = 20, 40, 80, 160, 500, 1000. " is the pheromonedegradation rate, u is the foraging ants’ chemotactic sensitivity (see Table 2). A star? indicates simulations where, for fixed ", the food removal efficiency is greatest. Awhite star indicates simulations where, for fixed u, the food removal efficiency isgreatest (see Table 4).
22
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Analytical results (with R. Alonso and Th. Goudon)
Fast pheromone diffusion ant foraging system∂tu −∆u + div
(u∇p
)= −uc + wN
∂tw −∆w + div(w ∇v
)= uc − wN
−∆p = w − p
• Pheromone diffusion is fast relative to ants’ movement
• Food source c(x) is very large or renewable: no depletion
• N(x) is the (given) nest function
• ∇v is the direction to the nest
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Theorem (R. Alonso, PA, Th. Goudon, to appear in M3AS)
Let T > 0, 0 ≤ (u0,w0) ∈ L1 ∩ L2+(Ω), Ω ⊂ R2. There exists a unique
nonnegative weak solution for the fast pheromone diffusion ant foragingsystem. For any 0 < t ≤ T :
(i) Lγ-integrability
‖w(t)‖γ + ‖u(t)‖γ ≤ C(
1 +1
t(1/γ′)+
), γ ∈ [1,∞) ,
(ii) L∞-integrability
‖w(t)‖∞ + ‖u(t)‖∞ ≤ C(
1 +1
t1+
).
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Proposition (a priori estimates)
Let (u,w) be a classical nonnegative solution of the fast pheromonediffusion ant foraging system. Then, for any γ ∈ [1,∞] we have theestimate
‖w(t)‖γ + ‖u(t)‖γ ≤ C(
1 +1
t(1/γ′)+
), t > 0 ,
with constant independent of time. Furthermore, for any γ ∈ [1,∞] thisestimate can be upgraded to∫
Ωwγ (t) dx +
∫Ωuγ(t) dx ≤ C (m0, ‖u0‖γ , ‖w0‖γ ) , t > 0 ,
if ‖u0‖γ , ‖w0‖γ <∞.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Why no blow-up?
• Usual chemotaxis system:∂tρ−∆ρ = − div(ρ∇c),
−∆c = ρ− c.
• − div(ρ∇c) = −∇ρ · ∇c − ρ∆c = −∇ρ · ∇c + ρ2 − ρc
• Formally,
∂tρ−∆u +∇ρ · ∇c + ρc = ρ2.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Why no blow-up?
∂tu −∆u + div
(u∇p
)= −uc + wN
∂tw −∆w + div(w ∇v
)= uc − wN
−∆p = w − p
• − div(u∇p
)= −∇u · ∇p − u∆p ≤ −∇u · ∇p + uw
• ∂tu −∆u +∇u · ∇p ≤ −uc + wN + uw
• But w is a solution to a parabolic equation; uw is better than u2.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A priori estimates
• Tools: Gagliardo–Nirenberg–Sobolev inequality in 2-d:
∫Ωξα+1 ≤ C (Ω)‖ξ‖1
∫Ω|ξα|+ |∇ξα/2|2 , ∀α ≥ 1 ,
• Mass conservation: ∫Ωu(t) +
∫Ωw(t) = m0
• Lebesgue interpolation: r > 1,
‖u‖r ≤ ‖u‖1−θ1 ‖u‖θr+δ, θ ∈ (0, 1)
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A priori estimates
• 1 < γ < α < 1 + γ <∞,
• d
dt
∫Ωuγ dx + C
∫Ωuγ+1 dx .
∫Ωuγ dx +
∫Ωwγ dx +
∫Ωwγ+1 dx ,
• d
dt
∫Ωwα dx + C
∫Ωwα+1 dx .
∫Ωuα dx +
∫Ωwα dx
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A priori estimates
• 1 < γ < α < γ + 1 <∞
• If |Ω| <∞, use wγ+1 ≤ C + εwα+1, etc
• d
dt
∫Ωuγ dx + C
∫Ωuγ+1 dx . C + ε
∫Ωuγ+1 dx + ε
∫Ωwα+1 dx ,
• d
dt
∫Ωwα dx + C
∫Ωwα+1 dx . C + ε
∫Ωuγ+1 dx + ε
∫Ωwα+1 dx
⇓
• d
dt
∫Ωuγ + wα dx + C
∫Ωuγ+1 + wα+1 dx ≤ C
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A priori estimates
• Use ‖u‖γ ≤ C‖u‖θγ+1 with the right θ to find
d
dtZ + C Z
αα−1 ≤ C
with
Z (t) =
∫Ωuγ +
∫Ωwα.
ODE comparison result
⇓
Z (t) ≤ C(
1 +1
tα−1
)which is essentially the desired result
‖w(t)‖γ + ‖u(t)‖γ ≤ C(
1 +1
t(1/γ′)+
), t > 0 .
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
L∞ estimate
‖w(t)‖∞ + ‖u(t)‖∞ ≤ C(
1 +1
t1+
)• L∞ estimates by a DiGiorgi energy method for the level sets (Moseriteration also works in some cases). Consider
∂tw −∆w +∇ · (w∇v) = f , in [0,T ]× Ω ,
• For M > 0 define the level sets and the sequence of levels and times
wk = (w − λk) 1w>λk ,
λk =(1− 1/2k
)M , tk =
(1− 1/2k+1
)t? , t? > 0.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
L∞ estimate
wk = (w − λk) 1w>λk
• Energy functional for the level sets
Wk := supt∈[tk ,T ]
∫Ωw2k dx +
∫ T
tk
∫Ω
∣∣∇wk
∣∣2 dx dt,
• We have ∂tw −∆w +∇ · (w∇v) = f ⇒
d
dt
∫Ωw2k dx +
∫Ω
∣∣∇wk
∣∣2 dx ≤ ∫Ω
∣∣w∇v ∣∣2 1w>λkdx + 2
∫Ωf+ w dx .
• Estimate the right-hand side using Wk−1.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
L∞ estimate
Note that if wk > 0 then wk−1 ≥ (λk − λk−1)M = 2−kM, and, as aconsequence,
1w>λk ≤(2k
Mwk−1
)a, ∀ a ≥ 0 . (1)
This and a Sobolev inequality allows to bound terms with wk using(wk−1)a.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
L∞ estimate
• Prove a recurrence relation Wk ≤ C (M, t?)W ak−1, a > 1, giving Wk → 0
• From
Wk ≥1
T − t?
∫ T
t?
∫Ωw2(t, x) 1w(t,x)≥M(1−1/2k ) dx dt
we conclude that ∫ T
t?
∫Ωw2(t, x) 1w(t,x)≥M dx dt = 0,
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
L∞ estimate
and so0 ≤ w(t, x) ≤ M a. e. (t?,T )× Ω.
Moreover, M ≤ C1
t1+?
giving
‖w(t)‖∞ + ‖u(t)‖∞ ≤ C(
1 +1
t1+
).
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Stability, uniqueness
• Existence of classical short-time solution follows by a contractionargument, which can be extended to t =∞ by the a priori estimates.
Stability
Given two initial data (ui ,o ,wi ,o) ∈ L1 ∩ L2+(Ω), with i = 1, 2, then
solutions (ui ,wi ) satisfy the estimate
‖u1(t)− u2(t)‖2 + ‖w1(t)− w2(t)‖2
≤(‖u1,o − u2,o‖2 + ‖w1,o − w2,o‖2
)eC(mo)t , t ≥ 0 .
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Stability, uniqueness
• u = u1 − u2, ...
• Use equations and previous estimates to get
d
dt‖w‖2
2 + ‖∇w‖22 ≤ C
(‖w‖2
2 + ‖u‖22
),
andd
dt‖u‖2
2 +1
2‖∇u‖2
2 ≤ C(
1 +1√t
)(‖u‖2
2 + ‖w‖22
).
• With Z (t) := ‖u‖22 + ‖w‖2
2,
dZ (t)
dt≤ C
(1 +
1√t
)Z (t) .
• Integrate to get the stability result.
• Allows to construct weak solution by approximation.Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
An Individual-Based Model of ant trail following
“The exact nature of how individual ants move on the trail andrespond to pheromone remains unknown.” A. Perna et al. (2012) Individual
Rules for Trail Pattern Formation in Argentine Ants (Linepithema humile). PLoS Comput Biol 8(7)
• At microscopic level, the continuous model only considers attractiontowards higher concentrations of pheromone.
• Trail-following must involve something more, since in nature trails donot have pheromone gradient along the trail itself.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
An Individual-Based Model of ant trail following
• Reaction-diffusion models are at one end of a modeling spectrum:macroscopic models.
• At the other end are discrete or individual-based (IBM) models.
• Individual ants are modeled by the law of motiond
dtX (t) = V (t),
d
dtV (t) = −1
τ
(V − F (X ,V )
)• τ is a relaxation time and F (X ,V ) is the desired velocity.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
An Individual-Based Model of ant trail following
• Traditional approach: the desired velocity F is proportional to thegradient of an attracting chemical.
• Our assumption: Ants are attracted to the presence of pheromone on acircular sector representing sensing area.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
An Individual-Based Model of ant trail following
• Sensing area:
B(V , `, θ∗) = X ∈ R2 : ∠(V ,X ) ∈ (−θ∗, θ∗), ‖X‖ ≤ `.
• Desired velocity:
F (X ,V ) =
∫B(V ,`,θ∗)
X ′c(t,X + X ′) dX ′
• c(t,X ) is the pheromone concentration.
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
An Individual-Based Model of ant trail following
Weber’s Law:
Individual response ' L− R
L + R,
where L is stimulus detected on left side, and R on right side.
F (X ,V ) = λ
∫BX ′c(t,X + X ′) dX ′∫Bc(t,X + X ′) dX ′
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
An Individual-Based Model of ant trail following
• How to remove the singularity of F (X ,V ) = λ
∫B X ′c(t,X + X ′) dX ′∫B c(t,X + X ′) dX ′
when the pheromone is zero, and introduce a pheromone detectionthreshold?
• Ants move at near-constant speed.
• Pheromone below a small threshold c∗ is not detected.
• With constant pheromone, the desired velocity F (X ,V ) simplifies to thenatural velocity
CV
‖V ‖.
⇒ Replacing c(t,X ) with c(t,X ) := max(c(t,X ), c∗) removes thesingularity and makes the ant move at the natural velocity whenpheromone is below threshold.
“pheromone below the threshold is treated as constant pheromone”Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A linearized model
F (X ,V ) = λ
∫BX ′c(t,X + X ′) dX ′∫Bc(t,X + X ′) dX ′
.
Taylor develop: c(t,X + X ′) ' c(t,X ) + X ′ · ∇c(t,X ). Then
Flin(X ,V ) = λ
c(t,X )
∫B
X ′ dX ′ +
∫B
X ′ ⊗ X ′ dX ′∇c(t,X )
c(t,X )|B|+∇c(t,X ) ·∫B
X ′ dX ′
= λc(t,X )`
2
3sin θ∗ V
‖V‖ +`2
4
(θ∗I +
1
2sin 2θ∗A(Θ)
)∇c(t,X )
c(t,X )θ∗ +∇c(t,X ) · `2
3sin θ∗ V
‖V‖
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
A linearized model
• Θ is the angle of the ant’s current velocity, V . So the (linearized)desired velocity is
Flin(X ,V ) = λc(t,X )`
2
3sin θ∗ V
‖V ‖ +`2
4
(θ∗I +
1
2sin 2θ∗A(Θ)
)∇c(t,X )
c(t,X )θ∗ +∇c(t,X ) · `2
3sin θ∗ V
‖V ‖
with
A(θ∗) =
(cos 2Θ sin 2Θ
sin 2Θ −cos 2Θ
),
V = ‖V ‖(cos Θ, sin Θ).
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Some results
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics
Thank you for your attention!
http://www.im.ufrj.br/paulo/
Paulo Amorim - UFRJ - Rio de Janeiro Ant Foraging Dynamics