Bi-stability in cooperative transport by ants in the presence of obstacles

To cooperatively carry large food items to the nest, individual ants conform their efforts and coordinate their motion. Throughout this expedition, collective motion is driven both by internal interactions between the carrying ants and a response to newly arrived informed ants that orient the cargo towards the nest. During the transport process, the carrying group must overcome obstacles that block their path to the nest. Here, we investigate the dynamics of cooperative transport, when the motion of the ants is frustrated by a linear obstacle that obstructs the motion of the cargo. The obstacle contains a narrow opening that serves as the only available passage to the nest, and through which single ants can pass but not with the cargo. We provide an analytical model for the ant-cargo system in the constrained environment that predicts a bi-stable dynamic behavior between an oscillatory mode of motion along the obstacle and a convergent mode of motion near the opening. Using both experiments and simulations, we show how for small cargo sizes, the system exhibits spontaneous transitions between these two modes of motion due to fluctuations in the applied force on the cargo. The bi-stability provides two possible problem solving strategies for overcoming the obstacle, either by attempting to pass through the opening, or take large excursions to circumvent the obstacle.

and N ≡ ntot 2 we can describe the system as a one step jump process where at any given epoch, the system at state n can jump forward or backward (n → n + 1 or n → n − 1) with transition rates of α n and β n respectively within the sample space of n ∈ [0, N ].
The gain term α n is given by contributions from pullers at the back and lifters at the front: The probability of finding the system in state n at time t + dt is given by: P n (t + dt) = α n−1 P n−1 (t)dt + β n+1 P n+1 (t)dt + (1 − α n dt + β n dt)P n (t) (S4) which yields the master equation for the given jump process: By definition, the mean value of n can be written as: and therefore the rate of change in n(t) is given by: [(n + 1)α n P n (t) + (n − 1)β n P n (t) − n(α n + β n )P n (t)] (S10) Recognizing both terms on the r.h.s. of (S11) as the weighted averages of the gain and loss terms: and by neglecting fluctuations around the mean we obtain a deterministic equation for the rate: where by plugging in (S2,S3) into (S13) we obtain the rate equation: The equations of motion of the simplified model Starting by decomposing the force vector into an internal and external parts: The internal part will be the difference between the number of puller ants on both sides of the cargo lattice, and will also account the reduction of friction due to the lifter ants where where Θ is the Heaviside step function, f s is the static friction force due to the contact of the cargo with the surface, β is the magnitude of friction force reduced by a lifter ant, and n l is the number of lifter ants.
Eq.S17 implies that each lifter ant reduces a factor of β from the static frictional force f s , until enough lifters lifted the cargo above the surface (βn l > f s ), such that h(f s , n l ) → 0. For simplicity, we consider throughout the model that the cargo is fully lifted above the surface, and neglect the contribution of the lifter ants.
The external part of the force balance S55 is given by: In the limit of → 0 (tanh x → sign(x)), the informed ants act as a restoring force ∀x.
The c.o.m. velocity is given by normalizing f tot by the cargo mechanical response coefficient γ: and therefore the acceleration is given by: By plugging in the rate equation (S14) into the acceleration (S20) we obtain: For for a fully occupied lattice, the number of ants on both sides is given by: and by adding and subtracting (S19) from (S22,S23) we obtain: Therefore, by plugging in (S24,S25) into (S20) we obtain: where in (S27) we rescale the parameters with force units by γ:

COMPARISON TO THE TETHERED CARGO MODEL
In [1] the dynamics of a cargo that is constrained by a tether was investigated. It was found that the cargo is pulled in the direction of the nest until the tether is taught, and then begins to perform oscillations. Unlike our system, the informed ants pull the cargo in a direction that is mostly orthogonal to the allowed motion. The equations of motion for the both models are given by: The difference between the two models is the external force and its derivative (the emboldened terms).
While for the rigid obstacle case the restoring force is localized to x = 0, in the tethered cargo case it is dependent on the angle θ, and therefore effectively directed (in a non uniform manner) to a scent trail with width of 2Lsinθ max , where L is the tether length and θ max is the maximal aperture the tethered cargo can take with respect to the origin. Comparison between the solution of both models shows that when ≥ 0.5 both models coincide (Fig.A) at a parameter range where there is no bi-stability. Blue -tethered cargo. Red -rigid obstacle. Black -separatrix (linear obstacle solution).

STABILITY ANALYSIS DERIVATIONS
a. Linear stability The dynamical system is given by: Finding the fixed points:ẋ Linearizing (S30) and (S31) around the fixed point (x * , v * ) = (0, 0): The eigenvalues of (S34,S35) are given by: where: The fixed point will change stability when µ = 0, which corresponds to the point where Re[λ] = 0.
Therefore, the critical f ind for the Hopf bifurcation is given by:

. Estimation of the homoclinic bifurcation
The acceleration of system is given by: At the transition between phases (i) and (ii) two nullclines merge, and the free motion changes to relaxation oscillations [2] .The transition point is an extrema of (S40). By treating the system separately for each half space of x ( tanh x → sign(x)) we obtain: and by equating (S41) and (S42) we obtain the velocity at the transition: The approximation of the critical point of bifurcation is given by pluggin in (S43) into (S41):

RELATION BETWEEN CARGO SIZE AND INDIVIDUALITY
The number of ants N that can occupy a circular cargo with radius R is given by where L is the average width of an ant, and R is the radius of the circular cargo.
The number of pullers that occupy the cargo can be written as where b is a fraction. By Plugging Eq.S46 into Eq.S45, we obtain that the number of pullers that are attached to the cargo, is proportional to the cargo size The role switching rate is given by where f tot is the total applied force, p i is the body axis unit vector of an individual ant labeled by i, and F ind is the individuality parameter.
When the cargo is in a persistent motion, the total force f tot can be approximated by and the body axis unit vector of each ant can be approximated by where f 0 is the magnitude of force applied by a single puller. What Eqs. S49,S50 imply, is that the total force is given by the puller ants, that pull the cargo in a single persistent direction.
By plugging Eq.S49,S50 into Eq.S48 we obtain And by using the relation between the number of pullers and the cargo size (Eq.S47) we obtain Therefore an increase in R is equivalent to an effective decrease of F ind , and a decrease in R is equivalent to an effective increase of F ind . For the circular cargoes we use in the experiments, we have a linear relation between the mass and the radius. i.e m ∼ R (Fig.B).

FULL 2D MATHEMATICAL MODEL
For a full 2D description we model a ring-like cargo as a circular lattice with n max sites (Fig.Ca) [3].
Each site i can be occupied by a puller, a lifter, or remain empty and is denoted by: The total force and c.o.m. velocity are given by: where the force f 0 exerted by a single puller is considered equal for all the ants, and p i is the body axis unit vector of an ant on site i: where θ i is the angle of site i around the cargo with respect to the chosen frame of reference, and φ i is the local orientation of the ant with respect to θ i .
The static friction is parameterized by the coefficient is given by f s , which is reduced with respect to the lifter activity by:f where βf 0 is the reduction in friction force by a single lifter, and is equal for all sites of a cargo with radial symmetry. The angular velocity of the cargo is given by: where γ rot is the cargo rotation response coefficient and τ kin is the kinetic torque friction acting in the same manner as (S58).
The dynamics are stochastic and simulated using a Gillespie algorithm [4], where ants can attach, detach, role switch, reorient (change its angle φ i ) and forget (transition from informed to uninformed).
The general dynamic scheme of the 2D model is described in Fig

DISCUSSION OF THE INTERACTIONS OF SINGLE ANTS WITH THE OBSTACLE EDGES
In the experimental setup we used, the cargo did not reach the side edges. Although we attribute this fact to the informed ants that pull towards the hole [5], ants were free to forage everywhere inside the obstacle. Since according to our model single ants can hold information that can be valuable to the group, we ask if single ants that arrived to the side edges, can act as "informed" and take the cargo away from these edges. In Fig.D we provide an example from an analysis that counts the number of ants near the obstacle edges and inside the area of oscillations where the cargo moved.
The analysis shows that the number of ants near the edges is much smaller compared to the number of ants along the rest of the obstacle. Throughout all the experiments that we have conducted, we have not found any case where a single ant that reached the obstacle edges, changed its direction of motion towards the cargo and attached to it.