Mobility can promote the evolution of cooperation via emergent self-assortment dynamics

The evolution of costly cooperation, where cooperators pay a personal cost to benefit others, requires that cooperators interact more frequently with other cooperators. This condition, called positive assortment, is known to occur in spatially-structured viscous populations, where individuals typically have low mobility and limited dispersal. However many social organisms across taxa, from cells and bacteria, to birds, fish and ungulates, are mobile, and live in populations with considerable inter-group mixing. In the absence of information regarding others’ traits or conditional strategies, such mixing may inhibit assortment and limit the potential for cooperation to evolve. Here we employ spatially-explicit individual-based evolutionary simulations to incorporate costs and benefits of two coevolving costly traits: cooperative and local cohesive tendencies. We demonstrate that, despite possessing no information about others’ traits or payoffs, mobility (via self-propulsion or environmental forcing) facilitates assortment of cooperators via a dynamically evolving difference in the cohesive tendencies of cooperators and defectors. We show analytically that this assortment can also be viewed in a multilevel selection framework, where selection for cooperation among emergent groups can overcome selection against cooperators within the groups. As a result of these dynamics, we find an oscillatory pattern of cooperation and defection that maintains cooperation even in the absence of well known mechanisms such as kin interactions, reciprocity, local dispersal or conditional strategies that require information on others’ strategies or payoffs. Our results offer insights into differential adhesion based mechanisms for positive assortment and reveal the possibility of cooperative aggregations in dynamic fission-fusion populations.


S1.2 Evolvable traits: Cooperative and cohesive interac-13 tions 14
Each individual i has two evolvable traits: (i) a 'cooperative tendency' modelled 15 as a binary variable, denoted by ω c,i . Individuals with ω c,i = 1 always cooperate 16 and those with ω c,i = 0 always defect. (ii) The second trait is a continuous vari-17 able 'cohesive tendency', denoted by ω s,i ∈ [0, ∞). Individuals exhibit collective 18 movement because of this cohesive tendency.

S1.3 Organismal mobility and cohesive interactions 20
To model organismal mobility, we consider two extreme scenarios: 21 1. An 'active' scenario modelling self-propelled individuals, such as birds, fish, 22 mammals and flagellated microbes. Here, the medium has no influence on 23 organismal movement and individuals actively display 'local flocking inter-24 actions' (attraction towards, and alignment with the direction of motion of 25 their neighbours). Here, the cohesive tendency is related to the distance 26 up to which an individual looks for neighbours to flock with. We call this 27 distance as the 'local flocking radius' R s,i . The cohesive tendency is then de-28 fined as ω s,i = R s,i − R r , where R r is a measure of the body size or personal 29 space of the individual, as will be described in the next section. 1 2. A 'passive' scenario, where individuals show no active movement, but are 31 propelled by the medium that they live in, as in non-swimming bacteria or 32 other microbes where the fluidity of the medium dominates individual motil-33 ity. Here, the cohesive tendency is related to stickiness properties between 34 individuals when they are in close contact. The cohesive tendency is the 35 stickiness of the particles. We denote the stickiness by γ i to avoid potential 36 confusion with the slightly different definition of cohesive interactions in the 37 self-propelled particle model.
where c i (t) and c j (t) are the position vectors of individuals i and j respectively, and 50 d ij is the distance between them. This tendency for repulsion at short distances 51 may be thought of as tendency of individuals to avoid collision with one another 52 and that R r could represent individuals' body size. Moving away from individuals 53 within R r takes precedence over all other movement decisions, so the next step 54 (attraction and alignment) is skipped. 55 2) Attraction and Alignment: If there are no individuals within R r of the 56 focal individual, but there are individuals within a distance of R s,i (≥ R r ), the 57 focal individual will exhibit local flocking interactions with its neighbours. The 58 direction of motion due to these local interactions is a weighted average of the 59 direction of attraction towards neighbours (d a,i ) and the direction of alignment 60 with their direction of motion (d o,i ), calculated as follows: where v j (t) is the velocity of individual j. The desired direction of motion is then 62 calculated as follows: If there are no other individuals within a distance R s,i , then both d a,i (t) and d o,i (t) 64 are zero, and the desired direction is the same as the previous direction, with some 65 added error as described in the next point. The parameters k a and k o represent the 66 tendencies to attract and align with neighbours respectively, such that k a + k o ≤ 1 67 and are chosen based on the empirical work of [2].

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3) Constraints: Once the desired direction of motion is computed, two con-69 straints are imposed to make the movement rules more realistic. i) Copying er-70 ror: Since individuals may make mistakes in copying the directions and velocities 71 of other individuals, a small vector error η ce is added to the desired direction. 72 Each component of this vector error is normally distributed with mean zero and 73 variance σ 2 ce . ii) Turning rate constraint: An individual cannot turn instanta-74 neously, but at a maximum rate ω max . The maximum angle that it can turn 75 in a single time step is then θ max = ω max ∆t. The final direction taken is then 76 d i (t + ∆t) = turn(d s,i (t + ∆t) + η ce ) (OR turn(d r,i (t + ∆t) + η ce ), in case of re-77 pulsion). Here turn() means the vector is turned towards the desired direction up 78 to maximum turning of θ max .

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The position in the next step is then calculated as We fix the values of k a and k o throughout the simulation, but evolve the local 81 flocking tendency (R s ) across generations based on pay-off structures described 82 in sections S1. 6  Passive individuals (or particles) are completely driven by the medium. An indi-88 vidual will stick to the other individual with a straight γ i when it comes in contact 89 others. We modify the above movement model and adapt it to simulate this be-90 haviour. To model the medium, we simulate the velocity field of a turbulent fluid 91 with various levels of turbulence, following [3]. 92 S1.5.1 A synthetic turbulence model 93 We model a two-dimensional, isotropic, homogenous fluid flow with zero mean 94 velocity. We assume that the flow is divergence-free (as there are no fluid sources 95 in the area of interest), and can therefore be represented by a potential function, 96 ψ. The streamlines of the velocity field then v follow the contour lines of ψ, as The potential function ψ, is assumed to follow the following stochastic partial 98 differential equation: with  λ kŴk (t)e ikx/L (S1.7) Substituting this expansion in Eq (S1.6) leads to a system of Ornstein-Uhlenbeck 105 equations: which can then be evolved in Fourier space exactly using the theory of stochastic 107 integration [4]: where Z k are random numbers sampled from the N (0, 1), with the constraint that 109 Z k = Z * −k , so that ψ is real valued. 110 We solve these set of equations by initializing ψ as a delta function, so thatψ(k) = 111 1 ∀ k. 112 S1.5.2 Passive (tracer) particles in fluid medium 113 We assume that the inertia of particles can be ignored and thus, are carried by 114 the flow along its streamlines. The movement and cohesive interactions among 115 particles is implemented in a simple way by modifying the movement model for 116 active particles.

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If two individuals are within a distance R r of each other, they repel from each 118 other and their desired direction of motion given by where the summation is over all individuals within this short distance of repulsion. 120 If there are no individuals within the short repulsion area, individuals will be 121 attracted towards neighbors who are within a distance of R s . In such a case, the 122 desired direction motion is given by where η ce represents a random vector with mean zero and variance σ 2 ce .

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The final speed and the direction of motion of the particle is then given by where v f (t) is the flow velocity and γ i is the strength of cohesive interactions 126 arising from stickiness or adhesive property of the individual i. Note that we do 127 add a small noise to the individual's desired direction of the motion but due to 128 lack of interim of individuals, no turning rate constraint is applied. This model 129 can capture a range of behaviours where individuals trace the fluid when solitary, 130 but stick together and move as a group when they come very close. We also note 131 that the evolvable trait in this model is γ i and we fix the radius of interaction to 132 a constant. 133 S1.6 Payoff structure 134 Individuals move according to the movement rules (of either the active or passive 135 system) for n m steps. Depending on the strength of cohesive interactions, individ-136 uals may be found in groups. We define any two individuals as belonging to the 137 same group if they are within a distance R g from each other. We identify groups 138 using a standard union-find algorithm [5]. We assume that individuals perform 139 cooperative interactions with other individuals within their group at the n th m time 140 step.

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In the active system, if a group g has n g individuals of which k g are cooperators, 142 then all cooperators in that group receive a payoff of whereas all defectors of that group receive a payoff of where b is the benefit received from cooperators in the group (excluding self), c is 145 the cost of cooperation and c s R 2 s,i (= c s (R r + ω s,i ) 2 ) is the cost of cohesive interac-146 tions. Thus, in the active system, increasing the cost of cohesion also effectively 147 increases baseline fitness by c s R 2 r . Note that solitary cooperators get a payoff of 148 −c, and solitary defectors get zero payoff. We discuss the derivation of this payoff 149 structure and its modifications in section S5.1.

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In the passive system, R s,i in the above equations is replaced with γ i .

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The fitness of each individual is a certain baseline value (V 0 ) plus payoffs arising 152 from cooperative and flocking interactions: We subtract min i (V c/d,i ) so that even the least fit individual has a positive fitness 154 V 0 . This allows us to control the strength of selection by varying the value of V 0 . 155 A low value corresponds to strong selection, and a high value to weak selection. 156 S1.7 Reproduction and dispersal 157 Individuals compete globally for reproduction, i.e. they reproduce with probabil-158 ity of reproduction proportional to their relative fitness in the entire population. 159 Reproduction is asexual and synchronous. A roulette-wheel algorithm is used to 160 generate N offspring for the next generation. All parents are immediately removed 161 from the system after offspring are created. When an individual reproduces, it 162 passes on it's two traits (ω s,i and ω c,i ) with a small mutation rate to its offspring. 163 For the trait ω s,i , the mutation is implemented as an addition of a normally dis-164 tributed noise η µs with mean zero and standard deviation σ µs . For the binary 165 cooperative trait, ω c,i is flipped with a probability p µc . The offspring are dispersed 166 to random locations in space and with random orientation.
Furthermore, the average cohesive tendency of cooperators and defectors is calcu-197 lated as follows: 198 ω sc = 1 pN N i=0 ω s,i ω c,i (S1.18) The 'evolved values' of p and ω s are obtained by averaging p and ω s over all 199 generations in a simulation (leaving out the first T trans = 500 generations to allow 200 transients to die down), and again averaging over the n rep replicates.