Leadership through influence: what mechanisms allow leaders to steer a swarm?
aa r X i v : . [ q - b i o . P E ] F e b Leadership through influence: what mechanismsallow leaders to steer a swarm?
Sara Bernardi ∗ , Raluca Eftimie , and Kevin J. Painter Dipartimento di Scienze Matematiche (DISMA), Politecnico di Torino, Corso Ducadegli Abruzzi 24, 10129 Torino, Italy Laboratoire de math´ematiques de Besan¸con, UMR-CNRS 6623 Universit´e deBourgogne Franche-Comt´e, 16 Route de Gray, 25000 Besan¸con, France Dipartimento Interateneo di Scienze, Progetto e Politiche del Territorio (DIST),Politecnico di Torino, Viale Pier Andrea Mattioli, 39 10125 Torino, Italy
Abstract
Collective migration of cells and animals often relies on a specialisedset of “leaders”, whose role is to steer a population of naive followerstowards some target. We formulate a continuous model to understandthe dynamics and structure of such groups, splitting a population intoseparate follower and leader types with distinct orientation responses. Weincorporate “leader influence” via three principal mechanisms: a bias inthe orientation of leaders according to the destination, distinct speeds ofmovement and distinct levels of conspicuousness. Using a combination ofanalysis and numerical computation on a sequence of models of increasingcomplexity, we assess the extent to which leaders successfully shepherdthe swarm. While all three mechanisms can lead to a successfully steeredswarm, parameter regime is crucial with non successful choices generatinga variety of unsuccessful attempts, including movement away from thetarget, swarm splitting or swarm dispersal.
Keywords
Collective migration, Follower-leader, Swarming, Nonlocal PDEs
Subject class (MSC 2020)
Collective migration underlies numerous processes, including the migration ofcells during morphogenesis and cancer progression [19, 20], social phenomenasuch as pedestrian flow and crowding [22, 25, 9], and the coordinated movementsof animal swarms, flocks and schools [13, 38].In many cases, effective migration may demand the presence or emergenceof leaders , for example as an evolved strategy for herding the population to a ∗ Corresponding author: [email protected] etc . Atthe cellular level examples include epithelial wound healing, where a set of so-called leader cells at the tissue boundary appear to guide a migrating cell group[37], embryonic neural crest cell invasion, where trail-blazing pioneers lead fol-lowers in the rear [31], and kidney morphogenesis, where the lumen forms asleading cells leave the epithelialised tube in their wake [2]. Collective invasion ofbreast cancer appears to be driven by a specialised population defined by theirexpression of basal epithelial genes [8].Leadership is also found in various migrating animal groups, for examplearising from a cohort of braver or more knowledgeable individuals: faced bypoor feeding grounds, post-reproductive females take on an apparent leadershiprole in the guidance of a killer whale pod, their experience offering a reserveof ecological knowledge, [5]. As our principal motivation we consider honeybeeswarms, which form as a colony outgrows its nest site. At this point the queenand two-thirds of the colony depart (leaving a daughter to succeed her) and tem-porarily bivouac nearby, for example on a tree branch. Over the following hoursto days, a relatively small subpopulation of scout bees ( ∼ −
5% of the 10,000+strong swarm) scour the surroundings for a suitable new nest location, poten-tially several kilometres distant. The quality of a potential site is broadcast toother swarm members and, once consensus is obtained, the entire colony movesto the new dwelling. Consequently, guidance of 1000s of naive insects (includingthe queen) is entrusted to a relatively small number of informed scouts [32]. Ob-servations suggest that scouts perform a sequence of high-velocity movementstowards the nest site through the upper swarm [3, 30, 21, 32], “streaking” thatconceivably increases their conspicuousness and communicates the nest direc-tion.Understanding the collective and coordinated dynamics of migrating groupsdemands analytical reasoning. The mathematical and computational literaturein this field encompasses a particularly wide range of approaches. Microscopic,agent-based or individual-based models describe a group as a collection of in-dividual agents, where the evolution of each particle is tracked over time. Ben-efiting from their capacity to provide a quite detailed description of an agent’sdynamics, they offer a relatively natural tool to investigate collective phenomena(see, for instance, [11, 12, 14, 17, 24, 23, 4]).However, as the number of component individuals become large (as would betypical for many cancerous populations, large animal groups etc ), microscopicmethods become computationally expensive and macroscopic approaches maybecome necessary. Various continuous models have been proposed to understandthe collective migration dynamics of interacting populations, with nonlocal PDE frameworks becoming increasingly popular; models falling into this class havebeen developed in the context of both ecological and cellular movement, e.g. see[27, 1, 35, 16, 15]. Their nonlocal nature stems from accounting for the influenceof neighbours on the movements of an individual, and their relative novelty hasalso become a source of significant mathematical interest (see [7] for a review).The aim of this paper is to investigate the impact of informed leaders onnaive followers, using a nonlocal PDE model that builds on the hyperbolic PDE2pproach developed in [16]. In particular, we will explore the extent to whichthe presence of leaders can result in a steered swarm , defined as a populationacquiring and maintaining a spatial compact profile that is consistently steered towards a target known only to the leaders. Motivated by real-world case stud-ies (in particular, bee swarming as described above) we assume leaders attemptto influence the swarm using one or more of three mechanisms: (i) leaders pref-erentially choose the direction of the target; (ii) leaders move more quickly whenmoving towards the target; (iii) leaders alter their conspicuousness according tothe target direction. In Section 2, we introduce the full follower-leader model,along with two simple submodels – a leader-only and a follower-only system– designed to reveal insights into the behaviour of the full system. Section 3explores the dynamics of the submodels, via a combination of linear stabilityand numerical simulation. Section 4 subsequently addresses the full system, inparticular the effectiveness of the different biases. We conclude with a discussionand an outlook of future investigations.
We assume a heterogeneous swarm composed from distinct populations of knowl-edgeable leaders and naive followers. Both orient according to their interactionswith other swarm members, as detailed below, but leaders have “knowledge” ofthe target and therefore the direction in which the swarm should be steered. Forconvenience we will restrict here to one space dimension, assume fixed speedsand account for direction through separately tracking positively (+) and neg-atively ( − ) oriented populations. Without loss of generality we assume theleaders aim to herd the swarm in the (+) direction, influencing via: • Bias 1, orientation.
Leaders preferentially choose the target direction. • Bias 2, speed.
Leaders alter their speed according to the target direction. • Bias 3, conspicuousness.
Leaders alter their conspicuousness accordingto the target direction.Setting u ± ( x, t ) and v ± ( x, t ) to respectively denote the densities of followersand leaders at position x ∈ Ω ⊂ R and time t ∈ [0 , ∞ ), the governing equations3re as follows: ∂u + ∂t + γ ∂u + ∂x = − λ u + u + + λ u − u − ,∂u − ∂t − γ ∂u − ∂x = + λ u + u + − λ u − u − ,∂v + ∂t + β + ∂v + ∂x = − λ v + v + + λ v − v − ,∂v − ∂t − β − ∂v − ∂x = + λ v + v + − λ v − v − ,u ± ( x,
0) = u ± ( x ) ,v ± ( x,
0) = v ± ( x ) . (1)In its general setting the model is formulated under the assumption of an infinite1D line. For the simulations later we consider a bounded interval Ω = [0 , L ], butwrapped onto the ring (periodic boundary conditions) to minimise the influenceof boundaries. Initial conditions will be specified later.In the above model, followers move with a fixed speed (set at γ ). Leadershave potentially distinct speeds, β ± , according to whether bias 2 is in opera-tion; for example, in the example of bee swarming, scouts engage in streakingand increase their speed when moving towards the new nest site [32]. Switchingbetween directions is accounted for via the right hand side terms, where λ u + denotes the rate at which a follower ( u ) turns from (+) to ( − ), with similar def-initions for λ u − , λ v ± . Note that the current model excludes switching betweenfollower and leader status, although it is of course possible to account for suchbehaviour through additional role-switching transfer functions.The turning rate functions are based on interactions between swarm mem-bers where, accounting for the “first principles of swarming” [6], we combine repulsion (preventing collision between swarm members), attraction (preventingloss of contact and swarm dispersal) and alignment (choosing a direction accord-ing to those assumed by neighbours and/or external bias). Figure 1 summarisesthe general principals upon which the model is founded.The turning rate functions have the following form: λ i ± = λ + λ f ( y i ± ) , for i ∈ { u, v } and where f ( y ) = 0 . . y − y ). This assumes the turningrate smoothly and monotonically increases from a baseline to maximum valueaccording to the level of perceived signal , measured separately for (+) and ( − )follower and leader populations in y u ± and y v ± . If y is chosen in such a waythat f (0) ≪ λ and λ can be regarded as the baseline turningrate and the highly biased turning rate respectively. For positively-moving fol-lowers at position x and time t , y u + ( x, t ) combines the repulsive, attractive andalignment interactions with their neighbours into a single measure that dictatesthe turning rate, with similar interpretations for y u − , y v ± . Specifically, we set y u ± = Q u ± r + Q u ± a + Q u ± l (2)4igure 1: Assumptions underlying the turning behaviour of swarm members.Top row: attraction and repulsion are assumed to act equally on followers (bluecircles) and leaders (red circles). Repulsion acts over shorter ranges, pushingindividuals away from each other if they are too close; attraction acts over largerdistances, pulling individuals together if they become too separated. Bottomrow: alignment is distinct for followers and leaders. Followers do not know thetarget but are influenced by the orientation of the oncoming swarm, reorientingwhen they perceive the oncoming swarm is moving in the opposite direction.Leaders ignore the alignment of the swarm, biasing instead according to thetarget direction.and similarly for y v ± . Q r , Q a and Q l integrate the perceived positional anddirectional information from neighbours located at a distance s ∈ (0 , ∞ ) fromthe generic individual placed at ( x, t ).For simplicity we will assume here that followers and leaders are only dis-tinguished by their alignment response: repulsion/attraction are taken as “uni-versal” and act to keep the overall population together and avoid collisions. Anoteworthy consequence of this is that a leader is not bound to choose the di-rection of the target: for example, if there is a danger of losing contact withthe swarm the leader should be inclined to return to the fold. We adopt thefollowing standard choices. Q u ± r = Q v ± r = q r Z ∞ K r ( s ) ( u ( x ± s ) + v ( x ± s ) − u ( x ∓ s ) − v ( x ∓ s )) ds ,Q u ± a = Q v ± a = − q a Z ∞ K a ( s ) ( u ( x ± s ) + v ( x ± s ) − u ( x ∓ s ) − v ( x ∓ s )) ds . In the above, K i ( s ), i = { a, r } , denote interaction kernels and parameters q a q r represent the magnitude of the attraction and repulsion contributions,respectively. The attractive and repulsive terms depend on the total density ofthe cohort at a certain position, regardless of flight orientation, i.e. u ( x ± s, t ) = u + ( x ± s, t )+ u − ( x ± s, t ) and similarly v ( x ± s, t ) = v + ( x ± s, t )+ v − ( x ± s, t ). Foran individual flying in the direction of a large swarm (i.e. towards overall highertotal population densities), the contribution to y from Q r will be positive (hence,an increased likelihood of turning away) and from Q a will be negative (henceless likely to turn away). Whether the combined contribution is then positiveor negative depends on the individual parameters and the precise shape of thetotal density distribution.The alignment contribution is of the general form Q i ± l = q l Z ∞ K l ( s ) P i ( u ± , v ± ) ds, (3)for i ∈ { u, v } and where K l ( s ) and q l respectively denote the alignment ker-nel and the magnitude of the synchronization. The functions P u ( u ± , v ± ) and P v ( u ± , v ± ) respectively represent how the swarm influences alignment for thefollower and leader populations. Choices for Q l , i.e. the specification of P i ( u ± , v ± ),form the point of distinction for the various models and are described below, seeTable 1 for a summary of the models interactions. As we will see in Section 2.2,the latter may simply take into account a fixed preferred direction, i.e. modelinga case where a population knows where it wants to go.Interaction kernels are given by the following translated Gaussian functions K i ( s ) = 1 p πm i exp (cid:18) − ( s − s i ) m i (cid:19) , i = r, a, l s ∈ [0 , ∞ ) , (4)where s r , s a and s l are half the length of the repulsion, attraction and alignmentranges, respectively. The constants m i , i = r, a, l , are chosen to ensure > , ∞ ) (specifically, m i = s i , i = r, a, l ). This allows a high level approximation of the integral defined on[0 , ∞ ) to that defined on the whole real line. The full follower-leader model assumes the following leader alignment Q v ± l = ∓ q l Z ∞ K l ( s ) ε ds = constant (5)where we call ε the orientation bias parameter . Leaders ignore other swarmmembers for alignment, receiving instead a (spatially uniform and constant)alignment bias if the orientation bias is operating. Invoking the honeybeesexample, scouts have generally agreed on the new nest at swarm take-off. Gen-eralisations could include letting ε explicitly depend on a variable factor orincluding an influence of alignment from other swarm members.6lignment of followers is taken to be Q u ± l = q l Z ∞ K l ( s ) (cid:0) u ∓ ( x ± s ) + α ∓ v ∓ ( x ± s ) − u ± ( x ∓ s ) − α ± v ± ( x ∓ s ) (cid:1) ds . (6)This dictates that a follower will be more likely to turn when it detects, withinthe region into which is moving, a large number of individuals moving in theopposite direction. Other plausible choices can be considered, however we choosethe present form for its consistency with that assumed in [16]. Note that α ± areweighting parameters that distinctly weight the leader conspicuousness, bias 3 .Completely inconspicuous leaders would correspond to α ± = 0 while if leadersare completely indistinguishable from followers α ± = 1. If leaders engage inbehaviour that raises (lowers) their conspicuousness when flying towards (awayfrom) the destination we would choose α + > α − < A leader-only model can be obtained by setting follower populations to zero( u ± ( x, t ) = 0). As noted, attraction/repulsion social interactions are main-tained, but the alignment bias is independent of the population. The targetdirection is potentially favoured through bias 1 ( ε ) and bias 2 ( β + = β − ,differential speeds). The model reduces to ∂v + ∂t + β + ∂v + ∂x = − λ v + v + + λ v − v − ,∂v − ∂t − β − ∂v − ∂x = + λ v + v + − λ v − v − ,v ± ( x,
0) = v ± ( x ) , (7)where λ v ± = λ + λ h . . y v ± − y ) i , with y v ± = Q v ± r + Q v ± a + Q v ± l . The interaction contributions are given by Q v ± r = q r Z ∞ K r ( s ) ( v ( x ± s ) − v ( x ∓ s )) ds , (8) Q v ± a = − q a Z ∞ K a ( s ) ( v ( x ± s ) − v ( x ∓ s )) ds , (9) Q v ± l = ∓ q l Z ∞ K l ( s ) εds = constant . (10) We obtain a follower-only model by ignoring dynamic evolution of the leaders.Specifically, we stipulate fixed and uniform leader populations, i.e. v + ( x, t ) and7able 1: Summary of the interactions involved in the models. “Full” denotesFull follower-leader model; “LO” denotes Leaders Only model; “FO” denotesFollowers Only model. Full LO FOPop. composition
Leaders (L) Followers (F) Leaders (L) Followers (F)
Attraction to
F+L F+L F+L F+L
Repulsion to
F+L F+L F+L F+L
Alignment to implicit F + L implicit F + implicitorientation (weighted orientation leaderbias ε with α ± ) bias ε bias ηv − ( x, t ) are constant in space and time. A leader contribution to attraction andrepulsion is eliminated while their contribution to follower alignment is reducedto a fixed and constant bias, which we refer to as an implicit leader bias andrepresent by parameter η : large η corresponds to highly influential leaders. Theresulting model is given by ∂u + ∂t + γ ∂u + ∂x = − λ u + u + + λ u − u − ,∂u − ∂t − γ ∂u − ∂x = + λ u + u + − λ u − u − ,u ± ( x,
0) = u ± ( x ) , (11)where λ u ± = λ + λ h . . y u ± − y ) i , with y u ± = Q u ± r + Q u ± a + Q u ± l . and interaction terms Q u ± r = q r Z ∞ K r ( s ) ( u ( x ± s ) − u ( x ∓ s )) ds , (12) Q u ± a = − q a Z ∞ K a ( s ) ( u ( x ± s ) − u ( x ∓ s )) ds , (13) Q u ± l = q l Z ∞ K l ( s ) (cid:0) u ∓ ( x ± s ) − u ± ( x ∓ s ) ∓ η (cid:1) ds . (14) Given its complexity, the model has a large parameter set and we thereforefix many at standard values, based on previous studies [16] and listed in Ap-8able 2: Table of parameters varied throughout this study. The parametersthat are fixed throughout this study are summarised in Table 3 (see AppendixA). “LO” denotes Leaders Only model; “FO” denotes Followers Only model.
Grouping Parameter Description Model
Bias: α + alignment due to (+) oriented leaders Full α − alignment due to ( − ) oriented leaders Full η implicit leader bias FO ε implicit orientation bias LO, Full β + speed of (+) moving leaders LO, Full β − speed of ( − ) moving leaders LO, FullPop. size: A u mean follower density FO, Full A v mean leader density LO, Full M u maximum initial follower density Full M v maximum initial leader density FullInteraction: q r repulsion strength All q l alignment strength All q a attraction strength AllOthers: λ baseline turning rate All λ bias turning rate Allpendix A. The fixed parameters include the follower speed γ as well as theinteraction ranges s r , s l , s a , fixed to generate “short-range repulsion, mid-rangealignment and long-range attraction”, a common assumption in biological mod-els of swarming behaviour [33, 6, 17]. Similarly, the more technical parameters y , m l , m a , m r are also chosen according to [16], see Appendix A.Consequently, we focus on a smaller set of key parameters that distinguishleader/follower movement, listed in Table 2 along with the models to whichthey belong. In particular, we highlight the bias parameters that stipulate alevel of attempted leader influence. We also remark that model formulationslead to conservation of follower and leader populations, generating two furtherpopulation size parameters. As a final note, we generally restrict to alignment-attractive dominated regimes, i.e. q a , q l ≫ q r .9 Dynamics of Leaders Only and Followers Onlymodels
We first analyse the dynamics of the simplified models, via linear stability anal-ysis and numerical simulation. Note that details of the numerical scheme areprovided in Appendix B.
In this model, all swarm members have some knowledge of their target and biastheir movement through two mechanisms: bias 1 , orientation according to thetarget and parametrised by ε ≥
0, and bias 2 , differential speed of movement,i.e. β + ≥ β − . We first examine the form and stability of spatially homogeneous steady state(HSS) solutions, v + ( x, t ) = v ∗ and v − ( x, t ) = v ∗∗ , for the leader-only model(7-10). Conservation of mass leads to A v = v ∗ + v ∗∗ , where A v is the sum ofinitial population densities averaged over space ( A v = (cid:10) v +0 ( x ) + v − ( x ) (cid:11) ). Thesteady state equation is obtained by solving h ( v ∗ , q l , λ, A v , ε ) = 0 , (15)where h ( v ∗ , q l , λ, A v , ε ) = − v ∗ (1 + λ tanh( − εq l − y ))+( A v − v ∗ )(1 + λ tanh(2 εq l − y ))and λ = 0 . λ . λ + λ . (16)From Eq. 15, we obtain a single HSS solution v ∗ = A v [1 + λ tanh(2 εq l − y )]2 + λ tanh( − εq l − y ) + λ tanh(2 εq l − y ) . (17)For q l = 0 (no alignment) or ε = 0 (no bias 1 ) we obtain an unaligned HSS( v ∗ , v ∗∗ ) = (cid:0) A v , A v (cid:1) , i.e. a population equally distributed into those movingin ( ± ) directions. Assuming ε >
0, dominating alignment ( q l → ∞ ) leads tosteady state ( v ∗ , v ∗∗ ) = ( A v (1 + λ ) / , A v (1 − λ ) / q l > bias 1 , i.e. ε → ∞ . Intuitively, the introduction of biaseliminates symmetry, with ε > ε is illustratedin Figure 2A. Unlike bias 1 , introduction of differential leader speed does notalter the HSS solution, since h does not depend on β ± , see Figure 2B.10o assess stability and the potential for pattern formation we perform a stan-dard linear stability analysis. Specifically, we examine the growth from homoge-neous and inhomogeneous perturbations of the HSS at ( v ∗ , v ∗∗ ) = ( v ∗ , A v − v ∗ ).Note that here it is convenient to extend K r and K a to odd kernels on the wholereal line, i.e. Q v ± r = q r Z + ∞−∞ K r ( s ) v ( x ± s ) ds,Q v ± a = − q a Z + ∞−∞ K a ( s ) v ( x ± s ) ds. We set v + ( x, t ) = v ∗ + v p ( x, t ) and v − ( x, t ) = v ∗∗ + v m ( x, t ), where v p ( x, t )and v m ( x, t ) each denote small perturbations. We substitute into (7), neglectnon-linear terms in v p and v m and look for solutions v p,m ∝ e σt + ikx . Here, k is referred to the wavenumber (or spatial eigenvalue) while σ is the growth rate(or temporal eigenvalue). A few rearrangements lead to the expression σ + ( k ) = C ( k ) + p C ( k ) − D ( k )2 , (18)where σ + ( k ) is used to denote the growth rate with largest real part. In theabove C ( k ) = ( β − − β + ) ik − λ − λ − . λ [tanh( − q l ε − y ) + tanh(2 q l ε − y )] ,D ( k ) = 4 β + β − k + 4 ikλ ( β + − β − )+2 λ ik { β + (1 + tanh(2 q l ε − y )) − β − (1 + tanh( − q l ε − y ))+ v ∗ [1 − tanh ( − q l ε − y )][( − q r ˆ K + r ( k ) + q a ˆ K + a ( k ))( β + + β − )]+ v ∗∗ [1 − tanh (2 q l ε − y )][( q r ˆ K − r ( k ) − q a ˆ K − a ( k ))( β + + β − )] } , where ˆ K ± j ( k ) , j = r, a, l denote the Fourier transform of the kernel K j ( s ), i.e.ˆ K ± j ( k ) = Z + ∞−∞ K j ( s ) e ± iks ds = exp (cid:18) ± is j k − k m l (cid:19) , j = r, a, l. The HSS is unstable (stable) to homogeneous perturbations if ℜ ( σ + (0)) > ℜ ( σ + (0)) ≤
0) and unstable to inhomogeneous perturbations if ℜ ( σ + ( k )) > k > ℜ ( σ + ( k )) > k ∈ R + ). Any k for which ℜ ( σ + ( k )) > ℜ ( σ + (0)) >
0. Solutions areexpected to diverge from the HSS both with and without movement.(S2) Stable to homogeneous and inhomogeneous perturbations, i.e. ℜ ( σ + ( k )) < , ∀ k ≥
0. We expect small (homogeneous or inhomogeneous) pertur-bations to decay and solutions that evolve to the HSS.11S3) Stationary patterns, HSS stable to homogeneous perturbations and unsta-ble to inhomogeneous perturbations. Specifically, we have ℜ ( σ + (0)) ≤ ∃ ˜ k > ℜ ( σ + (˜ k )) > k , ℑ ( σ + (˜ k )) = 0.(S4) Dynamic patterns, as (S3), but ℑ ( σ + (˜ k )) = 0 for at least some of theunstable wavenumbers.(S3) and (S4) both indicate a Turing-type instability [36], i.e. symmetry break-ing in which a spatial pattern emerges from quasi-homogeneous initial condi-tions. The presence of wavenumbers where ℑ ( σ + (˜ k )) = 0 implies growing pat-terns that oscillate in both space and time, potentially generating a dynamicpattern (e.g. a travelling swarm). These are, though, predictions based on so-lutions to the linearised system and nonlinear dynamics are likely to introducefurther complexity.Key results from the analysis are summarised in Figure 2, indicating thatboth the HSS and its stability change with bias parameters ε (or q l ), the ra-tio β + /β − and q a . As noted above, increasing ε (or q l ) generates a HSS with( ± ) distributions increasingly favouring the target direction. Variations in the β + /β − ratio do not alter the HSS value but do impact on the stability. Un-der both biases 1 and 2, the stability nature changes at key threshold values,critically depending on the strength of attraction, q a . For low q a the HSS isstable for all values of ε and/or β + /β − : attraction is insufficient to cluster thepopulation and it remains dispersed. There may be biased movement towardsthe target, but the population remains in a uniformly dispersed/non-swarmingstate.For larger q a , however, the HSS becomes unstable under inhomogeneousperturbations. A Turing-type instability occurs and emergence of a spatialpattern is expected. The predicted pattern critically depends on the bias. For anunbiased scenario ( ε = 0 and β + /β − = 1) we have stability class (S3) and predicta stationary pattern, see dark green asterisks in Figures 2A and 2B. Simulationscorroborate this prediction (see Figure 2C), where we observe stationary clusterformation. Each cluster is weighted equally between ( ± ) directed populationsand the overall cluster is fixed in position. Note, however, that the nonlocalelements of the model generate a degree of intercluster communication and,over longer timescales, clusters may attract each other and merge.Introducing bias 1 ( ε >
0) or bias 2 ( β + /β − > D ( k ) and/or C ( k ) – and the instability is of type (S4). In this casea dynamic component is predicted, with simulations substantiating this, cf.Figures 2D and 2E. The forming clusters are asymmetrically distributed between( ± ) directed movement and, overall, we observe steered swarming: clusters movein the direction determined by the bias. Notably, clusters move at distinct speedsaccording to their size, so that clusters collide and merge. Eventually, a singlesteered swarm has formed and migrates with fixed speed and shape (a travellingpulse). The simultaneous action of biases 1 and 2 generates similar behaviour,Figure 2F, with the combined action creating faster movement towards thetarget. 12igure 2: Dynamics of the leader-only model. (A-B) Bifurcation plots showingHSS (proportion moving rightwards) and stability under parameter variation.(A) Bias 1 , i.e. variation of ε under low (top panel, q a = 0 .
5) and high (bottompanel, q a = 50) attraction ( bias 2 inactive, β + = β − = 0 . Bias 2 ,varying β + for fixed β − = 0 .
1, under low (top panel, q a = 0 .
5) and high (bottompanel, q a = 50) attraction ( bias 1 inactive, ε = 0). Solid blue and dashed blacklines denote stability class S2 and S4 respectively, dark green asterisks indicatestability class S3. (C-F) Space-time density map showing evolving total leaderdensity, under: (C) unbiased case, generating a stationary patterns; (D) bias 1 ,obtained for ε = 0 . bias 2 inactive, β + = β − = 0 . bias 2 , obtained for β + /β − = 0 . / . bias 1 inactive, ε = 0),generating target directed swarms; (F) biases 1 and 2 , obtained for ε = 0 . β + /β − = 0 . / .
1, generating target directed swarms with enhanced speed.(C-F) ICs are perturbations of ( v ∗ , v ∗∗ ) = (2 , v ∗ , v ∗∗ ) = (3 . , . q r = 0 . q l = 7 .
5, ((C-F) q a = 7 . A v = 4 , λ = 0 . , λ = 0 . . Summarising, the leader-only model illustrates the distinct contributionsfrom different model elements: (i) attraction is crucial to aggregate a dispersedpopulation; (ii) assuming sufficient attraction, either bias 1 or bias 2 is suffi-cient to propel the swarm in the direction of the target, with increased swarmspeed if both biases act together. We next examine the follower-only model. Interaction occurs through attrac-tion, repulsion and alignment, with an additional uniform alignment bias parametrisedby η and corresponding to implicit perception of a leader population. Proceeding as before, we explore the form of spatially homogeneous steady statesolutions. Conservation of the total follower population leads to A u = u ∗ + u ∗∗ ,13here A u is the average (over space) of the sum of initial population densities, A u = (cid:10) u +0 ( x ) + u − ( x ) (cid:11) . Steady states will be given by h ( u ∗ , q l , λ, A u , η ) = 0 , (19)where h ( u ∗ , q l , λ, A u , η ) = − u ∗ (1 + λ tanh( A u q l − u ∗ q l − ηq l − y ))+( A u − u ∗ )(1 + λ tanh( − A u q l + 2 u ∗ q l + ηq l − y ))and λ = 0 . λ . λ + λ . (20)The zero-bias scenario ( η = 0) has been analysed in depth previously, see[16], and we restrict to a brief summary. First, a single unaligned HSS exists at( u ∗ , u ∗∗ ) = ( A u , A u , i.e. both directions equally favoured. Dominating alignment ( q l → ∞ ) generatestwo further HSS at ( u ∗ , u ∗∗ ) = ( A u (1 ∓ λ ) / , A u (1 ± λ ) / aligned HSScorresponds to a population where alignment induces the population to favourone direction. A typical structure for the bifurcation diagram is illustrated inFigure 3A: a central branch corresponding to the unaligned HSS and upperand lower aligned branches. For the chosen parameters, these branches areconnected via a further set of intermediate (unstable) branches. Thus, as q l increases the number of steady states shifts between 1, 5 and 3 steady states(see also Figure 10 of Appendix C.2).The symmetric structure of η = 0 is lost for η = 0, even under small val-ues: see Figures 3B-C. The aligned HSS branch corresponding to the targetdirection is more likely to be selected, the other branch is shifted rightwards(Figure 3B) and for larger η disappears entirely (Figure 3C). Overall, the ex-ternal bias is amplified by follower to follower alignment and the populationbecomes predominantly oriented in the target direction.We extend to a spatial linear stability analysis, applying the same processas in section 3.1.1 to obtain the following dispersion relation σ + ( k ) = C ( k ) + p C ( k ) − D ( k )2 , (21)where C ( k ) = (cid:16) λ u + u − − λ u + u + (cid:17) u + + (cid:16) λ u − u + − λ u − u − (cid:17) u − − λ u − − λ u + , (22) D ( k ) = 4 γ k +4 γik h(cid:16) − λ u + u − − λ u + u + (cid:17) u + + (cid:16) λ u − u − + λ u − u + (cid:17) u − + λ u − − λ u + i . (23)In the above, λ u ± u ± denote the partial derivatives of λ u ± with respect to u ± and subsequently evaluated at the HSS ( u ∗ , u ∗∗ ). For reference we provide the14xplicit forms in Appendix C.1, yet intricacy of the dispersion relation restrictsus to a numerical approach. Stability is again classified into one of the 4 classesdescribed earlier.The diagrams shown in Figure 3A-C reveal a complex bifurcation structureand potentially diverse dynamics according to parameter selection and initialcondition. Indeed, this has already been highlighted in depth for the unbiased( η = 0) model in [16], where various complex spatiotemporal pattern forms havebeen revealed. For example, Figure 11 in Appendix C.3 illustrate transitioningbetween stationary and dynamic aggregates as the key parameter q l is altered.Note that moving aggregates can be generated without any incorporated bias,though if the population begins quasi-symmetric either direction will be selectedwith equal likelihood.Here we focus on the extent to which introduction of a bias influences thedynamics of aggregate structures, with Figure 3D-G providing a representativesequence. We begin with an unbiased scenario, setting η = 0 and choosingparameters from a region predicted to lead to stationary patterning. We initiatepopulations in quasi-symmetric fashion, setting u + ( x,
0) = A u (1 + r u ( x ))2 , u − ( x,
0) = A u (1 − r u ( x ))2 , where r u ( x ) denotes a small random perturbation. As expected from the sta-bility analysis, a stationary cluster forms (see Figure 3D) with its shape andposition maintained by a symmetric distribution of ( ± ) directed populations. In-troducing bias, though, disrupts the symmetry and Turing instabilities falls intothe dynamic pattern class. Moreover, even a marginal alignment bias stronglyselects clusters that move in the direction of the bias, e.g. see Figures 3E. Start-ing from a symmetric or nonaligned initial set-up, bias slightly tilts followerstowards the target. Follower-follower alignment snowballs, eventually resultingin a cluster moving towards the target. Increasing the bias magnitude increasesswarm speed, Figure 3F.As for the 100% leader model, there is a clear relationship between clusterspeed and cluster size. This is illustrated in Figure 3G, where the initial sym-metry breaking process generates two clusters of slightly different size, Figure3H(bottom). Both clusters move in the target direction, but the smaller clusteris considerably faster. The clusters eventually collide and merge to form aneven larger and slower cluster, see Figure 3H(top). Note that, in principle it isalso possible to obtain a swarm migrating oppose the target direction, e.g. byheavily favouring biasing the initial conditions. Simulations, though, suggestthat such situations are highly unlikely to occur in practice.Introducing bias can even trigger symmetry breaking, as shown in Figure 4.To highlight this, we neglect attractive and repulsive interactions ( q a = q r = 0)and focus solely on alignment. Initially setting η = 0, remaining parametersare specified such that the unaligned HSS (i.e. u ∗ = u ∗∗ ) is stable to bothhomogeneous and inhomogeneous perturbations: a typical dispersion relation isprovided in Figure 4A (top), showing the absence of wavenumbers with positivegrowth rates and the corresponding simulation confirms the absence of pattern15igure 3: Dynamics of the follower-only model. (A-C) Bifurcation plots showingHSS (proportion moving rightwards) and stability under parameter variation.Bifurcation parameter is q l and the resulting bifurcation plots are shown for(A) η = 0 (unbiased), (B) η = 0 .
04, (C) η = 4. Other parameters set at q r = 0 , q a = 0 . , A u = 2 , λ = 0 . , λ = 3 .
6. Stability classes plotted as S1:dotted red, S2: solid blue, S3: dark green asterisks, S4: dashed black. (D-F)Space-time plot showing the evolving total follower density under variation of η : (D) η = 0 (unbiased), (E) η = 0 .
04, (F) η = 4. Stronger biases lead tofaster swarm movement towards the target. Other parameters set as in (A-C) with q l = 0 .
4. ICs are perturbations of ( u ∗ , u ∗∗ ) = (1 , η = 0 . , q r = 0 . , q l = 1 , q a = 10 , A u = 2 , λ = 0 . , λ = 0 . q r = 0 , q l = 2 . , q a = 0 , A u = 2 , λ =0 . , λ = 3 .
6. (A) top row and (B) assume no bias ( η = 0), while (A) bottomrow and (C) consider the effect of a small alignment bias ( η = 0 . u ∗ , u ∗∗ ) = (1,1).formation, Figure 4B. Introducing bias ( η >
0) breaks symmetry, yielding anonzero range of wavenumbers with positive growth rates, Figure 4A (bottom).A pattern emerges which generates multiple clusters moving in the target direc-tion, Figure 4C.Summarising the analysis and numerics in this section, we emphasize that thefollower only submodel can display a range of aggregating/swarming behaviour,where the processes of alignment and attraction combine to generate one ormore cluster. The addition of bias breaks directional symmetry, eliminating theformation of stationary structures and generating clusters that move coherentlyin the target direction.
We turn attention to the full “follower-leader” model, formed from Equations(1-6) and where followers constitute a completely naive population. Our prin-cipal aim will be to understand whether a steered swarm can arise under leadergenerated bias. Relying principally on numerical simulation, we focus on twogeneral parameter regimes: (P1) strong attraction-strong alignment, and (P2)strong attraction-weak alignment. For simplicity we neglect repelling interac-tions ( q r = 0). A bias corresponding to the (+)-direction can occur through17arameter choices: • Bias 1 , ε >
0, orientation; • Bias 2 , β + > β − , speed; • Bias 3 , α + > α − , conspicuousness.Consequently, the unbiased case is ε = 0, α + = α − , and β + = β − . As discussedearlier, evidence is found for each bias in our honeybee swarming exemplar.Note that parameter regimes are selected such that linear stability analysis ofthe uniform solution in the unbiased case predicts Turing pattern formation. Steady state analysis proceeds as before: we look for the spatially homogeneoussteady states u + ( x, t ) = u ∗ , u − ( x, t ) = u ∗∗ and v + ( x, t ) = v ∗ , v − ( x, t ) = v ∗∗ ,noting that conservation ensures A u = u ∗ + u ∗∗ and A v = v ∗ + v ∗∗ , where A u and A v are as earlier described. Steady states for the full model satisfy h u ( u ∗ , q l , λ, A u , A v , α − , α + , y ) = 0 , (24) h v ( v ∗ , q l , λ, A v , ε, y ) = 0 , (25)where h u = − u ∗ (1 + λ tanh( A u q l − u ∗ q l + q l α − ( A v − v ∗ ) − q l α + v ∗ − y ))+( A u − u ∗ )(1 + λ tanh( − A u q l + 2 u ∗ q l + q l α + v ∗ − q l α − ( A v − v ∗ ) − y )) ,h v = − v ∗ (1 + λ tanh( − εq l − y )) + ( A v − v ∗ )(1 + λ tanh(2 εq l − y )) , and λ = 0 . λ . λ + λ . (26)Leader steady states correspond to those obtained previously for the leader-onlymodel. Hence, the proportion of leaders at HSS moving in the (+) directionincreases monotonically between A v / A v (1+ λ ) /
2, according to ε and/or q l ,(Figure 2A). This equivalence stems from the simplification that leaders ignoreothers with respect to alignment.In absence of alignment, i.e. q l = 0, we find a single unaligned HSS at( u ∗ , u ∗∗ , v ∗ , v ∗∗ ) = ( A u / , A u / , A v / , A v / q l = 0, follower steady statesare clearly more complex and we first consider the unbiased case ( ε = 0, α + = α − , β + = β − ). Here we have v ∗ = v ∗∗ = A v / h u = − u ∗ (1 + λ tanh( A u q l − u ∗ q l − y ))+( A u − u ∗ )(1 + λ tanh( − A u q l + 2 u ∗ q l − y )) . Leaders have no influence and follower steady states are as observed for thefollower-only model with η = 0. As described earlier, the number of followersteady states varies between 1, 3 and 5 (see Figure 10 of the Appendix C.2) with18ufficiently large alignment allowing followers to self-organise into a dominatingorientation.We next consider an extreme bias 1 ( ε → ∞ ), while excluding other bi-ases ( α = α + = α − , β + = β − ). Leaders favour the (+) direction, specifically( v ∗ , v ∗∗ ) = ( A v (1+ λ )2 , A v (1 − λ )2 ), and hence h u = − u ∗ (1 + λ tanh( A u q l − u ∗ q l − q l αA v λ − y ))+( A u − u ∗ )(1 + λ tanh( − A u q l + 2 u ∗ q l + q l αA v λ − y )) . The above has the same structure as for the follower-only model under externalbias, where η in Equation (19) is replaced by αA v λ . Consequently, for eitherincreasing leader to follower influence ( α ) or increasing leader population size( A v ), bifurcations occur as in Figure 3A-C: symmetric follower steady statesbecome asymmetric, favoured according to the bias.Differential speeds (bias 2, β + = β − ) do not impact on steady states and weturn instead to distinct conspicuousness, specifically extreme bias 3 ( α + /α − →∞ ) while eliminating bias 1 . Leader steady states remain symmetrical ( v ∗ = v ∗∗ = A v / u ∗ , u ∗∗ , v ∗ , v ∗∗ ) = ( A u (1 + λ ) / , A u (1 − λ ) / , A v / , A v / . The bifurcation diagrams in Figure 5 numerically confirm these results. Finally,we note that as q l → ∞ two further HSS’s arise at ( u ∗ , u ∗∗ , v ∗ , v ∗∗ ) = ( A u (1 ∓ λ ) / , A u (1 ± λ ) / , A v (1 + λ ) / , A v (1 − λ ) / The steady state analysis provides insight into whether different biases induceleft-right asymmetry, yet the emerging dynamics of spatial structures remainsunclear. We numerically explore the full spatial nonlinear problem, in particularits capacity to generate a steered swarm as described earlier. Simulations willbe conducted for two forms of initial condition.(IC1)
Unbiased and dispersed . Populations quasi-uniformly distributed in spaceand orientation. Letting A u and A v , respectively denote the mean totalfollower and leader densities, u + ( x,
0) = A u (1 + r u ( x ))2 , u − ( x,
0) = A u (1 − r u ( x ))2 ,v + ( x,
0) = A v (1 + r v ( x ))2 , v − ( x,
0) = A v (1 − r v ( x ))2 . (IC2) Unbiased and aggregated . Populations initially aggregated but unbiased inorientation. Letting M u and M v respectively denote the maximum initial19igure 5: Proportion of right-moving populations at steady state(s). (A,C)Effect of bias 1 , increasing ε , on position and number of equilibrium points( bias 2 and 3 inactive, β − = β + = 0 . , α − = α + = 1). (B,D) Effect of bias 3 ,increasing α + /α − , on position and number of equilibrium points, for α − = 0 . bias 1 and 2 inactive, ε = 0 and β − = β + = 0 . q r = 0 , q l = 10 , q a = 8), bottom rowcorresponds to (P2) strong attraction-weak alignment ( q r = 0 , q l = 1 , q a = 10).Other parameter values fixed at A u = A v = 1, λ = 0 . , λ = 0 . Figure 6: Dynamics of the full-model, unbiased case, obtained for ε = 0 , β − = β + = 0 . , α − = α + = 1. (A,C) Space-time evolution of densities under (IC1)for (A) P1, strong attraction-strong alignment, (C) P2, strong attraction-weakalignment. Non-white regions indicate where the total population > A u + A v ),i.e. “clustering” has occurred, with red/blue indicating local predominance ofleaders/followers respectively. Panels (B,D) show (top) population distributionand (bottom) population fluxes for solutions under (IC2) for (B) P1, strongattraction-strong alignment, (D) P2, strong attraction-weak alignment. (P1) q r = 0 , q l = 10 , q a = 8, (P2) q r = 0 , q l = 1 , q a = 10, with other parameter valueset as A u = A v = 1 (IC1), M u = M v = 12 .
61 (IC2), λ = 0 . , λ = 0 . u + ( x,
0) = M u e − x − x ) (1 + r u ( x ))2 , u − ( x,
0) = M u e − x − x ) (1 − r u ( x ))2 ,v + ( x,
0) = M v e − x − x ) (1 + r v ( x ))2 , v − ( x,
0) = M v e − x − x ) (1 − r v ( x ))2 . Note that r u ( x ) , r v ( x ) denote small (1%) random perturbations. (IC1) allowinvestigation into whether dispersed populations self-organise into swarms while(IC2) tests whether aggregated populations maintain a swarm profile. (IC2) areparticularly appropriate for bee swarming, where followers and leader scouts areinitially clustered together. We first explore the capacity for self-organisation in the unbiased scenario. Notethat each of the two principal parameter sets were selected to generate Turinginstabilities and Figure 6A and C demonstrate the patterning process under (P1)21igure 7: Impact of biases on swarm movement for the full model in a strongattraction-strong alignment regime (P1), strong attraction-weak alignmentregime (P2) and under (IC1). Populations plotted in space-time (colourmap asdescribed in Figure 6). (A,D)
Bias 1 , obtained for ε = 0 . bias 2 and 3 inac-tive, β − = β + = 0 . , α − = α + = 1); (B,E) bias 2 , obtained for β + /β − = 0 . / . bias 1 and 3 inactive, ε = 0 , α − = α + = 1); (C,F) bias 3 , obtained for α + /α − = 1 . / . bias 1 and 2 inactive, ε = 0 , β − = β + = 0 . q r = 0 , q l = 10 , q a = 8, (P2) q r = 0 , q l = 1 , q a = 10 and A u = A v = 1, λ = 0 . , λ = 0 . c = 0,Figure 6B and D. Swarms contain leaders concentrated at the swarm centre,with followers symmetrically dispersed either side. The distinct follower/leaderprofiles arise as leaders only interact through attraction, while followers receiveadditional alignment information. We further plot the fluxes , i.e. the quantities u + ( x, t ) − u − ( x, t ) and v + ( x, t ) − v − ( x, t ). In the stationary swarm profile, ( ± )movement is balanced such that the swarm maintains its position and shape,see Figure 6. We perform the same set of simulations, but extended to include one of thethree proposed mechanisms for leader bias. Simulation results under (IC1) These are leaders in name only, as in the unbiased scenario there is no directional bias inforce. bias 1 are illustrated in Figure 8A and Figure9A. Over a wide range of bias strengths, bias 1 generates steered swarming,with an increased speed in the target direction as ε increases. However twocaveats must be highlighted. First, under certain parameter combinations weunexpectedly observe swarms that move away from the the target, specificallyfor weaker biases in the weak alignment regime (see Figure 9A1). Second,excessive biases can lead to loss of swarm coherence and eventual dispersion (seeFigure 8A4). Thus, we conclude bias 1 is found to be only partially successfulin generating a steered swarm.We next consider bias 2 , i.e. increasing the ratio of leader speeds whenmoving towards or away from the target. Indicative simulations are plotted inFigures 8B and Figure 9B. Similar to bias 1 , successful steering only occurswithin a range of β + /β − values. First, as observed above under bias 1 , certainparameter regimes are capable of generating counter target directed swarms,for example see Figure 8B1 for a moderately faster v + population in the highattraction-high alignment regime. Second, while increasing the speed ratio canhelp generate steered swarms, excessively fast target-directed movements canlead to “swarm-splitting”, i.e. leaders that pull free from followers and leavethem stranded. This phenomenon is observed in Figure 8B4 at around T ≈ T ≈
80. Under periodic boundary conditions, runawayleaders eventually reconnect with the stranded followers, leading to a periodiccycle (see the illustrative example of Figure 9B3). Of course, in a real-worldscenario, leaders would simply leave followers behind.Representative swarm behaviours under modulation of bias 3 , i.e. wherewe modulate the relative conspicuous of leaders moving towards or away fromthe destination, are shown in Figures 8C and Figure 9C. Notably, this form ofbias was found to consistently generate a steered swarm in the target direction,over all tested ranges of α + /α − and for both two parameter regimes.As a final exploration we examined swarm movement with all biases appliedsimultaneously: typical results are shown in Figure 8D for the high attrac-tion/high alignment regime only. The application of multiple biases appears to23igure 8: Impact of biases on swarm movement for the full model in a strongalignment-strong attraction regime (P1) and under (IC2). Populations plot-ted in space-time (colourmap as described in Figure 6). Note that we appendeach plot with the swarm speed, for cases where a travelling wave solutionis (numerically) found. (A) Bias 1 , obtained for ε = (A1) 0.5, (A2) 1.0,(A3) 2.0, (A4) 3.0 (bias 2 and 3 inactive, β + = β − = 0 . , α + = α − = 1).(B) Bias 2 , obtained for β + /β − = (B1) 0.2/0.1, (B2) 0.3/0.1, (B3) 0.5/0.1,(B4) 0.6/0.1 (bias 1 and 3 inactive, ε = 0 , α + = α − = 1). (C) Bias 3 , ob-tained for α + /α − = (C1) 1/0.9, (C2) 1/0.575, (C3) 1/0.55, (C4) 1/0 (bias1 and 2 inactive, ε = 0 , β + = β − = 0 . ε = 0 . , β + /β − = 0 . / . , α + /α − = 1 / /
3, (D2) ε = 0 . , β + /β − =0 . / . , α + /α − = 1 / .
5, (D3) ε = 0 . , β + /β − = 0 . / . , α + /α − = 1 / . ε = 1 , β + /β − = 0 . / . , α + /α − = 1 / (1 / q r = 0 , q l = 10 , q a = 8, M u = M v = 12 . λ = 0 . , λ = 0 . ε and ratios of β + /β − .Yet escaping leaders can still occur when β + /β − becomes too large, e.g. seeFigure 8D4. 24igure 9: Impact of biases on swarm movement for the full model in a strongattraction-weak alignment regime (P2) and under (IC2). Populations plotted inspace-time (colourmap as described in Figure 6). Note that we append each plotwith the swarm speed, for cases where a travelling wave solution is (numerically)found. (A) Bias 1 , obtained for ε = (A1) 3.0, (A2) 5.0, (A3) 15.0 ( bias 2 and 3 inactive, β + = β − = 0 . , α + = α − = 1). (B) Bias 2 , obtained for β + /β − = (B1)0.2/0.1, (B2) 0.8/0.1, (B3) 1/0.1 ( bias 1 and 3 inactive, ε = 0 , α + = α − = 1).(C) Bias 3 , obtained for α + /α − = (C1) 1/0.9, (C2) 1/0 ( bias 1 and 2 inactive, ε = 0 , β + = β − = 0 . q r = 0 , q l = 10 , q a = 8, M u = M v = 12 . λ = 0 . , λ = 0 . Collective migration occurs when a population fashioned from interacting indi-viduals self-organise and move in coordinated fashion. Recently, much attentionhas focused on the presence of leaders and followers, essentially a division of thegroup into distinct fractions that are either informed and aim to steer or naiveand require steering, [18, 29, 28]. Here we have formulated a continuous modelto understand such phenomena, a non-local hyperbolic PDE system that ex-plicitly incorporates separate leader and follower populations that have distinctresponses to other swarm members. We considered distinct mechanisms throughwhich leaders attempt to influence the swarm. Specifically, taking inspirationfrom the guidance provided by scout bees, [32, 26], we focused on three differentmechanisms: a bias in the leader alignment according to the target ( bias 1 ),higher speed ( bias 2 ) when moving towards the target and greater conspicu-ousness ( bias 3 ) when moving towards the target.We initially focused on simpler models of greater analytical tractability.First, a 100% leader model composed only of informed members. Here onlybiases 1 or 2 operate, both proving effective at steering the group towards thedestination. Maintaining group cohesion is, unsurprisingly, contingent on suffi-ciently strong attraction. Second, we considered a 100% follower model: pop-ulation members were naive but received some alignment bias, e.g. due to animplicitly present leader population. The range of dynamics generated by this25odel is more complicated, stemming from the more sophisticated alignmentresponse. Nevertheless, introduction of bias acts to break the symmetry, signif-icantly favouring the target direction.The full follower-leader model is capable of forming and maintaining a swarmthat is consistently steered towards the destination, across a broad range of pa-rameter regimes. Nevertheless, when examined individually, the distinct biasesreveal varying levels of success at generating a steered swarm. First, biases1 and 2 can, somewhat surprisingly, generate a swarm that moves away fromthe target, even inside plausible parameter regimes. Second, introduction ofbiases can lead to eventual dispersal of the swarm, pushing the system outsidethe regime in which attraction maintains swarm cohesion. Third, significantvariation in speeds can lead to swarm splitting, where leaders split away fromthe group and leave followers stranded. Distinctly conspicuous leaders, how-ever, consistently generated swarms moving towards the target, although weacknowledge the generality of this statement is limited by the purely numericalnature of the study.The varying success of the different influence strategies may stem from themanner in which the biases act. Biases 1 and 2 only indirectly influence followers:they describe behaviours in which the leaders alter their response according tothe target direction, but do not directly enter the dynamics of the followerpopulation. Any influence they exert on followers is therefore through alteringthe distribution of the leader population with respect to the followers, e.g. avariation in velocity that tends to polarise the position of leaders.Furthermore, the guidance efficiency provided by biases 1 and 2 appears to berelated to the interactions parameter regime. On one hand, we observed counter-target directed swarms under bias 1 in the strong attraction-weak alignmentregime. From a mathematical perspective this is reasonable, as the alignmentstrength ( q l ) is directly proportional to the orientation bias ( ε ), see Eqs. 5 and10. We therefore speculate that bias 1 demands a sufficiently strong alignment.Otherwise, attraction dominates and the swarming group is directed accord-ingly, potentially against the target, as in Figure 9A1. On the other hand, bias2 can lead to swarming against the target under strong alignment-strong attrac-tion. In support of this, we remark that swarming direction derives from thetransport terms (depending on β + : β − ) and the competing social interactions.In summary, we speculate that bias 1 is favoured by interactions (specifically,alignment) while bias 2 is hindered by them. Bias 3, however, directly influencethe followers: weighting the follower alignment to favour the target direction.For swarming populations it is worth stressing that these various biases may wellact in concert: for example, in the context of bee swarms, speed variation maynot be the intended mechanism for generating movement towards the target,it may rather be a side effect of altering the conspicuousness of nest-orientedscouts.As noted above, biases act to alter the relative positions of leader and fol-lower populations, subtly weighting the interactions to break the symmetry ofthe system. Different parameter regimes lead to different follower/leader dis-tributions, which we broadly classify as pull or push systems: in the former,26eaders adopt a position at the front of the swarm, pulling the followers towardsthe target; in the latter, leaders are primarily concentrated in the rear, pushingthem towards the destination. Transitions in the follower-leader distributionare highly contingent on parameter choice: for example, in Figure 8(C2-C3) weobserved a sharp transition in the follower-leader distribution under a marginalvariation in conspicuousness, in turn generating a significantly faster swarm. Amore detailed analytical investigation into these transitions would be of signifi-cant interest, but lies outside the scope of the present study.The model here provides substantial insight into the mechanisms throughwhich informed leaders direct a swarm, yet its complexity has demanded cer-tain simplifications. For example, in this preliminary work we have restricted tofixed leader and follower populations – a reasonable approximation for, say, beeswarms, where a fixed subset of the population has explicit knowledge of thedestination. In other instances, follower-leader distinction may be less clearcutand potentially transferable: for example, within cell populations “leadership”may be a chemically acquired characteristic determined by signals transmittedby other cells or the environment, [2]. Extensions of the model in this directionwould require additional terms that account for the transfer between followerand leader status. We also note that the model here has focused on a simplifiedone-dimensional framework, though real-life collective migration phenomena aretypically two or three dimensional in structure (for example, in bee swarms thestreaker leaders adopt a position concentrated towards the upper portion of a3D swarm). Further potential adaptations could include incorporating environ-mental heterogeneity, such as the need to overcome environmental obstacles,or modelling other forms of interaction, such as “chase-and-run” phenomena inwhich one population attempts to escape a population of pursuers. The latteris certainly relevant in ecological instances, for instance predator-prey relation-ships, but extends to various cellular populations including neural crest andplacode cells [34]. While discrete models have been formulated to describe suchprocesses, e.g. [10], a complementary continuous approach may yield furtherinsights. Finally, the model lends itself to studying decision-making, i.e. whereswarming may lead to consensus where there are multiple informed leader pop-ulations each exhibiting their own preferred direction. A fundamental contribu-tion in this direction has been provided in [11] through a discrete description.Notwithstanding its simplifications, we believe the model presented here pro-vides a starting point for future investigations into the role of heterogeneity oncollective migration phenomena. Acknowledgements
This research was partially supported by the Italian Ministry of Education,University and Research (MIUR) through the “Dipartimenti di Eccellenza” Pro-gramme (2018-2022) – Dipartimento di Scienze Matematiche “G. L. Lagrange”,Politecnico di Torino (CUP: E11G18000350001). SB is member of GNFM(Gruppo Nazionale per la Fisica Matematica) of INdAM (Istituto Nazionaledi Alta Matematica), Italy. 27
Fixed Parameters
The parameters in Table 3 are common to all models and set at a fixed referencevalue, in accordance with those in [16].Table 3: Table of parameters with fixed values.Parameter Description Value [Unit] y shift of the turning function 2 s r attraction range 0.25 [L] s l alignment range 0.5 [L] s a attraction range 1 [L] m r width of repulsion kernel 0.25/8 [L] m l width of alignment kernel 0.5/8 [L] m a width of attraction kernel 1/8 [L] L domain length 10 [L] γ follower speed 0.1 [L/T] B Numerical Method
Numerical simulations are performed on a 1D spatial domain [0 , L ] with periodicboundary conditions imposed at x = 0 , L : u + (0 , t ) = u + ( L, t ) , u − (0 , t ) = u − ( L, t ) , and similarly for v ± . The numerical scheme invokes a Methods of Lines ap-proach, where initial discretisation over space yields a system of ordinary differ-ential equations that are subsequently integrated over time. Spatial movementterms are approximated with a first order upwind scheme. Switching terms de-mand approximation of the infinite attraction/repulsion/ alignment integrals.For this we exploit the Gaussian nature of the Kernel functions and approx-imate the integrals on finite domains [0 , i ], i = s r , s a , for attractive and re-pulsive kernels and [0 , s l ] for alignment. The integral itself is approximatedusing Simpson’s method. We finally remark that under the periodic boundaryconditions the integrals are wrapped around the domain.28 Stability analysis for follower-only model
C.1 Expressions for stability analysis
Basic algebra provides the following expressions for λ i ± i ± , for i ∈ { u, v } : λ u + u − = 0 . λ (cid:2) − tanh ( q l ( u ∗∗ − u ∗ ) − y ) (cid:3) (cid:16) q r ˆ K + r ( k ) − q a ˆ K + a ( k ) + q l ˆ K + l ( k ) (cid:17) , (27) λ u − u − = 0 . λ (cid:2) − tanh ( − q l ( u ∗∗ − u ∗ ) − y ) (cid:3) (cid:16) q r ˆ K − r ( k ) − q a ˆ K − a ( k ) − q l ˆ K + l ( k ) (cid:17) ,λ u + u + = 0 . λ (cid:2) − tanh ( q l ( u ∗∗ − u ∗ ) − y ) (cid:3) (cid:16) q r ˆ K + r ( k ) − q a ˆ K + a ( k ) − q l ˆ K − l ( k ) (cid:17) ,λ u − u + = 0 . λ (cid:2) − tanh ( − q l ( u ∗∗ − u ∗ ) − y ) (cid:3) (cid:16) q r ˆ K − r ( k ) − q a ˆ K − a ( k ) + q l ˆ K − l ( k ) (cid:17) , where ˆ K ± j ( k ) , j = r, a, l denote the Fourier transform of the kernel K j ( s ), i.e.ˆ K ± j ( k ) = Z + ∞−∞ K j ( s ) e ± iks ds = exp (cid:18) ± is j k − k m l (cid:19) , j = r, a, l. Substituting Eq. 27 into Eqs. 22 and 23 yields C ( k ) = − λ − λ − . λ [tanh( q l ( u ∗∗ − u ∗ − η ) − y ) + tanh( − q l ( u ∗∗ − u ∗ − η ) − y )]+ q l ( ˆ K + l ( k ) + ˆ K − l ( k )) { u ∗ (cid:2) . λ (1 − tanh ( q l ( u ∗∗ − u ∗ − η ) − y )) (cid:3) + u ∗∗ (cid:2) . λ (1 − tanh ( − q l ( u ∗∗ − u ∗ − η ) − y )) (cid:3) } D ( k ) = 4 γ k + 2 γλ ik { tanh( − q l ( u ∗∗ − u ∗ − η ) − y ) − tanh( q l ( u ∗∗ − u ∗ − η ) − y )+ u ∗ [1 − tanh ( q l ( u ∗∗ − u ∗ − η ) − y )]( − q r ˆ K + r ( k ) + 2 q a ˆ K + a ( k ) − q l ˆ K + l ( k ) + q l ˆ K − l ( k ))+ u ∗∗ [1 − tanh ( − q l ( u ∗∗ − u ∗ − η ) − y )](2 q r ˆ K − r ( k ) − q a ˆ K − a ( k ) − q l ˆ K + l ( k ) + q l ˆ K − l ( k )) } . C.2 Steady state variation in parameter space
When alignment impacts on the social interactions, i.e. q l = 0, Eq. 19 can haveone, three or five solutions, depending on the value of λ and η . Specifically, forsmaller η two-parameter numerical bifurcation diagrams in ( q l , λ ) space indi-cate a threshold value λ ∗ ∈ (0 ,
1) such that if λ > λ ∗ then there are up to threesolutions. Conversely, if λ < λ ∗ there are up to five solutions (Figure 10A,10B).As η increases, the parameter region resulting in 5 steady states reduces, com-pletely disappearing for η = 0 .
2, see Figures 10A, 10B, 10C. For η = 0 .
4, Eq.19 shows one or three solutions, see Figure 10D.
C.3 Effect of alignment on equilibrium points
The bifurcation diagram in Figure 3A shows that in the absence of any ex-ternal bias,various stationary and temporal patterns can emerge, as previouslydescribed [16]. For example, under weak alignment and sufficiently strong at-traction, pattern formation occurs whereby the population aggregates throughmutual attraction (Figure 11A). Under weak alignment, however, a symmetry29igure 10: Two parameter bifurcation diagram in ( q l , λ ) space, obtained for (A) η = 0, (B) η = 0 .
04, (C) η = 0 .
2, (D) η = 0 .
4. Dashed lines in (A,B) indicatethe value of λ , denoted as ˆ λ , resulting from λ = 0 . λ = 0 . A u = 2.is maintained between the proportions of ( ± ) populations and the aggregateremains stationary. For stronger alignment the HSS becomes stable and thepopulation remains uniformly dispersed across the domain, see Figure 11B, yetfor even stronger alignment two symmetric branches appear with both locallyundergoing a saddle point bifurcation and leading to the formation of mov-ing patterns (Figure 11C). Note, though, for parameters in this region andquasi-symmetrically distributed initial populations, patterns are equally likelyto favour the (+) or ( − ) directions. Within this alignment range, the centralequilibrium turns back to generate stationary aggregations (Figure 11D) andwhen the system crosses a critical value a pitchfork bifurcation arises: the so-lution ( u ∗ , u ∗∗ ) = (1 ,
1) loses its stability in the homogeneous space and thepopulation jumps to one of the stable branches.30igure 11: Pattern formation for the follower-only model under the unbiasedscenario, i.e. η = 0, obtained for (A) q l = 0 .
4, (B) q l = 1 .
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