arXiv:1611.09624v2 [q-bio.PE] 11 Jul 2017 fhwuiutu hsceitnemcaimis. Keywords: mechanism coexistence Rock-Paper- this the presen ubiquitous of are how generalization group of well-studied each a o for to spatial similar strategies of mechanism presence individual the and predictions in collective behavior mean-field which richer their its with Besides behavior strategies. resulting the compare and we ebr ftegopwiefeig concentrates feeding, be- while competition group the there acquir- the increases hand, of also other of process the it members the On as tween setbacks in etc. be are 22], that may [18, or conspecifics skills hunt hunting related to ing track- helps unable faster distracting, be tactics, for complex may be- chasing allows more from and 20], and carcass [19, ing [21] the predators location prevent other For helps food large by 18], capturing stolen therein). 17, of ing references 10, probability and [6, the [1] prey increases Ref. it discussed see groups widely example, review, been in has a Hunting and (for benefits 12], [16]. several coyotes others [11, bring 14], among may Brazil [13, [15], of birds badgers honeyguide south and and the men hunter in between honey example, dolphins for and Inter- well, [1]. as fishermen species [7], exist other crocodiles collaborations several and [6], species [10], hawks ants pair [5]), 9], [8, Tsavo the spiders of (also lions [2–4] man-eater lions coordinated of include exhibit collab- hunting that a collaborative animals called or of also comple- Examples is and [1]. it oration different involved, are When tem- behaviors and mentary a actions. spatially by similar correlated way employing coordinated porally prey predators cooperative, them, of a Among group in rate. attacked success be their may increase to utilize Introduction 1. Biol. Pop. c [Theor. cooperative, model a Lett-Auger-Gaillard and the in conditions extend several prey here on We grouped depends species or each to isolated advantageous attack may Predators Abstract Cazaubiel), rpitsbitdt Elsevier to submitted Preprint [email protected] hr r yido oaigsrtge htpredators that strategies foraging of myriad a are There ∗ mi addresses: 33086478 Email 51 +55 author, Corresponding olciesrtge n ylcdmnnei smercp asymmetric in dominance cyclic and strategies Collective [email protected] b a ylcdmnne okPprSisr ae olciehnig fl hunting, collective game, Rock-Paper-Scissors dominance, Cyclic nttt eFıia nvriaeFdrld i rnedo Grande Rio do Federal F´ısica, Universidade de Instituto cl oml u´rer,ItrainlCne fFundam of Center Sup´erieure, International Normale Ecole ´ [email protected] Jfro .Arenzon) J. (Jeferson net Cazaubiel Annette Aesnr .L¨utz), F. (Alessandra (Annette a,b lsadaF L¨utz F. Alessandra , games rd-ff[0 1.Pe n rdtr eemdldby mean-field modeled structure), spatial were (no predators mixed and fully Prey a mouths” assuming many 31]. “many-eyes, [30, others, the with trade-off as prey the to re- share referred to for having suc- sometimes of of competition expense probability the the greater at a cess increasing have being of predators of while risk cost sources, the the lowers grouping at that preyed specifi- assumed More prey is strategy. both it grouping cally, account for a disadvantages into choosing and takes predators advantages and model the LAG of The some (see, [38]). coevolved strategy using Ref. collective however, populations a each or individual of an fractions assumed either the were only predators in and and model, prey constant LAG of the abundance as the to which referred hereafter model, oretical inhsbe eiae omdlcodnt utn [36]. Lett hunting ago, coordinate decade model a to Over dedicated been has tion [35] (but well). benefits and 33] as [34] [32, costs groups involve may larger to sharing in deci- information addition improved collective be In predators, can and making 31]. prey sion [30, both members for resources factors, all the be those by and may shared individual prey be an of should than group spotted take a easily es- may Conversely, more and group intimidating the techniques. [29], and caping distracting 28] group of [27, probability of smaller advantage The have is [24–26]. others caught themselves while being feed individuals to when several time efficient by de- more more parallel is may Surveillance in may that tactics done [23]. territory Collective prey smaller etc. benefit prey, a also of to availability the food crease for search the aeter saueu olt elwt uhaproblem. a such with deal to tool useful a is theory game ept onigeprmna eut,mc esatten- less much results, experimental mounting Despite u,C 55,95190ProAer S Brazil RS, Alegre Porto 91501-970 15051, CP Sul, na hsc,4 u ’l,705Prs France Paris, 75005 d’Ulm, Rue 45 Physics, ental gnzto,w loso htteceitnepaein phase coexistence the that show also we rganization, ssal eas fa ffcie ylcdominance cyclic effective, an of because stable is t 65 b letv a.Wehrageaiu eairis behavior gregarious a Whether way. ollective eesnJ Arenzon J. Jeferson , csosgm ihfu pce,afrhrexample further a species, four with game Scissors o h ovligdniiso rdtradprey and predator of densities coevolving the for 6 20) osailydsrbtdpopulations distributed spatially to (2004)] 263 , ocking tal et eao-ryspatial redator-prey b, 3]itoue aethe- game a introduced [37] ∗ etme 2 2018 22, September approach, and the temporal evolution of both densities Definition and default value being described by replicator equations [39]. p probability of a predator in a group capturing a lone A complementary approach, based on a less coarse prey (0.5) grained description, explicitly considers the spatial distri- G gain per captured prey per unit of time (1) n number of predators in a group (3) bution of individuals and groups. The local interactions e number of prey captured by a group of predators (2) between them introduce correlations that may translate α preying efficiency reduction due to grouped prey into spatial organization favoring either grouping or iso- β preying efficiency reduction when hunting alone lated strategies, raising a number of questions. For in- γ reduction of prey resources due to aggregation (1) stance, do these strategies coexist within predators or prey F gain for isolated prey per unit of time (1) populations? If yes, is this coexistence asymptotically sta- ble? How does the existence of a local group induce or Table 1: Model parameters [37] along with the default value consid- prevent grouping behavior on neighboring individuals? Do ered here. gregarious individuals segregate, forming extended regions dominated by groups? In other words, how spatially het- erogeneous is the system? Does the replicator equation The first contribution comes from the interaction of these provide a good description for both the dynamics and the predators with the fraction y of prey that organize into asymptotic state? If not, when does it fail? If many strate- groups for defense. By better defending themselves, prey gies persist, which is the underlying mechanism that sus- reduce the hunting efficiency by a factor 0 ≤ α < 1; tains coexistence? We try to answer some of these ques- nonetheless, e prey are captured with probability p and tions with a version of the LAG model in which space is ex- the gain G per prey is shared among the n members in the plicitly taken into account through a square lattice whose group of predators. The second term is the gain when the sites represent a small sub-population. Each of the sites group attacks an isolated prey, whose density is 1 − y, and is large enough to contain only a single group of predators shares it among the n predators as well. When the remain- and prey at the same time. If any of these groups is ever ing 1 − x predators hunt solely, they are limited to a single disrupted, their members will resort to a solitary strategy, prey and an efficiency that is further reduced by a factor hunting or defending themselves alone. 0 ≤ β < 1, what is somehow compensated by not having The paper is organized as follows. We first review, in to share with others. This information is summarized in Section 2.1, the LAG model [37] and summarize the main the payoff matrix: results obtained with the replicator equation, and then eαpG/n pG/n describe, in Section 2.2, the agent based implementation A = . (1) αβpG βpG with local competition. The results obtained in the spatial   framework are presented in Sec. 3. Finally, we discuss our conclusions in Section 4. As isolated prey consume the available resources, the gain per unit time is, on average, F . Once aggregated, the 2. The Model resources are shared and the individual gain reduced by a factor 0 ≤ γ < 1. The fraction of prey that aggregates 2.1. Replicator Equations becomes less prone to be preyed on by a factor α. If the Lett et al [37] considered, within a game theoretical grouped prey are attacked by a group of predators, e prey framework, grouping strategies for prey and predators. are captured and Lett et al [37] considered that the payoff Both can choose between single and collective behavior coefficient is 1 − eαp (imposing eαp ≤ 1). On the other and each choice involves gains and losses for the individ- hand, a lone predator has its efficiency reduced by a factor uals, as discussed in the introduction. The relevant pa- β, thus the surviving probability is 1−βp or 1−αβp for an rameters of the model are defined in Table 1. The size of individual or a group of prey, respectively. The payoff for both populations is kept constant during the evolution of the fraction y of prey that remain grouped is then written the system; only the proportion of cooperative predators, as x(t), and the fraction of gregarious prey, y(t), evolve in Py = (1 − eαp)γF x + (1 − αβp)γF (1 − x). time (see, however, Ref. [38] for a version that also consid- ers population dynamics). Variations depend on how the A similar consideration can be done for isolated prey [37], subpopulation’s payoff compares with the average payoff whose payoff matrix is of the respective population. If collective behavior leads to a larger payoff than the average, the associated density (1 − eαp)γF (1 − αβp)γF B = . (2) increases, otherwise it decreases. This dynamics is then (1 − p)F (1 − βp)F described by the replicator equations [39].   For the fraction x of predators hunting collectively, the It is the difference between the payoff P and its average, payoff is [37] P , that drives the evolution of both x and y. Indeed, the eαpG pG replicator equations,x ˙ = x(Px − Px) andy ˙ = y(Py − Py), P = y + (1 − y). x n n which give the rate at which these two densities evolve in 2 −1 time, are [37] characteristic time is τ = α21 and as β → e/n, τ diverges as τ ∼ (βn − e)−1. Different transitions may depend on x˙ y y = (1 0)A − (x 1 − x)A other coefficients, Eqs. (5), but the exponent of τ at the x 1 − y 1 − y     (3) transition is always 1. y˙ x x = (1 0)B − (y 1 − y)B . y 1 − x 1 − x 1     Mean Field These equations describe an asymmetric game and can 0.8 01 be rewritten as [39]:

x˙ = x(1 − x)[α12(1 − y) − α21y] (4) 0.6 y˙ = y(1 − y)[β12(1 − x) − β21x], COEXISTENCE β where 0.4 11 α12 = −p(β − 1/n)G 0.2 α21 = αp(β − e/n)G 10 PSfrag replacements(5) β = [γ − 1+ βp(1 − αγ)]F 12 0 β21 = [1 − γ − p(1 − eαγ)]F. 0 0.2 0.4 0.6 0.8 1 Eqs. (4) have five fixed points: the vertices of the unit α square, where x(1−x)= y(1−y) = 0, and the coexistence Figure 1: Mean-field phase diagram obtained in Ref. [37], showing state: the 11, 01, 10 and coexistence phases. The vertical transition line is at α = 1/2 for 0 ≤ β ≤ 2/3 while there is a horizontal line at ∗ β x = 12 β = 2/3, ∀α, and another one at β = 1/3 for α ≥ 1/2. Notice that β12 + β21 (6) a coexistence phase replaces phase 00. ∗ α y = 12 . α12 + α21 Besides presenting general results, Lett et al [37] also We will use the notation 01, 11, 10 and 00 either for the discussed the particular case when there is no reduction asymptotic state of the whole system, i.e., x∞y∞ with in resource intake by the prey when they are grouped x∞ ≡ x(t → ∞) and y∞ ≡ y(t → ∞) or, in the spatial (γ = 1). In this case, their behavior is a response to the version to be discussed in Section 2.2, for the site vari- capture rate alone. Considering the values listed in Ta- ble 1 for the several model parameters (F = G = γ = 1, able describing the combined state of site i, xiyi. The p = 1/2, n = 3 and e = 2), we see that while β12 is al- asymptotic state is determined only by the signs of α12, ways positive, β21, α12 and α21 change signs at 2α = 1, α21, β12 and β21, as discussed in Refs. [37, 39] and sum- 3β = 1 and 3β = 2, respectively. These changes in sign marized here. Indeed, if α12α21 < 0 or β12β21 < 0, the densities of grouped predators x and grouped prey y will lead to different asymptotic behaviors (phases) and locate monotonically converge to an absorbent state in which at the transition lines between them, as can be seen in the least one of the populations is grouped, i.e, 01, 10 or 11. mean-field phase diagram of Fig. 1. As expected, prey are grouped when α is small, whatever the value of β. Simi- On the other hand, if α12α21 > 0 and β12β21 > 0, there are two different possibilities. From the linear stability larly, small values of β lead to cooperating predators for ∗ ∗ all values of α. Remarkably, instead of a 00 phase in which analysis [37, 39], (x ,y ) is a saddle point if α12β12 > 0, and the system ends up in one of the vertices of the unit neither species form groups, there is a coexistence phase where the densities of grouped animals oscillate in time square (for the choice of the parameters considered here, ∗ ∗ see below, this does not occur). On the other hand, if along closed orbits around the center point (x ,y ) given by Eq. (6). This occurs for 2α> 1 and intermediate values α12β12 < 0, the eigenvalues are imaginary and the system evolves along closed orbits around the center (x∗,y∗). In 1 < 3β < 2, where the eigenvalues of the Jacobian associ- other words, when this last condition is obeyed, based on ated with Eqs. (4) become purely imaginary (α12β12 < 0, a linear stability analysis, both strategies, grouped or not, as discussed in Refs. [37, 39]). coexist at all times with oscillating fractions of the popula- tion and their time averaged values, once in the stationary 2.2. Spatially Distributed Population regime, correspond to Eqs. (6). The above replicator equations predict that both x and The above description of the competition between col- y approach the asymptotic state 1 or 0 exponentially fast lective and individual strategies for both predators and (except in the coexistence phase). Consider, for instance, prey does not take into account possible spatial correla- phase 11. As discussed below, y attains the asymptotic tions and geometrical effects. Space is usually introduced state much faster than x. Then, taking y = 1 and expand- by considering an agent based model in which individu- ing for small 1 − x, we get x(t) ∝ 1 − exp(−α21t). The als are placed on a lattice or distributed on a continuous 3 region. We consider the former case with a unit cell cor- 1 responding to the size of the territory of the smallest vi- 0.8 β < 2/3 able group. Since both predators and prey coexist in each

) 0.6 t site, there are two variables (xi,yi), i = 1,...,N (where (

x 0.4 N = L2 is the total number of sites and L is the linear 0.2 length of the square lattice) that take only the values 1 or β > 2/3 1 0; the former when agents are grouped, the latter for inde- 1000 pendent individuals. Thus, the global quantities x(t) and 0.8

y(t) now correspond to the fraction of sites having xi =1 τ 100

) 0.6 and yi = 1, respectively. t ( 10 Differently from the mean-field description where pay- y 0.4 0.001 0.01 0.1 off was gained from interactions with all individuals in the 0.2 PSfrag replacements |β − 2/3| system, in the lattice version, interactions are local and 0 occur only between nearest neighbor sites (self-interaction 0 1 2 3 is also considered since each site has both predators and log t prey). At each step of the simulation, one site (i) and one of its neighbors (j) are randomly chosen. The preda- Figure 2: Behavior of both the fraction of predators hunting in tors (prey) on i interact with the prey (predators) both groups, x(t), and prey defending themselves collectively, y(t), as a on i and in the four nearest neighboring sites, accumulat- function of time on a single run with α = 0.2, L = 100 and several (i) (i) values of β. The transition from collectively hunting predators to ing the payoff Px (Py ). At the same time, both groups single, individual hunters occurs at β = e/n = 2/3. Notice that in j accumulate their payoffs as well. If more efficient, most of the prey are in the aggregated state at all times, with y(t) the strategies of site j are adopted with a probability pro- monotonically increasing from y(0) = 0.5 to y(∞) = 1. Predators, on the other hand, have a richer behavior (see text). As β → 2/3, portional to the difference of payoffs. For predators (and from both sides, a plateau at x ≃ 0.26 develops (the dashed, horizon- analogously for prey), this probability is tal line is only a guide to the eyes). Inset: Power-law behavior of the characteristic time τ such that |x − x∞| < 0.2 around the transition (j) (i) at β = 2/3. The straight lines have exponent 1, as predicted by the Px − Px replicator equation, although the coefficients differ by one order of Prob(xi ← xj ) = max max , 0 , (7) − " Px # magnitude. The top curve is for β → 2/3 while the bottom one is for β → 2/3+. max where Px is the maximum value of the accumulated pay- off of the predators for the chosen parameters. This rule is known to recover the replicator equation when going from initial state with x(0) = y(0) = 1/2. In this case, for the microscopic, agent based scale to the macroscopic, all values of β, prey remain mostly grouped at all times coarse grained level [40]. After N such attempts, time since y(t) > 0.5 (bottom panel). As predicted in mean- is updated by one unit (one Monte Carlo step, MCS). In field, there is a transition at β = 2/3 where predators Ref. [41], where some preliminary results were presented, change strategy (top panel): for β > 2/3, hunting alone we considered a slightly different dynamics in which the becomes more efficient and x∞ = 0. On the other hand, two chosen neighbors compare their payoff and that earn- when the cost of sharing the prey is compensated by more ing the smallest one adopted the strategy of the winner efficient preying, β < 2/3, we have x∞ = 1. Interestingly, (with the above probability). The difference is that, in for the initial state chosen here, the behavior of x(t) is not Ref. [41], either i or j would change strategy, while here, monotonic when (1+2α)/3(1+α) <β< 2/3: x(t) initially only i may be updated. decreases,x ˙(0) < 0, attaining a minimum value and then Notice that because the level of spatial description is still resumes the increase towards x∞ = 1. The location of this minimum corresponds to the time at which y crosses larger than the individual scale, as a first approximation ∗ we take the limit of high population viscosity, neglecting the point y , and the envelope of all minima follows the the motion of individuals and groups across different sites. plateau developed for β > 2/3 as this value is approached In this case, strategies only propagate by the limited dis- from above. In this latter region, the behavior follows persal introduced by the above reproduction (or imitation) a two-step curve: there is a first, fast approach to the rule. We further discuss this point in the conclusions. We plateau followed by the departure from it at a much longer now present our results for the spatial version of the LAG timescale. In this model, the fast relaxations occur as prey model and, when possible, compare with the mean-field organize themselves into groups (increasing y) while the predictions. later slow relaxation is a property of the predators alone and is caused by the orbit passing near an unstable fixed point, as can be understood from Eqs. (4). For values of 3. Results α in the interval (1 + 2α)/3(1 + α) <β < 2/3, if y < y∗, x decreases until y crosses the line at y∗, whose value Fig. 2 shows the temporal evolution of both x(t) and depends on both α and β. At this point, the coefficient y(t) for α = 0.2, several values of 0 <β < 1 and an ofx ˙ is zero and there is a minimum. Once y >y∗, x 4 resumes the increase and approaches the asymptotic state 1 L=100 − ∗ 200 x∞ = 1 exponentially fast. As β → 2/3 , y → 1 and 400 the minimum crosses over to an inflection point. Indeed, 800 ∞ 0.5 1600 for β > 2/3, x(t) decreases towards 0 after it crosses the x plateau. This behavior is seen both in simulations and in mean-field. Associated with the late exponential regime there is 1 a characteristic time that diverges as a phase transition −1 is approached. For example, in mean-field, τ ∼ α21 ∼ −1 (βn − e) for the 11-01 transition. In the simulations, as ∞

y 0.5 x approaches the limiting value x∞, τ is estimated as the time beyond which |x−x∞| <ǫ, where 0 <ǫ< 1 is chosen, for convenience, to be ǫ =0.2. The exponent measured in 0 the simulations is in agreement with the mean-field pre- 0.4 0.6 0.8 1 diction, as can be seen in the inset of Fig. 2. α 1 PSfrag replacements β < 1/3 Figure 4: Average asymptotic value of the fraction of predators hunt- 1/3 <β< 2/3 ing in group, x(t) (top), and prey defending themselves in group, y(t) β > 2/3 (bottom), for β = 0.4, as a function of α. The solid lines correspond to the fixed points, Eqs. (6), predicted in mean-field and discussed

) in Sec. 2.1. Notice that the transition from the 11 phase to the coex-

t 0.5 ( istence one is continuous for both quantities and occurs at a smaller x value of α than predicted in mean-field, αc ≃ 0.45. At a larger, size dependent value of α there is a transition to the 01 phase. As the system size L increases, the coexistence phase becomes broader. 1 Averages are over 10 samples.

MF PSfrag replacements discontinuous transition, respectively, at αc =0.5, when )

t 0.5 predators and prey are spatially distributed, the simula- (

y tion shows, instead, continuous transitions for both quan- tities at a smaller value, αc ≃ 0.45. This is not due to finite size effects since the curves, for the broad range of 0 0 1 2 3 system sizes considered here, collapse onto a single curve log t in this region, and the larger the system is, the wider the collapsed region becomes. The simulation results on the Figure 3: The same as in Fig. 2 but for α = 0.8. Two transitions are lattice disagree with the mean-field predictions both quan- present at β = 1/n = 1/3 and e/n = 2/3, with a coexistence phase titative and qualitatively. In the spatial model, instead of in between where both x∞ and y∞ attain a plateau. a monotonic decay, y∞ has a minimum at α ≃ 0.59 for β = 0.4, even in the limit of very large systems. The ex- Differently from the α = 0.2 case, for α = 0.8 there istence of this minimum is remarkable because one would is, in agreement with the replicator equation, a coexis- expect that as α increases, predators become more efficient tence phase where both strategies may persistently coex- against grouped prey and the fraction of the latter would ist. Fig. 3 shows that, depending on α and β, there is an decrease. Moreover, both x∞ and y∞ present strong fi- initial, transient regime in which both x and y oscillate, nite size effects in this phase. Above a size dependent getting very close to the absorbing states. Because of the value of α, x is absorbed onto the group disrupted state stochastic nature of the finite system, it may eventually (x∞ = 0) and, immediately after, y∞ evolves toward 1. end in one of those absorbing states during the oscillating Fig. 5 also shows the behavior of x∞ and y∞, but for a regime (although the required time may be exponentially fixed value of α. In the interval 1/3 <β< 2/3, coming large, see later discussion), otherwise the amplitude of the from the 10 phase, the system first goes through the 00 oscillations decreases and a mixed fixed point is attained. and 01 phases (the former is not present in the mean-field A natural question is how close this fixed point is to the case), before crossing the wider coexistence region. Then mean-field prediction, Eqs. (4). Since it is the intermediate it crosses region 11 and eventually arrives again at the region in which coexistence may occur that presents new, 01 phase when β = 2/3. All these behaviors and transi- non trivial behavior, we now discuss it in more detail. tions are summarized in the phase diagram of Fig. 6, for The mean-field monotonic behavior can be observed in L = 100. The coexistence region, that is reentrant and ex- the solid lines of Fig. 4 as the system enters the coexistence ists also for α< 1/2, increases with the system size (as can region at a fixed β =0.4 from the 11 phase. While mean- be observed in Figs. 4 and 5). The small regions shrink for field predicts that x∞ and y∞ present a continuous and increasing L, but the convergence to the L → ∞ behavior 5 1

∞ 0.5 x

00 10 01 11 1 (a) (b) (c) ∞

y 0.5 L=100 Figure 7: Snapshots using a color code for the combined xiyi strategy 200 for β = 0.5 and α = 0.6 (left), 0.7 (middle) and 0.8 (right), all inside 400 the coexistence phase, at t = 213 MCS and on an L = 200 lattice. 800 0 Notice that the strategies organize in intertwined domains, whose 0.3 0.4 0.5 0.6 0.7 characteristic size depends on α and β. In this case, both 00 and 01 increases as the system approaches the border with the 01 phase. β PSfrag replacements Figure 5: The same as Fig. 4 but with α = 0.8 and β as the variable. Averages are over 10-20 samples, depending on the size. these four strategies organize into domains whose sizes de- pend on both α and β. While the coexistence persists, the four strategies survive by spatially organizing them- (e.g., whether 00 and 10 phases completely disappear or selves in nested domains whose borders move, invading not), is very slow. A more detailed analysis can be done other domains in a cyclic way. In Fig. 8a we show the (see Refs. [42–44] and references therein) by measuring the direction of these invasions by placing, in the initial state time it takes for the coexistence state to be absorbed into (t = 0, top row), one strategy inside, and another outside one of the homogeneous states (what eventually will occur a circular patch (a flat interface would do as well). The for a finite system because of the stochastic nature of the bottom row shows the corresponding state after 40 Monte dynamics) and how it depends on L. Indeed, inside the Carlo steps. By swapping positions, the invasion direction coexistence region, this characteristic time increases ex- is reversed, indicating that it is not a simple curvature ponentially with the system size while it is much smaller driven dynamics but, instead, involves a domination rela- (logarithmically) for the other phases. tion. Taking into account the six combinations of Fig. 8a, the interaction graph shown in Fig. 8b summarizes how 0.7 the different strategies interact (this particular orientation 01 2d of the arrows may change for other points in the coexis- tence phase [41]). Similar cyclic dominance behavior has 0.6 11 10 been extensively studied in predator-prey models with in- teraction graphs with three (Rock-Paper-Scissors) or more COEXISTENCE species [40, 45] with intransitive (sub-)loops. Indeed, the 0.5

β topology of the interaction graph in Fig. 8b is closely re- 01 lated to the one in Ref. [44]. The invasion is either direct as in the first four columns (or, equivalently, along the 0.4 perimeter of the interaction graph) or, as seen in the last two columns, involves the creation of an intermediate do- PSfrag replacements 00 10 main. For example, a patch of 00s invades 11 (fifth column 0.3 0.4 0.6 0.8 1 in Fig. 8a) by first disrupting the prey organization. The strategy 10 thus created, invades 11 and, in turn, is in- α vaded by 00. In those cases that the invasion proceeds through an intermediate domain (strategies along the di- Figure 6: Zoomed in phase diagram, obtained with L = 100, showing the rich behavior when agents are spatially distributed (compare with agonal of Fig. 8b are not neutral), we use a dashed arrow the pure coexistence phase existing in the region 1/2 ≤ α ≤ 1 and in the interaction graph. This cyclic dominance among the 1/3 ≤ β ≤ 2/3 in the mean-field diagram, Fig. 1). As L increases, combined strategies of prey and predators is the mecha- the coexistence region gets bigger while the phases around it shrink nism underlying the persistence of the coexistence state in (in particular the small ones like 10 and 00). Whether they disappear or remain very small when L →∞ is not clear. this region of the phase diagram. While the solid lines shown in the interaction graph of In the coexistence phase, any of the four combinations Fig. 8b are valid under broader conditions, the diagonal, of lone and collective strategies for both predators and dashed ones are not. For example, if only 00 and 10 sites prey may be present at each site (xiyi): 00, 01, 10 and 11. are present (Fig. 8a, first column), prey will never change Fig. 7 shows, for β = 0.5 and different values of α, how to a collective strategy (11 and 01 sites will not appear). 6 10 11 01 00 11 01 00 10 11 01 00 10 00 10

01 11

(a) (b)

Figure 8: a) Strategy dominance for β = 0.4 and α = 0.7 starting from an initial, circular patch (top row) and evolving for 40 MCS (bottom row). In some cases, denoted by the solid arrows in the graph at the right, the initial patch increases in size. For the interactions along the diagonal (dashed arrows), the invasion occurs in two steps. For example (fifth column), 00 first disrupts the aggregated prey, forming an intermediate 10 cluster, and then grows into it. Notice that for the 10 invading 01 (last column), besides the strategy 11 that intermediates the invasion, there are some groups of 00 growing inside the 10 patch (at the interface 01-10, all combinations may be created and, sometimes, migrate toward the interior of the circle). b) Interaction graph showing the direction of the invasion front for each of the four strategies.

Only predators may change strategy in this case and be- increase (remember that 01, as shown in Fig. 8b, prey on cause they earn their payoffs from isolated prey, this is easy 00). There is, for a given size L, a value of α where those to evaluate and does not depend on the specific geometry domains are comparable to L and it seems that this finite of either the lattice or the initial configuration. Indeed, for size effect triggers the transition, originating the strong a 00 site comparing its payoff with a 10 neighbor, while the dependence of its location on L. If we move toward the former earns 5β/2, the latter receives 5/6. The 00 strat- other phases, a similar behavior occurs, with the dominant egy will then invade the 10 population if β > 1/3. If we strategy of the approaching phase increasing in density. repeat this analysis for the first four columns of Fig. 8a, The important point is that, as the system increases, the we get: 10 invades 11 if α> 1/2, 11 invades 01 if β < 2/3 coexistence phase becomes larger and the other phases de- and, finally, 01 invades 00 if α < 1. These are the inter- crease in size. Whether this trend eventually leads to a actions around the graph of Fig. 8b, represented by solid coexistence phase occupying the whole region is an open lines. Thus, as expected, by changing the values of α and question, beyond our current computational capabilities. β, the arrows may change orientation as well. Indeed, the possible combinations of these arrows agree with the re- 4. Conclusions and discussion gions shown in the mean-field phase diagram of Fig. 1. In particular, the one with α > 1/2 and 1/3 <β< 2/3 is where the four arrows are clockwise oriented, dominance The foraging behavior of predators and the correspond- is cyclic and coexistence may ensue. Diagonal interactions ing defensive response from prey is fundamental to under- are much more complex. All four states may now be cre- stand how small, spatially distributed animal communities ated from initial states with only 00 and 11 (or 10 and 01) organize and eventually engage in more complex forms of and the earned payoffs depend on the number of neighbors sociability [46–48]. It thus becomes important, when mod- having each strategy, what changes from site to site and eling such behavior, to go beyond mean-field where a fully evolves in time, leading to the rich phase diagram inside mixed, infinite size system is considered and the spatial the coexistence region, shown in Fig. 6. Indeed, it is im- structure, with the correlations it implies, is missing. We portant to emphasize that the invasion graph in Fig. 8b considered here a finite dimensional stochastic version of was obtained for the particular values β =0.4 and α =0.7 the model introduced by Lett et al [37] in which short within that time window. Other points inside the coexis- range interactions between predators and prey are taken tence phase may show similar behavior albeit with some into account. In this way, local spatial correlations that arrows reversed and strategies switching roles while keep- change, to some extent, the foraging behavior predicted ing some of the sub-loops intransitive and the coexistence by the replicator (mean-field) equation, and the accompa- stable [41]. An intriguing characteristic of the phase dia- nying new dynamical behavior are introduced. The game gram, Fig. 6, is that the coexistence phase is surrounded theoretical framework in this model considers two strate- by several smaller phases, absent in that region within gies, collective or individual, for both hunting predators mean-field. This behavior can be better understood ob- and defensive prey. The advantages and disadvantages serving the patterns in Fig. 7. In that case, as α increases of each option are modeled by a set of independent pa- and β is kept fixed, i.e., one approaches the frontier with rameters from which we considered variations in only two: the 01 phase, the densities of both 00 and 01 strategies how the probability p of a group of predators capturing a lone prey is reduced when prey are grouped (αp) or when 7 the predators hunt alone (βp). We then study in detail further check how robust the results are when also chang- this particular case, discussed only within mean-field in ing the updating rule, replacing Eq. (7) by, for example, Ref. [37], in the limit of high population viscosity in which the Fermi rule (see Ref. [40] for a review on the several strategy dispersion is a slow process, solely occurring due possible dynamics). to the newborn limited dispersal driven by the pairwise Prey and predators usually engage in somewhat coordi- updating rule between nearest neighbors patches. nated chase and escape interactions [49–54] that also allow The combined strategies of both predators and prey them to explore and profit from neighboring patches. It present four possible states in our binary version. There is thus interesting to check whether, and to what extent, are three phases with either prey or predators (or both) be- the properties of the model, in particular in the coexis- having collectively (10, 01 and 11), while in the 00 phase tence region, change in the presence of mobile individu- there are no groups. Results from extensive simulations als [43]. Chasing and escaping behaviors may be quite on the square lattice confirm part of the mean-field phase complex depending on the physical and cognitive capabil- diagram, with a richer structure around the coexistence ities of each individual and involve space and time corre- phase. More importantly, our results unveil the under- lations between their displacements and changes in veloc- lying mechanism for the coexistence of these strategies. ity [55]. Simple movement rules, if not completely ran- Specifically, in this phase, this model is an example of an dom, are likely to generate repeatable (and, because of asymmetric game presenting cyclic dominance between the that, exploitable) patterns of behavior, while those involv- above four combined strategies. Starting, say, with a pop- ing higher levels of variability and complexity somehow ulation of collective predators preying on lone individuals involve more advanced cognitive skills. By studying such (i.e., most of the sites are 10 while the other strategies, mobility patterns, one could get a better understanding albeit present, have small densities), free riders have the of how important those cognitive abilities are in defining advantage of not having to share their prey with the other hunting strategies [1]. Moreover, these patterns are also members of the group and the 00 strategy invades 10. At relevant for the demographic distribution of both preda- this point, if prey organizes into groups they may better tors and prey since the shuffling of strategies, depending defend themselves, and thus 01 replaces 00. Finally, the on how random the mobility is [56], may decrease the spa- lone predators get better off preying collectively and 11 tial correlation and destroy local structures, changing the dominates 01. Finally, 10 invades 11 because it is better spatial organization of prey and predators [47]. for predators to go after grouped prey, since the number Several other extensions of this model are possible. For of captured prey (e) compensates for the loss of efficiency simplicity, we assumed that the size of a group is constant (α). Thus, it is better for aggregated prey to stay alone. and homogeneous throughout the population. However, The strategies 11, 01, 11 and 00 form spatially extended it can also be considered as a dynamical parameter coe- domains whose borders are not static and move accord- volving together with the collectivist trait. It is possible ingly with the interaction graph (e.g., Fig. 8b). to further explore the possibilities offered by the spatial Besides considering simple models for this less explored setup, for example, having heterogeneous parameters de- foraging variability, our results emphasize the importance pending on the landscape (due to an uneven distribution of studying both the asymptotic states and the dynam- of resources or to the variability of the species), what may ics towards them in a finite size population. It is not induce an optimal, intermediate level of collective forag- possible to disentangle the strong finite size effects ob- ing corresponding to mixed strategies [57–59], sometimes served in the asymptotic state of the stochastic model hunting in group, sometimes alone. Another important from the dynamical evolution since it is the very exis- question is how the size of the hunting groups respond to tence of orbits that, by closely approaching the absorbing an increase or to stochastic fluctuations in the size of a states, makes the system prone to be captured by them swarm of prey (and vice-versa). Although we focus here as a consequence of fluctuations. Since actual populations on the discrete, binary situation, it is also possible to have are far from the thermodynamical limit (infinite popula- larger patches with enough individuals to form more than tion), the results obtained for intermediate sizes become one group, such that the variables describing the local pop- relevant. Timescales may be very large and the transient ulations may have multiple allowed values or even become coexistence may extend to times much larger than those continuous. In addition, the dynamics considered here al- that are relevant in practice. Moreover, the differences lows neither a species with a smaller payoff to be the win- between the mean-field and the spatial version show the ner nor the reintroduction of an already extinct species. importance of trying several complementary approaches Removing such constraints (what can be considered a kind even when studying quite simplified models. In our simu- of external noise) may be an effective way of taking into lations, the microscopic updating rule, Eq. (7), is applied account some of the missing ingredients of the model. It to one site i chosen at random, i.e., both xi and yi may be would thus be interesting to see how robust the results are updated in each single step. We checked that the results in the presence of such noise. Finally, the short-range in- are essentially the same when they are independently cho- teractions present in the model are not able to synchronize sen or, as in Ref. 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