Physics Procedia 00 (2009) 000–000 www.elsevier.com/locate/procedia

Pattern formation in spatial games Luo-Luo Jiang 1,2 , Wen-Xu Wang 2 , and Bing-Hong Wang 1,* 1

Department of Modern Physics, University of Scienceand Technology of China, Hefei 230026, China 2

Department of Electrical Engineering, Arizona State University, Tempe, Arizona 85287, USA

Elsevier use only: Received date here; revised date here; accepted date here

Abstract

Pattern formation is crucial for maintaining biodiversity and cooperation in nature and human society. We review recent works on the formation of spatial patterns in the framework of evolutionary games, involving rock-paper-scissors game and social dilemmas. The rock-paper-scissors game, as a basic paradigm for characterizing nonhierarchical cyclical competition in ecosystems, has been widely used for studying biodiversity sustained by species coexistence. Spiral waves and target waves have been observed in spatially dispersed population with mobility, which has been deemed play significant roles in species coexistence. For social dilemmas, the formation of cooperator clusters is a key factor for the survival of cooperators who can resist the invasion of defectors in the pattern of clusters. Several underlying mechanisms contributing to the formation of various spatial patterns will be reviewed in spatial games. © 2009 Published by Elsevier B.V. PACS: 87.23.Cc, 05.10.Ln, 02.50.Le, 87.23.Ge Keywords: Spatial game, diversity, cooperation, spirals waves, target waves

* Corresponding author. Tel.: +86-551-3607407. E-mail address: [email protected].

Author name / Physics Procedia 00 (2009) 000–000

1. Introduction Biodiversity and cooperation are fundamental to ecologic and social systems. Understanding mechanisms that facilitate factors for biodiversity and cooperation is an important issue which is frequently addressed in the framework of the evolutionary game theory. Individuals compete with others to maximize its offspring in ecologic systems, while to obtain largest payoffs in social systems. Spatial patterns in spatially dispersed populations have been found to play significant role in the maintenances of species coexistence and cooperation. Species coexistence and relevant coevolutionary dynamics, as a key factor to maintain biodiversity, have motivated growing interests [1-7], ranging from theoretical biology to nonlinear physics. Recent experiments have demonstrated the importance of non-hierarchical, cyclic competitions to coexistence, such as in carcinogenic microbes [1], certain lizard populations [8], mutant strains of yeast [9] and coral reef invertebrates [10]. The coevolutionary dynamics driven by such competitions can be well captured by the rock-paper-scissors (RPS) game, particularly associated with spatial dispersal to stabilize coexistence [6]. Quite recently, population mobility, as a basic feature of ecosystems, has been incorporated into spatial cyclic competitions to better mimic individual activity [11]. A critical mobility has been identified, below which species can coexist in the pattern of entangled travelling spiral waves; when mobility exceeds the critical value, biodiversity is lost. Inspired by this model, some effort has been devoted to such coevolutionary dynamics, such as investigating noise and correlation [12, 13], instability of spatial pattern [14, 15], and conservation law for total density [16]. It is demonstrated spatial pattern formation greatly affect biodiversity. For social dilemmas, as one of the representative games, prisoner’s dilemma game (PDG) seizes the characteristics of the conflict between the selfish individuals and the collective interests. In PDG, when most of the individuals take the cooperation strategy, the collective interests is optimized, but as to an individual, if it defects when its opponents cooperate, it will profit much greater, and meanwhile its opponents will profit little or none. Thus, due to the selection pressure routed in the fundamental Darwinian assumption, more and more individuals will choose to defect, and as a result the level of cooperation will decrease. There are many mechanisms that can promote the cooperation of PDG such as repeated interaction [17], spatial extensions [18], reciprocity [19], and partly randomly contacts [20]. The heterogeneity of payoffs had also been found playing a crucial role in promoting cooperation in PDG [21]. The common characteristic is that the cooperation strategy spreads through cooperator clusters, which form spatial patterns.

2. Spiral and target waves in RPS games According to the previous works [11-15], PRS games involve three species, marking A, B, and C, and nodes of an L × L square lattice present mobile individuals belonging to one of these three species. Each node can either host one individual of a given species or it can be vacant. Vacant sites are also the so-called resource sites. Within the model three processes are possible, namely predation, reproduction and exchange, whereby these occur only between neighboring nodes. Predation: species A eliminates species B at a rate 1, whereby the node previously hosting species B becomes vacant. In the same manner, species B can eliminate species C, and species C can eliminate species A, thus forming a closed loop of dominance between them. Reproduction:

Author name / Physics Procedia 00 (2009) 000–000

individuals can place an offspring at a neighboring vacant node at a rate 1. Exchange: two individuals, including vacant sites, can exchange their position at a rate α , thus introducing the mobility of the participants. Therefore, the mobility can be defined: M = α /(2 L ) . 2

1.0 × 10−5

5.0 × 10−5

1.0 × 10−4

Mc

5.0 × 10−4

Fig. 1 Pattern formation for different mobility，L = 500. Biodiversity is promoted for mobility below the value

M c , and spirals emerge. With increasing M (from left to right), the spiral structures grow, and outgrow the system size at the critical mobility M c . Then Biodiversity disappears, in which uniform populations remain and which one species surviving is randomly determined.

(a)

(b)

(c)

Fig. 2 Spirals (up) and anti-spirals (bottom) for different mobility, (a)M= 1.0 × 10−5 , (b) M= 5.0 ×10−5 , and (c) M= 1.0 ×10−4 ，L = 500.

We performed extensive computer simulations of the stochastic system with different initial conditions. For random initial condition in which three species as well empty sites randomly distribute on the lattice, when the mobility of the individuals is low, diversity with all species coexisting and self-organized patterns of spirals are observed, as shown in Fig. 1. With increasing mobility M, the structures spirals grow in size, and disappear for M> M c , and the

M c ≈ 4.5 ×10−4 [11]. In the absence of spirals, the system adopts a uniform state where only one

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species survives, while the others have died out, and which species remains is determined in a random process. In case the so-called heterogeneous initial conditions are used the initial setup of the three species and empty sites are the following: three roundish areas with radius of 10.5 are occupied by the three species respectively. It is interesting to note that the different direction of initial location of species leads different revolving direction of spirals: clockwise location of species A, B, and C inducing clockwise revolving direction of spirals, contrariwise, anticlockwise location inducing anticlockwise direction, as shown in Fig. 2. Beside the different initial conditions, we also introduce a localized periodic current of the three competing species. In particular, the periodic current is applied over a small (compared to the overall system size) area located at the center of the spatial grid. The periodic current thus acts as a pacemaker on the population, trying to impose its rhythm on the spatiotemporal evolution of the three species. The current is defined as follows: at time t = 0 the nodes inside R=10.5, are populated by species A, at time t = T0 these nodes are populated by species B, at time t = 2T0 these nodes are populated by species C, at time t = 3T0 these nodes are again populated by species A, and continuing further in this manner. Importantly, during nT0 < t < mT0 , where m= n + 1, the evolution of species inside R is governed by the same Monte Carlo updating as outside. Moreover, from the definition of the current it follows that its period equals Tin = 3 T0 . In the following, we will consider the mobility M and the time T0 between successive replacements of a species inside R as the two crucial parameters effecting the emergence of target waves in the examined model. It is of interest to investigate the impact of other values of these two parameters on pattern formation as well. Fig. 3(a) shows the typical snapshots of the spatial grid obtained for different mobility M of individuals after a long simulation time. Evidently, low mobility fails to evoke −5

target patterns. In the leftmost panel of Fig. 3(a), where M = 10 , turbulent spirals dominate −4

over the entire lattice. For M = 10 , on the other hand, the locally introduced inhomogeneities due to the periodic current are appropriately enhanced to eventually result in the emergence of target waves, as depicted in the middle panel of Fig. 3 (a). Increasing the mobility further to

M = 10−3 evokes a transition from target to spiral waves, as depicted in the rightmost panel of Fig. 3 (a), which is due to the strong mixing of individuals prohibiting the stability of the relatively stationary target wave pattern and instead favoring the more dynamically evolving spiral waves. From the phase diagram presented in Fig. 3(b), it follows that the region of coherent target waves is relatively small, and that the phenomenon thus results from a rather subtle interplay between the localized periodic current and the overall dynamics of the three species.

Author name / Physics Procedia 00 (2009) 000–000

Fig. 3 Characteristic snapshots of the spatial grid for M = 0.00001 (left), M =0.0001 (middle) and M = 0.001 (right), obtained after long transients have been discarded. In all the three panels, yellow, red and blue squares denote species A, B and C, respectively, whereas the gray squares depict vacant sites. Employed parameter values are: T0 = 150 and L = 500 (Figures comes from Ref. [15]).

3. Cooperator cluster in PD games In classical PDG, an agent updates its strategy according to the following rule: the agent i plays PDG with its neighbors, then randomly selects a neighbor j, and adopts its strategy with probability Gi → j =

1 , where T characterizes the stochastic noise. For T=0, 1 + exp[( Pi − Pj ) / T ]

the individual always adopt the best strategy determinately, while irrational changes are allowed for T>0. In numerical simulation, noise level is often set as T=0.1 because a few irrational behavior is common in real economic systems. With the probability defined above, if the selected neighbor j obtains more payoff than node i, node i will adopt the neighbor’s strategy with larger probability, but if the neighbor j obtains less payoff than node i, the probability will be much smaller. It is worth noting that the parameter b has great effects on the cooperation process and with b’s increasing, more and more agents would be defectors. We present a scheme that regulates the total payoffs continuously and try to find the optimal regulation strength. In our regulation α

scheme, we define the regulated payoffs Wi = Pi , where

α is the regulation parameter which

determines the regulation strength and when α =1, our model degenerates into the classical PDG. Therefor, we replaced the payoffs Pi and Pj by the regulated payoffs Wi and W j , and we get generalized probability Gi → j =

1 . 1 + exp[(Wi − W j ) / T ]

Figure 4 (a) shows the cooperation fraction ρC as a function of b at different values of α . It displays that ρC decreases monotonically with the increasing of b, no matter what α is. Most interestingly, the cooperation is greatly affected by the parameter α for fixed b: in a large region of α , ρC

will be increased, indicating the reduction in heterogeneity of payoffs will improve

Author name / Physics Procedia 00 (2009) 000–000

the cooperation. It is worth noting that there is at least one optimal value of α , where ρC takes its maximum, larger, or smaller α will cause the decreasing of ρC . Thus, to quantify the effects of α on the promotion of cooperation for different b, we present the dependence of ρC on α in Fig. 4 (b). It is clearly seen that with α ’s decreasing from 1, ρC will increase prominently and at the point

α ≈ 0.5 ρC reaches its maximum, and after that the value ρC will decrease

until no cooperators in the system. It has been confirmed that intermediate α promotes cooperation, while too small or too large value of α has the reverse effects. In Fig. 5, three typical snapshots are displayed to show how the system will be when α takes small, intermediate, and larger values. Clearly, most of the cooperators are not distributed in isolation but form some clusters. When the payoffs are not regulated ( α =1.0) or regulated too much ( α =0.3), there are only a few cooperator clusters in the system; but when α ≈ 0.5 , there will emerge so many cooperator clusters that the cooperation is remarkably promoted.

0.6

(a)

α=0.3 α=0.4 α=0.5 α=0.8 α=1.0

ρC

0.4

0.2

0.0 1.00

(b)

b=1.005 b=1.01 b=1.015 b=1.02

0.4

ρC

0.6

0.2

1.01

1.02

1.03

1.04

1.05

0.0 1.06 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

α

b Fig. 4 Fraction of cooperation

ρC

as a function of b in (a) and

α

in (b). b is fixed at 1.01 in (a) (Figures comes

from Ref. [21]).

Fig. 5 For panels (a), (b), and (c), typical snapshots of the distribution of cooperators (light gray) and

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defectors (black) on a square 100 × 100 lattice obtained for a different value of

α by b =1.01. (a) α =0.3, (b)

α =0.5, and (c) α =1.0 (Figures comes from Ref. [21]).

4. Conclusion In conclusion, self-organized spatial-temporal patterns have been widely observed in spatially dispersed populations. A variety of spatial patterns play key roles in the maintenance of biodiversity and resolving the social dilemmas of profit versus cooperation. Pattern formation can have potential implications in diverse spatially extended systems, such as reaction-diffusion systems. Understanding the underlying mechanisms relevant to the formation of various spatially pattern is challenging yet, met by scientific communities from different contents. ————————— [1]B. Kerr, M. A. Riley, M. W. Feldman and B. J. M. Bohannan, Nature (London),418, 171 (2002). [2]T. L. Czara, R. F. Hoekstra, and L. Pagie, Proc. Natl. Acad. Sci. U.S.A., 99 , 786 (2002). [3]Y.-C. Lai and Y.-R. Liu, Phys. Rev. Lett. , 94, 038102 (2005). [4]A. Traulsen, J. C. Claussen and C. Hauert , Phys. Rev. Lett., 95, 238701 (2005). [5]J. C. Claussen and A. Traulsen, Phys. Rev. Lett., 100, 058104 (2008). [6] G. Szabo and G. Fath, Phys. Rep., 446, 97 (2007). [7]M. Berr, T. Reichenbach, M. Schottenloher and E. Frey, Phys. Rev. Lett., 102, 048102 (2009). [8]B. Sinervo and C. M. Lively, Nature (London), 380, 240 (1996). [9]C. E. Paquin and J. Adams, Nature (London), 306, 368 (1983). [10]J. B. C. Jackson and L. Buss, Proc. Natl. Acad. Sci. U.S.A., 72, 5160 (1975). [11]T. Reichenbach, M. Mobilia and E. Frey, Nature (London), 448, 1046, (2007). [12]T. Reichenbach, M. Mobilia and E. Frey, Phys. Rev. Lett., 99, 238105 (2007). [13]T. Reichenbach, M. Mobilia and E. Frey, J. Theor. Biol., 254, 368 (2008). [14]T. Reichenbach and E. Frey, Phys. Rev. Lett., 101, 058102 (2008). [15]Luo-Luo Jiang, Tao Zhou, Matjaz Perc, Xin Huang, and Bing-Hong Wang, New J. Phys., 11, 103001 (2009). [16]M. Peltomaki and M. Alava, Phys. Rev. E, 78, 031906 (2008). [17] R. Axelrod, The Evolution of Cooperation (Basic Books, New York, 1984). [18] M. A. Nowak and R. M. May, Nature (London) 359, 826 (1992). [19] M. A. Nowak and K. Sigmund, Nature (London) 437, 1291 (2005). [20] J. Ren, W.-X. Wang, and F. Qi, Phys. Rev. E 75, 045101 (2007). [21] Luo-Luo Jiang ,Ming Zhao, Han-Xin Yang, Joseph Wakeling, Bing-Hong Wang, and Tao Zhou, Phys. Rev. E, 80, 031144 (2009).