Spatial Evolutionary Games with Weak Selection

Spatial evolutionary games with weak selection

Mridu Nandaa,1 and Richard Durrettb,1,2

aNorth Carolina School of Science and Mathematics, Durham, NC 27705; and bMath Department, Duke University, Durham NC, 27708-0320

Contributed by Richard Durrett, April 24, 2017 (sent for review December 20, 2016; reviewed by David Basanta and Jacob Gardinier Scott) Recently, a rigorous mathematical theory has been developed for To motivate our study of evolutionary games, we introduce two spatial games with weak selection, i.e., when the payoff differ- examples that will be used to illustrate the theory that has been ences between strategies are small. The key to the analysis is that developed. when space and time are suitably rescaled, the spatial model converges to the solution of a partial differential equation (PDE). This Examples approach can be used to analyze all 2 × 2 games, but there are A Public Goods Game in Pancreatic Cancer. In this system (4), some a number of 3 × 3 games for which the behavior of the limit- cells (type 2’s) produce insulin-like growth factor-II, while other ing PDE is not known. In this paper, we give rules for determin- cells (type 1’s) free-ride on the growth factors produced by ing the behavior of a large class of 3 × 3 games and check their other cells. Since the 1’s do not have to spend metabolic energy validity using simulation. In words, the effect of space is equiv- producing the growth factor, they have a higher growth rate. This alent to making changes in the payoff matrix, and once this is leads to the following very simple 2 × 2 game. done, the behavior of the spatial game can be predicted from the behavior of the replicator equation for the modified game. We 1 2 say predicted here because in some cases the behavior of the spa- 1 0 λ. [2] tial game is different from that of the replicator equation for the 2 1 1 modified game. For example, if a rock–paper–scissors game has a replicator equation that spirals out to the boundary, space stabilizes the system and produces an equilibrium. In words, 2’s give birth at rate 1, independent of what is around them, while 1’s give birth at rate equal to λ times the fraction cancer modeling | public goods game | bone cancer | rock–paper–scissors of neighbors that are of type 1. If λ > 1 there is a mixed strategy equilibrium for the game ρ2 = 1/λ, ρ1 = 1 − 1/λ, which is the limit of the solution to the replicator equation when 0 < volutionary games are often studied by assuming that the u1 < 1. Epopulation is homogeneously mixing, i.e., each individual interacts equally with all the others. In this case, the frequen- Multiple Myeloma. Normal bone remodeling is a consequence cies of strategies evolve according to the replicator equation. See, of a dynamic balance between osteoclast (OC )-mediated bone e.g., Hofbauer and Sigmund’s book (1). If ui is the frequency of resorption and bone formation due to osteoblast (OB) activity. players using strategy i, then Multiple myeloma (MM ) cells disrupt this balance in two ways. dui = ui (Fi − F¯ ), [1] (i) MM cells produce a variety of cytokines that stimulate the dt growth of the OC population. P where Fi = j Gi,j uj is the fitness of strategy i, Gi,j is the pay- (ii) Secretion of DKK 1 by MM cells inhibits OB activity. i off for playing strategy against an opponent who plays strat- OC cells produce osteoblast-activating factors that stimulate j F¯ = P u F egy , and i i i is the average fitness. The homogeneous the growth of MM cells, whereas MM cells are not affected by mixing assumption is not satisfied for the evolutionary games that the presence of OB cells. These considerations led Dingli et al. arise in ecology or modeling solid cancer tumors, so it is impor- (5) to the following game matrix. Here, a, b, c, d, e > 0. tant to understand how spatial structure changes the outcome of games. The goal of this paper is to facilitate applications of spatial evolutionary games by giving rules to determine the limiting Significance behavior of a large class of 3 × 3 games. Our spatial games will take place on the 3D integer lattice Game theory, which was invented to study strategic and eco- 3 Z . The theory (2, 3) has been developed under the assump- nomic decisions of humans, has for many years been used in tion that the interactions between an individual and its neigh- ecology and more recently in cancer modeling. In the applica- bors are given by an irreducible probability kernel p(x) on tions to ecology and cancer, the system is not homogeneously 3 Z with p(0) = 0, that is finite range, symmetric p(x) = p(−x), mixing, so it is important to understand how space changes and has covariance matrix σ2I . Here we will restrict our atten- the outcome of evolutionary games. Here, we present rules tion to the nearest neighbor case, in which p(x) = 1/6 for that can be used to determine the behavior of a wide class x = (1, 0, 0), (−1, 0, 0),... (0, 0, −1). of three-strategy games. In short, the behavior can be pre- To describe the dynamics we let ξt (x) be the strategy used by dicted from the behavior of the replicator equation for a mod- the individual at x at time t and let ified equation. This theory will be useful for a number of applications. X ψt (x) = G(ξt (x), ξt (y))p(y − x) Author contributions: M.N. and R.D. designed research; M.N. performed research; and y R.D. wrote the paper. be the fitness of x at time t. In birth–death dynamics, site x gives Reviewers: D.B., H. Lee Moffitt Cancer Center and Research Institute; and J.G.S., birth at rate ψt (x) and sends its offspring to replace the individ- Cleveland Clinic. ual at y with probability p(y − x). In death–birth dynamics, the The authors declare no conflict of interest. individual at x dies at rate 1 and is replaced by a copy of the one 1M.N. and R.D. contributed equally to this work. at y with probability proportional to p(y − x)ψt (y). The theory 2To whom correspondence should be addressed. Email: [email protected]. developed in ref. 3 can be applied to both cases. However, to save This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. space we will only consider the birth–death case. 1073/pnas.1620852114/-/DCSupplemental.

6046–6051 | PNAS | June 6, 2017 | vol. 114 | no. 23 www.pnas.org/cgi/doi/10.1073/pnas.1620852114 Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 w al .Smlto eut o h ulcgosgame goods public the λ for results Simulation 1. 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APPLIED ECOLOGY MATHEMATICS 0.6 4. Rock–Paper–Scissors

Suppose that the βi > 0 and the αi < 0 in Eq. 6. In this situation, there is an interior ﬁxed point with all coordinates positive. Theo- 0.5 rem 7.7.2 in Hofbauer and Sigmund (1) describes the asymptotic behavior of the replicator equations for these games. 0.4 1 Theorem 2. Let ∆ = β1β2β3 + α1α2α3. If ∆ > 0, solutions con- 2 verge to the ﬁxed point (stable spiral). If ∆ < 0, their distance from 0.3 the boundary tends to 0 (unstable spiral). If ∆ = 0, there is a one- 3 parameter family of periodic orbits. 0.2 In ref. 3, the following result is proved, which covers some sit- uations with stable spirals.

0.1 Theorem 3. Suppose that the modified three-strategy game H has (i) zeros on the diagonal, (ii) an interior equilibrium ρ, and that H is almost constant sum: Hij + Hji = γ + ηij with γ > 0 and 0 maxi,j |ηi,j | < γ/2. Then there is coexistence and furthermore for 0 200 400 600 800 1000 any δ > 0 if ε < ε0(δ) and µ is any stationary distribution con- centrating on configurations with infinitely many 1’s, 2’s, and 3’s Fig. 2. Frequencies of the three strategies versus time in stable rock– we have paper–scissors game 1 + 0.2G2.

sup |µ(ξ(x) = i) − ρi | < δ. • If λ < 4/3, then b¯ < 0, so strategy 2 domiinates strategy 1, x 2 1, and 2’s take over the system. • If λ > 4, then c¯ < 0, so 1 2, and 1’s take over the system. In words, the equilibrium frequencies are close to those of the • If 4/3 < λ < 4, then ρ =c ¯/(b¯ +c ¯) ∈ (0, 1) is an attracting fixed replicator equation for the modified game. point and there is coexistence in the spatial model. Turning to simulation, we first consider the constant sum game (which is covered by Theorem 3). Note that in contrast to ref. 4, one does not need the growth rate of 1’s to be a nonlinear function of the fraction of 2’s to G2 1 2 3 H2 1 2 3 maintain coexistence. 1 0 4 −3 1 0 6.5 −7.5 . To check the theoretical prediction, we turn to simulation. 2 −1 0 5 2 −3.5 0 8.5 Simulations were done using a standard algorithm for simulating 3 6 −2 0 3 10.5 −5.5 0 continuous-time Markov chains. Details can be found in a 1994 survey paper by Durrett and Levin (16). The method is described Note that H2 is again constant sum. It has ∆ > 0, so the replica- in SI Appendix, Section 1. tor equation spirals in to the fixed point, and Theorem 3 implies Table 1 gives the equilibrium frequencies of strategy 1 for var- there is coexistence in the spatial game with weak selection. The ious values of λ and w. Note that the agreement with the limiting simulation of the spatial game 1 + 0.2G given in Fig. 2 confirms result is very good when w = 1/10 and good when w = 1/2. Fig. this. Note that the observed densities of the three strategies are 1 gives the final configuration of the simulation when λ = 3. Here close to the theoretical prediction from the replicator equation and in later figures, we will quantify the spatial structure by com- for H2. (0.31, 0.43, 0.25). puting the nearest-neighbor correlation For our second example, we consider a game G3 for which fij − fi fj the modified game H3 has ∆ < 0, and hence the solution to the ρij = , σi σj replicator equation spirals out to the boundary.

where fij is the fraction of nearest neighbor pairs with states i, p and j , fi is the frequency of type i and σi = fi (1 − fi ) is the SD. 3. Three-Strategy Games 0.8 We will suppose that the game is written in the form 0.7 1 2 3 1 0 α β 0.6 G = 3 2 . [6] 2 β3 0 α1 3 α2 β1 0 0.5 The subscripts indicate the strategy that has been left out in 1 0.4 the various 2 × 2 subgames. 2 2 × 2 3 In the case, there are only four possibilities: 1 dominates 0.3 2, 2 dominates 1, stable mixed strategy fixed point, and unstable mixed strategy fixed point. Bomze (15, 17) lists more than 40 pos- 0.2 sibilities for the 3 × 3 case. Here, we do not consider games with unstable edge fixed points and only consider generic examples 0.1 in which the six off-diagonal entries are nonzero and distinct, so we end up with 11 cases described in SI Appendix, Section S2. 0 In the next three sections, we consider games with “rock–paper– 0 200 400 600 800 1000 scissors” relationships between the strategies and two examples in which the replicator equation has two locally attracting fixed Fig. 3. Frequencies of the three strategies versus time in the unstable rock– points (bistability). paper–scissors game 1 + 0.2G3.

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APPLIED ECOLOGY MATHEMATICS 0.6 analyzed rigorously using the methods of ref. 3. The main con- tribution is to describe a procedure for determining the behavior of spatial three-strategy games with weak selection, when the 0.5 game matrix G has no unstable edge fixed points. One first forms the modified game Hij = (1 + θ)Gij − θGj ,i , where θ is a con- 0.4 stant that depends on the spatial structure, but not on the entries in the game matrix. θ ≈ 1/2 in the three-dimensional nearest- 1 neighbor case. The behavior of the spatial game with matrix G 0.3 2 can then be predicted from that of the replicator equation for H 3 . We say predicted because in some cases the behavior is not the same. 0.2 For three-strategy games without unstable edge fixed points, there are four major types: 0.1 1. When there are 1, 2, or 3 stable edge fixed points and they can all be invaded, there is coexistence in the spatial evo- 0 lutionary game when selection is small. This was proved in 0 500 1000 1500 2000 2500 ref. 3 2. As first observed by Durrett and Levin (20), when the replica- Fig. 6. Frequencies versus time in a 100×100×100 simulation of the weak tor equation is bistable, i.e., the limit depends on the start- selection multiple myeloma game 1 + (1/3)G4, with a = e = 2, d = 1, and ing point, the spatial game has a stronger equilibrium that b = c = 1.25. Note that the frequencies first get close to the unstable fixed is the limit for generic initial conditions. In two strategy point at (0.531, 0.306, 0.1633) and then start heading toward the boundary games, the victorious strategy is determined by the direc- equilibrium. tion of movement of the traveling wave solution of the PDE. For three-strategy games, we do not know how to prove the existence of such traveling waves or compute of ref. 3. The dramatic differences between the properties of their speeds, but simulations suggest that the same result the spatial game and the replicator equation cast doubt on the holds. proposed insights into therapy that emerge from the analysis 3. In the case of rock–paper–scissors games, there is coexistence of Dingli et al. (see the discussion that begins on page 1,134 when the replicator equation converges to the interior fixed of ref. 5). point. This was proved in ref. 3 when the game is “almost Here, our interest in this model is as an example with bista- constant sum.” It is somewhat surprising that when the repli- A, B, C , E > 0 DC /BE > F /A bility. Suppose that and , which cator equation trajectories that spiral out to the boundary, holds for the original game entries. Results from ref. 3, which are space exerts a stabilizing effect and the three strategies coex- SI Appendix described in , Section S4, show that there are three ist. This result has also been found recently by Ryser and cases: Murgas (10). 4. Last, and least interesting, is the situation in which the replica- Case 1. C /E > 1 − F /A. The replicator equation converges to tor equation converges to a boundary fixed point. Simulations the 1,3 equilibrium. show (SI Appendix, Section S5) that the same behavior occurs in the spatial evolutionary game. Case 2. 1−F /A > C /E > 1−DC /BE. There is an interior fixed point that is a saddle point, and the replicator equation exhibits bistability.

Case 3. 1 − DC /BE > C /E. The replicator equation converges to the 1,2 equilibrium. In our simulation of case 2 we take a = e = 2, d = 1, and vary b = c. The perturbed game is very simple in this case: A = E = a = e, B = C = b = c, D = 1.5, F = 0.5. Since B = C = c the condition for case 2 is 0.5 c 1.5 1 − > > 1 − , 2 2 2 or 1.5 > c > 0.5. When c = 1.5, the 1,3 equilibrium wins. When c = 1, the 1,2 equilibrium wins. In principle we could find the value of c where the limiting equilibrium changes by showing that the limiting system of PDE has traveling wave solutions and find- ing the parameter value where the velocity changes sign, but this seems to be difficult mathematical problem. When c = 1.25, the 1,2 equilibrium wins (Fig. 6), but takes a long time to do so, suggesting that this value is near the point where the limiting state changes. Fig. 7, which shows the final state of the simulation, shows that we get separation into regions that look like the two possible equilibria, and then the 1,2 regions grow and take over. Fig. 7. Picture of final configuration for the simulation in Fig. 6. Note that 7. Summary the blues (1’s) are spread throughout the space, while the reds (2’s) and In this paper, we have used simulation and heuristic arguments whites (3’s) are segregated. The correlations ρ11 = 0.1342, ρ22 = 0.1600, and to make predictions about the behavior of games that cannot be ρ33 = 0.1497 are smaller than those in Fig. 4.

6050 | www.pnas.org/cgi/doi/10.1073/pnas.1620852114 Nanda and Durrett Downloaded by guest on October 2, 2021 Downloaded by guest on October 2, 2021 ad n Durrett and Nanda etdhr r eie ntelmtta h selection the pre- that have limit we the in results derived The are structure conclusions. here spatial equa- correct sented of replicator obtain impact have the the to consider of order we to that in important papers from is behav- the it differs the so cases all game tion, many spatial in In the 10. games ref. of the in ior example of one for all except analyze cited can to here presented used work the be however, points; fixed boundary ble .SirikA relkM(03 plcto feouinr ae omdln car- modeling to games evolutionary of Application (2013) M Krzeslak A, Swierniak 9. .BsnaD ta.(02 netgtn rsaecne uo-toaitrcin:Clin- interactions: tumor-stroma theory cancer prostate game Investigating (2012) Evolutionary al. et D, (2008) Basanta A 8. Deutsch interactions H, of Hatzikirou consequences M, Simon the D, Modeling Basanta (1997) 7. WF Bodmer IPM, Tomlinson 6. pheno- Cancer (2009) JM Pacheco S, Segbroeck van FC, Santos FACC, Chalub D, Dingli 5. main- production IGF-II for Heterogeneity (2015) G Christofori DA, Ferraro M, Archetti coefficients. 4. selection small with diffusion games reaction evolutionary Spatial and (2014) perturbations R Durrett model Voter 3. (2011) E Perkins R, Durrett T, Cox 2. (1998) K Sigmund J, Hofbauer 1. okrmist edn ntresrtg ae ihunsta- with games three-strategy on done be to remains Work cinogenesis. cladbooia nihsfo neouinr game. evolutionary an from insights biological and ical invasion. and progression glioma in glycolysis 980–987. of role the elucidates cells. tumor between cells. malignant and normal between game evolutionary Cancer J an Br of outcome the as type cancer. pancreatic USA neuroendocrine Sci in dynamics goods public by tained Probab J Ast equations. UK). Cambridge, Press, Univ bridge 112:1833–1838. 19:121. 101:1130–1136. ahBoc Eng Biosci Math rsu 4.arXiv:1103.1676. 340. erisque ´ rJCancer J Br 10:873–911. vltoayGmsadPplto Dynamics Population and Games Evolutionary 78:157–160. rJCancer J Br 106:174–181. w rcNt Acad Natl Proc elProlif Cell → Electron but 0, (Cam- 41: 5 oz M(93 ok-otraeuto n elctrdynamics. replicator and equation Lotka-Volterra (1983) IM populations. structured Bomze in strategies 15. Multiple (2011) struc- MA in Nowak selection N, Strategy Wage (2009) CE, Tarnita MA Nowak F, 14. Feng T, Antal H, graphs. on Ohtsuki tumor games CE, Evolutionary during Tarnita (2006) mutations MA 13. passenger Nowak H, and Ohtsuki driver of 12. Accumulation (2010) al game. et evolutionary I, spatial Bozic a as 11. remodeling Bone (2017) KA Murgas R, Ryser 10. 0 urt ,LvnS 19)Teiprac fbigdsrt adspatial). (and discrete being of importance The coexistence. (1994) SA global Levin R, and Durrett dependence frequency 20. Local (1998) al et competition. J, interspecific Molofsky of aspects 19. Spatial (1998) SA Levin the R, in Durrett issues New 18. dynamics: replicator and equation Lotka-Volterra appli- (1983) ecological IM to Bomze guide user’s A 17. models: spatial Stohastic (1994) SA Levin R, Durrett 16. a enprilyspotdb S rnsDS1025fo h proba- the biology. from mathematical 1505215 R.D. from opportunity. DMS 16164838 this Grants DMS her NSF and program by giving bility supported for partially Math been and the has Science North improve at for us program School helped Science Computational have Carolina in that Research the comments thanks their M.N. paper. for Scott Jacob and Ryser, ACKNOWLEDGMENTS. accurate are conclusions the cases when many in that show simulations rcNt cdSiUSA Sci Acad Natl Proc populations. tured progression. Biol Biol Biol Popul Biol classification. cations. 48:201–211. 418:16–26. 46:363–394. 53:30–43. w 0 = hlTasRScLondon Soc R Trans Phil 55:270–282. .1 rcNt cdSiUSA Sci Acad Natl Proc ilCybern Biol ree 0.25. even or ho Biol Theor J 108:2334–2337. etakDvdBsna re anthe,Marc Kaznatcheev, Artem Basanta, David thank We PNAS 72:447–453. 259:570–581. 343:329–350. | ue6 2017 6, June 107:18545–18550. | o.114 vol. ho Biol Theor J | o 23 no. ho Popul Theor ilCybern Biol ho Popul Theor 243:86–97. | Thoer J Theor 6051

APPLIED ECOLOGY MATHEMATICS