Bifurcation Analysis of Three-Strategy Imitative Dynamics with Mutations

Hu, Wenjun; Tian, Haiyan; Zhang, Gang

doi:https://doi.org/10.1155/2019/4134105

Complexity

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Special Issue

Dynamical Analysis of Biological Systems

View this Special Issue

Research Article | Open Access

Volume 2019 | Article ID 4134105 | https://doi.org/10.1155/2019/4134105

Bifurcation Analysis of Three-Strategy Imitative Dynamics with Mutations

Wenjun Hu,^1,2Haiyan Tian,¹and Gang Zhang¹

Academic Editor: Constantin Udriste

Received11 Apr 2019

Accepted13 Sept 2019

Published30 Oct 2019

Abstract

Evolutionary game dynamics is an important research, which is widely used in many fields such as social networks, biological systems, and cooperative behaviors. This paper focuses on the Hopf bifurcation in imitative dynamics of three strategies (Rock-Paper-Scissors) with mutations. First, we verify that there is a Hopf bifurcation in the imitative dynamics with no mutation. Then, we find that there is a critical value of mutation such that the system tends to an unstable limit cycle created in a subcritical Hopf bifurcation. Moreover, the Hopf bifurcation exists for other kinds of the considered mutation patterns. Finally, the theoretical results are verified by numerical simulations through Rock-Paper-Scissors game.

1. Introduction

Evolutionary game dynamics combines game theory and nonlinear dynamics to describe the evolution of the frequencies of strategies in one or more large population [1, 2]. It has edged into many fields such as networks population [3–6], economics [7, 8], biology [9, 10], management [11, 12], and cooperative behaviors [13–16]. There are many important evolutionary game dynamics such as replicator dynamics, imitative dynamics [17], best-response dynamics [18], and so on [19]. The Rock-Paper-Scissors (RPS) [20, 21] is a famous three-strategy game, which describes interactions among three competing species in ecology, sociological systems [22], and theoretical biology [23, 24]. Replicator dynamics is the best-known evolutionary dynamics, which was firstly defined by Taylor and Jonker [25], and has been researched in various fields [26, 27]. In practice, imitative dynamics is a generalized replicator dynamics, which investigates the spreading of strategies in the context of imitation instead of inheritance.

There are some research studies about the imitative dynamics [28–30]. Cheung [28] studied the imitative dynamics for games with continuous strategy space and obtained global convergence and local stability results for imitative dynamics. Wang et al. [29] investigated the imitation dynamics with delay, and they discussed the two-phenotype and three-phenotype model and obtained some relevant results for stability. Hu et al. [30] researched the imitative dynamics with discrete delay, and they discovered that the stability would be changed in the discrete delay dynamics and obtained some sufficient conditions. The emphasis of the literatures is the effect of delay in the imitative dynamics. However, the mutation is also a noticeable factor on the study of the stability of the evolutionary dynamics in reality.

Until now, many researchers have studied the effect of mutations in the replicator dynamics [31–34]. Mobilia [31] investigated the oscillatory dynamics in generic RPS games with mutations and found out the existence of the heteroclinic cycles in the RPS model. Nagatani et al. [32] studied a metapopulation model for RPS game with mutation, and they found that the mutation would lead to the phase transitions among three strategies. Toupo et al. [33, 34] researched the effect of mutations in the repeated prisoner’s dilemma game and RPS game, and they found that the mutations would result in the Hopf bifurcation in the replicator dynamics. Their research studies illustrate that mutation could change the stability of the dynamics, especially lead to bifurcation.

The bifurcation is an important behavior in dynamical systems [35], which has been researched by many scholars [36–39]. Wesson et al. [36, 37] investigated the Hopf bifurcation in two-strategy and three-strategy delayed replicator dynamics, and they demonstrated the existence of Hopf bifurcation and presented an analysis of the limit cycles through Lindstedt’s method. Nesrine et al. [38] researched the Hopf bifurcation in RPS game with distributed delays. Umezuki [39] studied the bifurcation of RPS game with discrete-time logit dynamics and showed that some bifurcations would destroy the coexistence of the attractors in the RPS game.

According to the previous literatures, there are few research studies about the bifurcation in imitative dynamics. In this paper, we aim to discuss the Hopf bifurcation in imitative dynamics with mutation. Our research will illustrate that (i) the imitative dynamics appears to be a Hopf bifurcation at the parameter γ in the RPS game; (ii) the stability would be changed in the mutative imitation dynamics; and (iii) a subcritical Hopf bifurcation would be exhibited in this dynamics.

The rest of this paper is organized as follows. Section 2 sets the imitation dynamics model without mutation and analyses the stability and bifurcation. Section 3 researches the Hopf bifurcations with mutations in the imitative dynamics. Section 4 gives numerical simulations of the equilibrium and an unstable periodic solution. Section 5 offers concluding remarks.

2. RPS Model without Mutation

2.1. Derivation

We consider a symmetric three-phenotype model with pure strategies Rock (R), Scissors (S), and Paper (P) and with payoff matrix:

The payoff matrix means that each strategy gets a payoff 1 when playing against itself, and the loser gets a payoff 0 while the winner gets . Let denote the frequency of and the expected payoff of with , where with and denotes the payoff of -individual plays against a individual in which .

The classic imitation dynamics tacitly supposes that an individual is randomly selected from the population and awarded the same opportunity to change the strategy. That is, when an individual using plays against an individual using , the imitation rate that the strategist switches to is denoted by for In the previous literature, it is assumed that the imitation rate depends on the expected payoffs and :where the function defines the imitation rule. Here, we take , i.e.,

For ease of notation, write . Under this condition, the imitation dynamics equations can be written as follows:

Since are the frequencies of the three strategies, the region of interest is the three-dimensional simplex in :

So, we can eliminate z using and the projection of into the plane: . In this case, equation (4) can be written as

2.2. Stability of Equilibria

System (6) has four equilibria:

In order to discuss the stability of these equilibria, we linearize equation (6). As a result, we can analyze the stability of each point through the eigenvalues of the Jacobian. The eigenvalues of the three corner equilibria can be calculated as shown in Table 1.

From above analysis, in the nonmutation RPS equation, each corner of S is a saddle point.

Next, we consider an important equilibrium ; first, we discuss the stability in the nonmutation system. Since there are two imaginary eigenvalues at this equilibrium point, we think there might be a Hopf bifurcation at .

2.3. Hopf Bifurcation

First, we introduce a lemma about Hopf bifurcation in the vector field.

Lemma 1 (see [40]). Suppose that systemhas an equilibrium at which the following properties are satisfied:(H1) has a simple pair of pure imaginary eigenvalues and no other eigenvalues with zero real parts(H2)(H3), where is the first Lyapunov coefficientThen, the system undergoes a Hopf bifurcation at .
The coefficient can be calculated as follows. On the center manifold, has the following form near the origin:where are nonlinear functions in x and y and are the eigenvalues of the linearized system around the equilibrium at the origin. Especially, at the bifurcation point (i.e., ), the coefficient is given by

Lemma 2 (see [41]). Consider the system form (8); for sufficiently small μ, the following four cases hold:(i): unstable equilibrium for and asymptotically stable equilibrium for , with unstable periodic orbit (i.e., subcritical) for (ii): unstable equilibrium for and asymptotically stable equilibrium for , with asymptotically stable periodic orbit (i.e., supercritical) for (iii): unstable equilibrium for and asymptotically stable equilibrium for , with unstable periodic orbit (i.e., subcritical) for (iv): unstable equilibrium for and asymptotically stable equilibrium for , with asymptotically stable periodic orbit (i.e., supercritical) for Next, we give a theorem to illustrate the bifurcation in dynamics (6).

Theorem 1. The imitation dynamics (6) exhibits a subcritical Hopf bifurcation at . Moreover, when , the interior equilibrium is locally stable, and it is unstable when .

Proof. (i)When , the sign of the real part of eigenvalues can be determined through Table 1. That is,(ii)When , as the formula in Lemma 1, we obtain the nonlinear function form,We can obtain the Lyapunov coefficient through Matlab as follows:According to Lemma 2, the Hopf bifurcation is subcritical at .

3. RPS Model with Mutations

In this section, we examine the imitative dynamics with all kinds of mutations, including global mutations, single mutation, double mutations, and so on.

3.1. Global Mutations in RPS Model

First, we discuss the global mutations in imitative dynamics. The relationship of mutations is shown in Figure 1.

In this case, the dynamics becomes the following form with mutant coefficient :

We give a theorem to illustrate the bifurcation in dynamics (14) as follows.

Theorem 2. The following two conclusions are established for imitative dynamics (14):(i)There exists a subcritical Hopf bifurcation at when . Moreover, for , the interior equilibrium is locally stable, and for , it is unstable(ii)The interior equilibrium is locally stable when

Proof. The Jacobian matrix of dynamics (14) at isand the conjugate complex eigenvalues areLet , then . One can obtain the following:(i): if , then the Jacobian matrix has a pair of pure complex eigenvalues(ii): is always correct, i.e., the interior equilibrium is locally stableSimilar to the proof in Theorem 1, the nonlinear functions can be obtained as follows:The Lyapunov coefficient can be calculated as follows:According to Lemma 2, the Hopf bifurcation is subcritical at .
The results in Theorem 1 and Theorem 2 show that the situation in dynamics (8) is different from dynamics (14). While the interior equilibrium is always unstable when in the former, the interior equilibrium is locally stable for when in the latter.

3.2. Other Mutations in RPS Model

In this section, we discuss the other mutations in imitative dynamics; the situation becomes complex as one adds more mutant pathways. For the ease of research, let us restrict attention to mutant forms that ensure as the inner equilibrium for all values of γ and μ.

Here, we discuss the following three kinds of mutations in RPS imitative dynamics (see Figure 2): (i) single mutation between two strategies; (ii) single-cycle mutations among three strategies; and (iii) double mutations between two strategies. These mutant forms are shown in Table 2. As the cycle symmetry of the RPS game, it suffices to consider one of the three possible single mutation and double mutations. In this case, we just consider the following representative mutations.

For the imitative dynamics with these three kinds of mutations, similar subcritical Hopf bifurcation at would be present. This is different from the stability in nonmutation imitative dynamics, which is always unstable when .

4. Numerical Simulations

In this section, we propose to compare the properties of the bifurcating periodic solution. Here, we report two simulation results for imitative dynamics with nonmutation and mutation, respectively.

Example 1. In the imitative dynamics (6), we takeinto the equation. Through the Matlab software, one can obtain the following results (see Figure 3).
In Figure 3, the numerical simulation shows that the interior equilibrium is asymptotically stable when (i.e., ). However, when , the system state tends to a unstable periodic solution, and when (i.e., ), the interior equilibrium is unstable.

(a)

(b)

(c)

Example 2. In the imitative dynamics (14), letThrough the Matlab software, one can obtain the following results (see Figures 4 and 5).
In Figure 4, the numerical simulation shows that the interior equilibrium is asymptotically stable when (i.e., ) for any value of μ, such as and .
In Figure 5, the numerical simulation shows the following: (i) the Hopf curve in dynamics (14), i.e., the criticality of μ, changes with γ; (ii) the interior equilibrium would be stable when (i.e., ); and (iii) when , the system state tends to a unstable periodic solution, and when , the interior equilibrium is stable and is unstable when . The marks in Figures 4 and 5 are same as the description in Figure 3.

(a)

(b)

(a)

(b)

(c)

(d)

5. Conclusion

In this paper, the stability of the interior equilibrium has been mainly investigated for imitative dynamics with mutations. Different from the result in replicator dynamics [31, 34], the stability of the interior equilibrium has been changed with the mutations, and a subcritical Hopf bifurcation appears.

For the imitative dynamics in the RPS game, the stability is changed at the parameter γ in the payoff matrix with no mutation, and the mutation μ in global mutation. In the dynamics with no mutation, the interior equilibrium is locally stable when and is unstable when , and a subcritical Hopf bifurcation appears at in given payoff matrix. In the imitative dynamics with global mutation, the interior equilibrium is stable when , and it is different from the case when . There is a subcritical Hopf bifurcation at , and the interior equilibrium is locally stable when and is unstable when .

If we change the number of parameters (i.e., from one to two), the stability and bifurcation would become much more complicated. Furthermore, some numerical examples have been given to illustrate the effectiveness of our results. As an extension to this work, we plan to discuss the imitation dynamics with delays and mutations.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this paper and read and approved the final manuscript.

Acknowledgments

The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NNSF) through grant no. 11562006, the Innovation Foundation for Graduate in Hebei Normal University (no. YJBS2019001), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province (no. 2019L0940), the Science Foundation of Hebei Normal University (no. L2018B01), and the Natural Science Foundation of Shanxi Province (no. 201801D121009).

References

J. M. Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, UK, 1982.
J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998.
M. R. D’Orsogna and M. Perc, “Statistical physics of crime: a review,” Physics of Life Reviews, vol. 12, pp. 1–21, 2015.
View at: Publisher Site | Google Scholar
D. Madeo and C. Mocenni, “Game interactions and dynamics on networked populations,” IEEE Transactions on Automatic Control, vol. 60, no. 7, pp. 1801–1810, 2015.
View at: Publisher Site | Google Scholar
Z. Wang, M. A. Andrews, Z.-X. Wu, L. Wang, and C. T. Bauch, “Coupled disease–behavior dynamics on complex networks: a review,” Physics of Life Reviews, vol. 15, pp. 1–29, 2015.
View at: Publisher Site | Google Scholar
J. M. Pacheco, A. Traulsen, and M. A. Nowak, “Coevolution of strategy and structure in complex networks with dynamical linking,” Physical Review Letters, vol. 97, no. 25, Article ID 258103, 2006.
View at: Publisher Site | Google Scholar
H. Xu, C. Tian, X. Xiao, and S. Fan, “Evolutionary investors’ power-based game on networks,” Applied Mathematics and Computation, vol. 330, pp. 125–133, 2018.
View at: Publisher Site | Google Scholar
H. Xu, S. Fan, C. Tian, and X. Xiao, “Evolutionary investor sharing game on networks,” Applied Mathematics and Computation, vol. 340, pp. 138–145, 2019.
View at: Publisher Site | Google Scholar
R. Liu and G. Liu, “Asymptotic behavior of a stochastic Two-Species competition model under the effect of disease,” vol. 2018, Article ID 3127404, 15 pages, 2018.
View at: Publisher Site | Google Scholar
J. Yang and X. Wang, “Threshold dynamics of an SIR model with nonlinear incidence rate and age-dependent susceptibility,” vol. 2018, Article ID 9613807, 15 pages, 2018.
View at: Publisher Site | Google Scholar
R. Lu, X. Wang, H. Yu, and D. Li, “Multiparty evolutionary game model in coal mine safety management and its application,” Complexity, vol. 2018, Article ID 9620142, 10 pages, 2018.
View at: Publisher Site | Google Scholar
X. Wang, R. Lu, H. Yu, and D. Li, “Stability of the evolutionary game system and control strategies of behavior instability in coal mine safety management,” Complexity, vol. 2019, Article ID 6987427, 14 pages, 2019.
View at: Publisher Site | Google Scholar
H.-X. Yang and X. Chen, “Promoting cooperation by punishing minority,” Applied Mathematics and Computation, vol. 316, pp. 460–466, 2018.
View at: Publisher Site | Google Scholar
J. Gómez-Gardeñes, M. Campillo, L. M. Floría, and Y. Moreno, “Dynamical organization of cooperation in complex topologies,” Physical Review Letters, vol. 98, no. 10, Article ID 108103, 2007.
View at: Publisher Site | Google Scholar
M. A. Nowak, A. Sasaki, C. Taylor, and D. Fudenberg, “Emergence of cooperation and evolutionary stability in finite populations,” Nature, vol. 428, no. 6983, pp. 646–650, 2004.
View at: Publisher Site | Google Scholar
M. A. Nowak, “Five rules for the evolution of cooperation,” Science, vol. 314, no. 5805, pp. 1560–1563, 2006.
View at: Publisher Site | Google Scholar
D. Vilone, J. J. Ramasco, Angel Sánchez, and M. S. Miguel, “Social and strategic imitation: the way to consensus,” Science Reports, vol. 2, Article ID 686, 2012.
View at: Publisher Site | Google Scholar
C. H. Hommes and M. I. Ochea, “Multiple equilibria and limit cycles in evolutionary games with Logit Dynamics,” Games and Economic Behavior, vol. 74, no. 1, pp. 434–441, 2012.
View at: Publisher Site | Google Scholar
R. Cressman and Y. Tao, “The replicator equation and other game dynamics,” Proceedings of the National Academy of Sciences, vol. 111, no. S3, pp. 10810–10817, 2014.
View at: Publisher Site | Google Scholar
B. Sinervo and C. M. Lively, “The rock-paper-scissors game and the evolution of alternative male strategies,” Nature, vol. 380, no. 6571, pp. 240–243, 1996.
View at: Publisher Site | Google Scholar
B. Xu, H. Zhou, and Z. Wang, “Cycle frequency in standard Rock-Paper-Scissors games: evidence from experimental economics,” Physica A: Statistical Mechanics and Its Applications, vol. 392, no. 20, pp. 4997–5005, 2013.
View at: Publisher Site | Google Scholar
T. Platkowski and J. Zakrzewski, “Asymptotically stable equilibrium and limit cycles in the RPS game in a population of players with complex personalities,” Physica A: Statistical Mechanics and Its Applications, vol. 390, no. 23-24, pp. 4219–4226, 2011.
View at: Publisher Site | Google Scholar
M. A. Nowak and K. Sigmund, “Evolutionary dynamics of biological games,” Science, vol. 303, no. 5659, pp. 793–799, 2004.
View at: Publisher Site | Google Scholar
E. Kukla and T. Platkowski, “Onset of limit cycles in population games with attractiveness driven strategy choice,” Chaos, Soliton and Fractals, vol. 56, pp. 77–82, 2013.
View at: Publisher Site | Google Scholar
P. D. Taylor and L. B. Jonker, “Evolutionary stable strategies and game dynamics,” Mathematical Biosciences, vol. 40, no. 1-2, pp. 145–156, 1978.
View at: Publisher Site | Google Scholar
Y. Tao and Z. Wang, “Effect of time delay and evolutionarily stable strategy,” Journal of Theoretical Biology, vol. 187, no. 1, pp. 111–116, 1997.
View at: Publisher Site | Google Scholar
J. S. Weitz, C. Eksin, K. Paarporn, S. P. Brown, and W. C. Ratcliff, “An oscillating tragedy of the commons in replicator dynamics with game environment feedback,” Proceedings of the National Academy of Sciences of the United Atates of America, vol. 113, no. 47, pp. E7518–E7525, 2016.
View at: Publisher Site | Google Scholar
M.-W. Cheung, “Imitative dynamics for games with continuous strategy space,” Games and Economic Behavior, vol. 99, pp. 206–223, 2016.
View at: Publisher Site | Google Scholar
S.-C. Wang, J.-R. Yu, S. Kurokawa, and Y. Tao, “Imitation dynamics with time delay,” Journal of Theoretical Biology, vol. 420, pp. 8–11, 2017.
View at: Publisher Site | Google Scholar
W. Hu, G. Zhang, and H. Tian, “The stability of imitation dynamics with discrete distributed delays,” Physica A: Statistical Mechanics and Its Applications, vol. 521, pp. 218–224, 2019.
View at: Publisher Site | Google Scholar
M. Mobilia, “Oscillatory dynamics in rock-paper-scissors games with mutations,” Journal of Theoretical Biology, vol. 264, no. 1, pp. 1–10, 2010.
View at: Publisher Site | Google Scholar
T. Nagatani, G. Ichinose, and K.-I. Tainaka, “Metapopulation model for rock-paper-scissors game: mutation affects paradoxical impacts,” Journal of Theoretical Biology, vol. 450, pp. 22–29, 2018.
View at: Publisher Site | Google Scholar
D. F. P. Toupo, D. G. Rand, and S. H. Strogatz, “Limit cycles sparked by mutation in the repeated prisoner’s Dilemma,” International Jounal of Bifurcation and Chaos, vol. 24, no. 12, Article ID 1430035, 2014.
View at: Publisher Site | Google Scholar
D. F. P. Toupo and S. H. Strogatz, “Nonlinear dynamics of the rock-paper-scissors game with mutations,” Physical Review E, vol. 91, no. 5, Article ID 052907, 2015.
View at: Publisher Site | Google Scholar
J. N. Webb, Game Theory Decisions, Interaction and Evolution, Springer Undergraduate Mathematics Series, New York, NY, USA, 2006.
E. Wesson and R. Rand, “Hopf bifurcations in delayed rock-paper-scissors replicator dynamics,” Dynamic Games and Applications, vol. 6, no. 1, pp. 139–156, 2016.
View at: Publisher Site | Google Scholar
E. Wessonn, R. Rand, and David Rand, “Hopf bifurcations in two-strategy delayed replicator dynamics,” International Jounal of Bifurcation and Chaos, vol. 26, no. 1, Article ID 1650006, 2016.
View at: Publisher Site | Google Scholar
B. K. Nesrine, E. A. Rachid, and H. Yezekael, “Hopf bifurcations in replicator dynamics with distributed delays,” 2017, https://arxiv.org/abs/1703.06721.
View at: Google Scholar
Y. Umezuki, “Bifurcation analysis of the rock-paper-scissors game with discrete-time logit dynamics,” Mathematical Social Sciences, vol. 95, pp. 54–65, 2018.
View at: Publisher Site | Google Scholar
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, NY, USA, 2002.
S. Wiggins, Intoduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, NY, USA, 2003.

Copyright

Copyright © 2019 Wenjun Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies