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Evolution of group behaviour in the ecological snowdrift game-level-headed and impatient delayed epidemic model under emergencies

Author

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  • Lv, Rongjian
  • Li, Hua
  • Sun, Qiubai
  • Gao, Qiyue
  • Li, Bowen

Abstract

Major emergencies generate negative emotions within a group. Therefore, the topic of the spread of emotions leading to the coexistence of cooperation and defection within a group has attracted much attention. We introduced the influence of the external environment and constructed an ecological snowdrift game model to explore the evolution of cooperation and defection in this paper. On this basis, the ecological snowdrift game-level-headed and impatient delayed epidemic model was developed. This model considers the differences in individual personalities by categorizing individuals into level-headed and impatient groups, and also incorporates a delay in the spread of panic within groups. We computed a threshold value for panic spread and analysed the qualitative behaviour of the epidemic model, such as locally and globally asymptotically stable equilibria. The results indicated that cooperation is more likely to emerge in a group as the probability of a cooperator encountering an individual with the same strategy increases. Comparison with the level-headed and impatient delayed epidemic model revealed that group behaviour (i.e., cooperation and defection) can influence the evolutionary process of populations within a group. Numerical simulations verified the effectiveness of the derived results. The proposed model was analysed and validated through a practical example and visualization for sentiment analysis using the Naive Bayes algorithm. The findings of the paper can help to further explore the evolution of group behaviour.

Suggested Citation

  • Lv, Rongjian & Li, Hua & Sun, Qiubai & Gao, Qiyue & Li, Bowen, 2026. "Evolution of group behaviour in the ecological snowdrift game-level-headed and impatient delayed epidemic model under emergencies," Applied Mathematics and Computation, Elsevier, vol. 509(C).
    • Handle: RePEc:eee:apmaco:v:509:y:2026:i:c:s0096300325003947
    DOI: 10.1016/j.amc.2025.129668
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    References listed on IDEAS

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    1. Shu, Feng & Li, Min & Liu, Xingwen, 2019. "Memory mechanism with weighting promotes cooperation in the evolutionary games," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 17-24.
    2. Yang, Xiqing & Wang, Wanxiong & Zhang, Feng & Qiao, Hongqiang, 2017. "Cooperation enhanced by habitat destruction in Prisoner’s Dilemma Games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 486(C), pages 668-673.
    3. Hu, Xiang & Liu, Xingwen, 2021. "Unfixed-neighbor-mechanism promotes cooperation in evolutionary snowdrift game on lattice," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 572(C).
    4. Pi, Jinxiu & Wang, Chun & Zhou, Die & Tang, Wei & Yang, Guanghui, 2024. "Evolutionary dynamics of N-person snowdrift game with two thresholds in well-mixed and structured populations," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    5. Liu, Yan-Ping & Wang, Lin & Zhang, Feng & Wang, Rui-Wu, 2020. "Diffusion sustains cooperation via forming diverse spatial patterns in prisoner's dilemma game," Applied Mathematics and Computation, Elsevier, vol. 375(C).
    6. Wang, Xianjia & Lv, Shaojie & Quan, Ji, 2017. "The evolution of cooperation in the Prisoner’s Dilemma and the Snowdrift game based on Particle Swarm Optimization," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 482(C), pages 286-295.
    7. Huo, Liang’an & Gu, Jiafeng, 2023. "The influence of individual emotions on the coupled model of unconfirmed information propagation and epidemic spreading in multilayer networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    8. Huang, He & Chen, Yahong & Yan, Zhijun, 2021. "Impacts of social distancing on the spread of infectious diseases with asymptomatic infection: A mathematical model," Applied Mathematics and Computation, Elsevier, vol. 398(C).
    9. Wu, Yu’e & Zhang, Zhipeng & Wang, Xinyu & Yan, Ming & Zhang, Qingfeng & Zhang, Shuhua, 2021. "Evolution of cooperation in the multigame on a two-layer square network," Applied Mathematics and Computation, Elsevier, vol. 400(C).
    10. Gao, Liyan & Pan, Qiuhui & He, Mingfeng, 2021. "Environmental-based defensive promotes cooperation in the prisoner’s dilemma game," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    11. Su, Qi & Li, Aming & Wang, Long, 2017. "Spatial structure favors cooperative behavior in the snowdrift game with multiple interactive dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 468(C), pages 299-306.
    12. Xu, Ting & Zhang, Juping & Jin, Zhen, 2025. "Analysis of final size and peak time for SIR epidemic model on simplicial complexes," Chaos, Solitons & Fractals, Elsevier, vol. 196(C).
    13. Cao, Mengxiao & Zhang, Guijuan & Wang, Mengsi & Lu, Dianjie & Liu, Hong, 2017. "A method of emotion contagion for crowd evacuation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 483(C), pages 250-258.
    14. Feng, Minyu & Han, Songlin & Li, Qin & Wu, Juan & Kurths, Jürgen, 2023. "Harmful strong agents and asymmetric interaction can promote the frequency of cooperation in the snowdrift game," Chaos, Solitons & Fractals, Elsevier, vol. 175(P2).
    15. Liu, Yaojun & Liu, Xingwen, 2024. "Promotion of cooperation in evolutionary snowdrift game with heterogeneous memories," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 633(C).
    16. Tyagi, Swati & Martha, Subash C. & Abbas, Syed & Debbouche, Amar, 2021. "Mathematical modeling and analysis for controlling the spread of infectious diseases," Chaos, Solitons & Fractals, Elsevier, vol. 144(C).
    17. Chen, Xingzhi & Tian, Baodan & Xu, Xin & Zhang, Hailan & Li, Dong, 2023. "A stochastic predator–prey system with modified LG-Holling type II functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 203(C), pages 449-485.
    18. Luo, Yijun & Huang, Changwei & Han, Wenchen, 2024. "The evolution of cooperation and global synchronization in the evolutionary Kuramoto dilemma combined with the prisoner's dilemma," Applied Mathematics and Computation, Elsevier, vol. 482(C).
    19. Ding, Haixin & Xie, Li, 2023. "Simulating rumor spreading and rebuttal strategy with rebuttal forgetting: An agent-based modeling approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 612(C).
    20. Yang, Yimei & Sun, Hao & Yang, Guangjing & Sun, Yanru, 2024. "Bilateral negotiation facilitates stable coexistence of cooperation with defection in Prisoner's Dilemma game," Applied Mathematics and Computation, Elsevier, vol. 471(C).
    21. Pu, Jia & Jia, Tao & Li, Ya, 2019. "Effects of time cost on the evolution of cooperation in snowdrift game," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 146-151.
    22. Huo, Feizhou & Li, Yufei & Li, Chao & Ma, Yaping, 2022. "An extended model describing pedestrian evacuation considering pedestrian crowding and stampede behavior," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    23. Wang, Mie & Kang, HongWei & Shen, Yong & Sun, XingPing & Chen, QingYi, 2021. "The role of alliance cooperation in spatial public goods game," Chaos, Solitons & Fractals, Elsevier, vol. 152(C).
    24. Bahbouhi, Jalal Eddine & Bouderba, Saif Islam & Elkouay, Abdelali & Moussa, Najem, 2024. "Effects of individual and collective decision rule on cooperation in public goods game," Applied Mathematics and Computation, Elsevier, vol. 469(C).
    25. Xie, Kai & Liu, Tingjin, 2024. "The regulation of good and evi promotes cooperation in public goods game," Applied Mathematics and Computation, Elsevier, vol. 478(C).
    26. Yang, Xiqing & Zhang, Feng & Wang, Wanxiong, 2019. "Predation promotes cooperation in Prisoner’s dilemma games," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 514(C), pages 20-24.
    27. Shu, Gang & Du, Xia & Li, Ya, 2016. "Surrounding information consideration promotes cooperation in Prisoner’s dilemma game," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 689-694.
    28. Shnip, A.I. & Trigger, S.A., 2024. "On the repeated epidemic waves," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 637(C).
    29. Tian, Yong & Ding, Xuejun, 2019. "Rumor spreading model with considering debunking behavior in emergencies," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
    30. Lv, Shaojie & Li, Jiaying & Zhao, Changheng, 2023. "The evolution of cooperation in voluntary public goods game with shared-punishment," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    31. Centres, P.M. & Perez-Morelo, D.J. & Guzman, R. & Reinaudi, L. & Gimenez, M.C., 2024. "Diffusion model for the spread of infectious diseases: SIR model with mobile agents," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 633(C).
    32. Zhao, Xiaowei & Xia, Haoxiang, 2023. "Information accuracy of migration and imitation influences the evolution of cooperation in spatial prisoner's dilemma," Chaos, Solitons & Fractals, Elsevier, vol. 176(C).
    33. Bahbouhi, Jalal Eddine & Elkouay, Abdelali & Bouderba, Saif Islam & Moussa, Najem, 2024. "The whale optimization algorithm and the evolution of cooperation in the spatial public goods game," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    34. Yang, Zhengzhi & Zheng, Lei & Perc, Matjaž & Li, Yumeng, 2024. "Interaction state Q-learning promotes cooperation in the spatial prisoner's dilemma game," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    35. Zhao, Shanshan & Pan, Qiuhui & Zhu, Wenqiang & He, Mingfeng, 2023. "How “punishing evil and promoting good” promotes cooperation in social dilemma," Applied Mathematics and Computation, Elsevier, vol. 438(C).
    36. Feng Zhang & Cang Hui, 2011. "Eco-Evolutionary Feedback and the Invasion of Cooperation in Prisoner's Dilemma Games," PLOS ONE, Public Library of Science, vol. 6(11), pages 1-7, November.
    37. Fu, Libi & Song, Weiguo & Lv, Wei & Lo, Siuming, 2014. "Simulation of emotional contagion using modified SIR model: A cellular automaton approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 405(C), pages 380-391.
    38. Huo, Feizhou & Li, Chao & Li, Yufei & Lv, Wei & Ma, Yaping, 2022. "An extended model for describing pedestrian evacuation considering the impact of obstacles on the visual view," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 604(C).
    39. Han, Xu & Xia, Haoxiang & Zhao, Xiaowei, 2024. "Memory–based adaptive interaction willingness enhances cooperation in spatial prisoner's dilemma," Applied Mathematics and Computation, Elsevier, vol. 476(C).
    40. Zhang, Lan & Huang, Changwei, 2023. "Preferential selection to promote cooperation on degree–degree correlation networks in spatial snowdrift games," Applied Mathematics and Computation, Elsevier, vol. 454(C).
    41. Chen, Jialin & Liu, Xingwen & Wang, Huazhang & Yang, Jun, 2022. "The disconnection-reconnection-elite mechanism enhances cooperation of evolutionary game on lattice," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    42. Zhang, Hong, 2023. "Evolution of cooperation with tag-based expulsion in spatial public goods game," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
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