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Dynamics of the Non-autonomous Boy-After-Girl System

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Abstract

In this paper, the non-autonomous boy after girl dynamical system was investigated. For the general case, we study some properties such as non-persistence, ultimately boundedness, permanence and globally asymptotical stability. For the periodic case, we study the existence of a periodic solution. For the almost periodic case, we study the existence, uniqueness and stability of almost periodic solution. Finally, we introduce several examples and their numerical simulations to verify our theoretical results.

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References

  1. Barbalat, I.: System d’equations differential d’oscillations nonlinearies. Rev. Roumaine Math. Pure Appl. 4(2), 267–270 (1959)

    MATH  Google Scholar 

  2. Chen, F.: On a periodic multi-species ecological model. Appl. Math. Comput. 171(1), 492–510 (2005)

    MathSciNet  MATH  Google Scholar 

  3. Chen, F.: Positive periodic solutions of neutral Lotka–Volterra system with feedback control. Appl. Math. Comput. 162(3), 1279–1302 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Chen, F., Li, Z., Chen, X., Laitochová, J.: Dynamic behaviors of a delay differential equation model of plankton allelopathy. J. Comput. Appl. Math. 206(2), 733–754 (2007)

    Article  MathSciNet  Google Scholar 

  5. Chen, F., Lin, F., Chen, X.: Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control. Appl. Math. Comput. 158(1), 45–68 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Chen, L., Chen, F., Chen, L.: Qualitative analysis of a predator-prey model with Holling type ii functional response incorporating a constant prey refuge. Nonlinear Anal. Real World Appl. 11(1), 246–252 (2010)

    Article  MathSciNet  Google Scholar 

  7. Chen, L., Song, X., Lu, Z.: Mathematical Models and Methods in Ecology (1988)

  8. Chen, X., Du, Z.: Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell–Varley type functional response and impulse. Qual. Theory Dyn. Syst. 17(1), 67–80 (2018)

    Article  MathSciNet  Google Scholar 

  9. Chen, Y.: Multiple periodic solutions of delayed predator-prey systems with type iv functional responses. Nonlinear Anal. Real World Appl. 5(1), 45–53 (2004)

    Article  MathSciNet  Google Scholar 

  10. Du, Z., Feng, Z.: Periodic solutions of a neutral impulsive predator-prey model with Beddington–Deangelis functional response with delays. J. Comput. Appl. Math. 258, 87–98 (2014)

    Article  MathSciNet  Google Scholar 

  11. Fan, M., Wang, K., Wong, P., Agarwal, R.: Periodicity and stability in periodic n-species Lotka–Volterra competition system with feedback controls and deviating arguments. Acta Math. Sinica 19(4), 801–822 (2003)

    Article  MathSciNet  Google Scholar 

  12. Fan, M., Wang, Q., Zou, X.: Dynamics of a non-autonomous ratio-dependent predator–prey system. Proc. Sect. A Math. R. Soc. Edinb. 133(1), 97 (2003)

    Article  MathSciNet  Google Scholar 

  13. Freedman, H.I., Wu, J.H.: Persistence and global asymptotic stability of single species dispersal models with stage structure. Q. Appl. Math. 49(2), 351–371 (1991)

    Article  MathSciNet  Google Scholar 

  14. Gaines, R., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations, vol. 568. Springer, Berlin (2006)

    MATH  Google Scholar 

  15. Liu, Q., Jiang, D., Hayat, T., Alsaedi, A.: Dynamics of a stochastic predator-prey model with stage structure for predator and Holling type ii functional response. J. Nonlinear Sci. 28(3), 1151–1187 (2018)

    Article  MathSciNet  Google Scholar 

  16. Lu, S., Ge, W.: Existence of positive periodic solutions for neutral population model with multiple delays. Appl. Math. Comput. 153(3), 885–902 (2004)

    MathSciNet  MATH  Google Scholar 

  17. Meng, X., Zhao, S., Feng, T., Zhang, T.: Dynamics of a novel nonlinear stochastic sis epidemic model with double epidemic hypothesis. J. Math. Anal. Appl. 433(1), 227–242 (2016)

    Article  MathSciNet  Google Scholar 

  18. Slyusarchuk, V.Y.: Conditions for the existence of almost periodic solutions of nonlinear difference equations with discrete argument. J. Math. Sci. 201, 391–399 (2014)

    Article  MathSciNet  Google Scholar 

  19. Slyusarchuk, V.Y.: Periodic and almost periodic solutions of difference equations in metric spaces. J. Math. Sci. 8, 87–394 (2016)

    Google Scholar 

  20. Slyusarchuk, V.Y.: Almost periodic solutions of differential equations. J. Math. Sci. 254, 287–304 (2021)

    Article  MathSciNet  Google Scholar 

  21. Song, Y., Tang, X.: Stability, steady-state bifurcations, and turing patterns in a predator-prey model with herd behavior and prey-taxis. Stud. Appl. Math. 139(3), 371–404 (2017)

    Article  MathSciNet  Google Scholar 

  22. Song, Y., Wu, S., Wang, H.: Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect. J. Differ. Equ. 267(11), 6316–6351 (2019)

    Article  MathSciNet  Google Scholar 

  23. Wei, Z., Xia, Y., Zhang, T.: Stability and bifurcation analysis of an Amensalism model with weak Allee effect. Qual. Theory Dyn. Syst. 19(1), 23 (2020)

    Article  MathSciNet  Google Scholar 

  24. Xia, Y., Cao, J.: Almost-periodic solutions for an ecological model with infinite delays. Proc. Edinb. Math. Soc. 50(1), 229–249 (2007)

    Article  MathSciNet  Google Scholar 

  25. Xia, Y., Cao, J., Cheng, S.: Multiple periodic solutions of a delayed stage-structured predator-prey model with non-monotone functional responses. Appl. Math. Model. 31(9), 1947–1959 (2007)

    Article  Google Scholar 

  26. Xu, C., Li, P.: The boy-after-girl dynamical model with time delay. Math. Practice Theory 45(22), 1–6 (2015)

    MathSciNet  Google Scholar 

  27. Xu, F., Cressman, R., Křivan, V.: Evolution of mobility in predator-prey systems. Discrete Continu. Dyn. Syst. B 19(10), 3397 (2014)

    Article  MathSciNet  Google Scholar 

  28. Ye, D., Fan, M., Zhang, W.: Periodic solutions of density dependent predator-prey systems with Holling type 2 functional response and infinite delays. ZAMM J. Appl. Math. Mech. 85(3), 213–221 (2005)

    Article  MathSciNet  Google Scholar 

  29. Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Differ. Equ. 246(5), 1944–1977 (2009)

    Article  MathSciNet  Google Scholar 

  30. Yoshizawa, T.: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, vol. 14. Springer, Berlin (2012)

    MATH  Google Scholar 

  31. Zhou, X., Ke, J.: The boy-after-girl mathematical model. Math. Practice Theory 42(12), 1–8 (2012)

    MathSciNet  Google Scholar 

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Acknowledgements

This paper was jointly supported from the National Natural Science Foundation of China under Grant (No. 11671176, 11931016), Natural Science Foundation of Zhejiang Province under Grant (No. LY20A010016).

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Authors and Affiliations

  1. Department of mathematics, College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua, 321004, China

    Salam M. Ghazi Al-Mohanna & Yong-Hui Xia

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  1. Salam M. Ghazi Al-Mohanna
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  2. Yong-Hui Xia
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Correspondence to Yong-Hui Xia.

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Al-Mohanna, S.M.G., Xia, YH. Dynamics of the Non-autonomous Boy-After-Girl System. Qual. Theory Dyn. Syst. 21, 52 (2022). https://doi.org/10.1007/s12346-022-00586-x

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