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Stability of a delayed competitive model with saturation effect and interval biological parameters

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Abstract

This work presents a delayed two-species competitive model with interval biological parameters, in which each interspecific competition term suffers delay and saturation effect. Sufficient criteria for the existence, local and global asymptotic stability of the unique positive equilibrium are established, respectively. Our theoretical and simulated results show that under appropriate conditions the fuzziness of biological parameters plays a critical role in the stability properties of the system while time delays have little influence.

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References

  1. Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74. Kluwer, Dordrecht (1992)

    Book  Google Scholar 

  2. Song, Y.L., Han, M.A., Peng, Y.H.: Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays. Chaos Solitons Fractals. 22, 1139–1148 (2004)

    Article  MathSciNet  Google Scholar 

  3. Chen, H.C., Ho, C.P.: Persistence and global stability on competition system with time-delay. Tunghai Sci. 5, 71–99 (2003)

    Google Scholar 

  4. Li, Q., Liu, Z.J., Yuan, S.L.: Cross-diffusion induced Turing instability for a competition model with saturation effect. Appl. Math. Comput. 347, 64–77 (2019)

    MathSciNet  MATH  Google Scholar 

  5. Alvarez, C., Lazer, A.: An application of topological degree to the periodic competing species problem. J. Aust. Math. Soc. 28, 202–219 (1986)

    Article  MathSciNet  Google Scholar 

  6. Ahmad, S.: Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations. J. Math. Anal. Appl. 127, 377–387 (1987)

    Article  MathSciNet  Google Scholar 

  7. Wang, W.D., Chen, L.S., Lu, Z.Y.: Globl stability of a competition model with periodic coefficients and time delays. Can. Appl. Math. Quart. 3, 365–378 (1995)

    MATH  Google Scholar 

  8. Fan, M., Wang, K., Jiang, D.Q.: Existence and global attractivity of positive periodic solutions of periodic \(n\)-species Lotka-Volterra competition systems with several deviating arguments. Math. Biosci. 160, 47–61 (1999)

    Article  MathSciNet  Google Scholar 

  9. Tang, X.H., Cao, D.M., Zou, X.F.: Global attractivity of positive periodic solution to periodic Lotka-Volterra competition systems with pure delay. J. Differ. Eq. 228, 580–610 (2006)

    Article  MathSciNet  Google Scholar 

  10. Liu, Z.J., Fan, M., Chen, L.S.: Globally asymptotic stability in two periodic delayed competitive systems. Appl. Math. Comput. 197, 271–287 (2008)

    MathSciNet  MATH  Google Scholar 

  11. Liu, Z.J., Tan, R.H., Chen, Y.P.: Modeling and analysis of a delayed competitive system with impulsive perturbations. Rocky Mountain J. Math. 38, 1505–1523 (2008)

    Article  MathSciNet  Google Scholar 

  12. Ahmad, S., Stamov, G.T.: Almost periodic solutions of N-dimensional impulsive competitive systems. Nonlinear Anal. Real World Appl. 10, 1846–1853 (2009)

    Article  MathSciNet  Google Scholar 

  13. Wang, Q.L., Liu, Z.J., Li, Z.X., Cheke, R.A.: Existence and global asymptotic stability of positive almost periodic solutions of a two-species competitive system. Int. J. Biomath. 7, 1450040 (2014)

    Article  MathSciNet  Google Scholar 

  14. Liu, B., Chen, L.S.: The periodic competing Lotka-Volterra model with impulsive effect. Math. Med. Biol. 21, 129–145 (2004)

    Article  Google Scholar 

  15. Liu, Z.J., Wu, J.H., Cheke, R.A.: Coexistence and partial extinction in a delay competitive system subject to impulsive harvesting and stocking. IMA J. Appl. Math. 75, 777–795 (2010)

    Article  MathSciNet  Google Scholar 

  16. Liu, M., Wang, K.: Asymptotic behavior of a stochastic nonautonomous Lotka–Volterra competitive system with impulsive perturbations. Math. Comput. Model. 57, 909–925 (2013)

    Article  MathSciNet  Google Scholar 

  17. Tan, R.H., Liu, Z.J., Guo, S.L., Xiang, H.L.: On a nonautonomous comepetitive system subject to stochastic and impulsive perturbations. Appl. Math. Comput. 256, 702–714 (2015)

    MathSciNet  MATH  Google Scholar 

  18. Hu, J., Liu, Z.J.: Incorporating coupling noises into a nonlinear competitive system with saturation effect. Int. J. Biomath. (2020). https://doi.org/10.1142/S1793524520500126

    Article  MathSciNet  MATH  Google Scholar 

  19. Tan, R.H., Xiang, H.L., Chen, Y.P., Liu, Z.J.: Dynamics behaviors of a delayed competitive system in a random environment. Int. J. Biomath. 8, 1550062 (2015)

    Article  MathSciNet  Google Scholar 

  20. Zhang, Z.Z., Tong, J.Y., Bao, J.H.: The stationary distribution of competitive Lotka-Volterra population systems with jumps. Abstr. Appl. Anal. 2014, Art. ID 820831 (2014)

  21. Zhao, Y., Yuan, S.L., Ma, J.L.: Survival and stationary distribution analysis of a stochastic competitive model of three species in a polluted environment. Bull. Math. Biol. 77, 1285–1326 (2015)

    Article  MathSciNet  Google Scholar 

  22. Bassanezi, R.C., Barros, L.C., Tonelli, P.A.: Attractors and asymptotic stability for fuzzy dynamical systems. Fuzzy Sets Syst. 113, 473–483 (2000)

    Article  MathSciNet  Google Scholar 

  23. Barros, L.C., Bassanezi, R.C., Tonelli, P.A.: Fuzzy modelling in population dynamics. Ecol. Model. 128, 27–33 (2000)

    Article  Google Scholar 

  24. Peixoto, M., Barros, L.C., Bassanezi, R.C.: Predator-prey fuzzy model. Ecol. Model. 214, 39–44 (2008)

    Article  Google Scholar 

  25. Sharma, S., Samanta, G.P.: Optimal harvesting of a two species competition model with imprecise biological parameters. Nonlinear Dyn. 77, 1101–1119 (2014)

    Article  MathSciNet  Google Scholar 

  26. Pal, D., Mahapatra, G.S., Samanta, G.P.: Optimal harvesting of prey-predator system with interval biological parameters. Math. Biosci. 241, 181–187 (2013)

    Article  MathSciNet  Google Scholar 

  27. Pal, D., Mahapatra, G.S., Samanta, G.P.: New approach for stability and bifurcation analysis on predator-prey harvesting model for interval biological parameters with time delays. Comput. Appl. Math. 37, 3145–3171 (2018)

    Article  MathSciNet  Google Scholar 

  28. Meng, X.Y., Wu, Y.Q.: Dynamical analysis of a fuzzy phytoplankton-zooplankton model with refuge, fishery protection and harvesting. J. Appl. Math. Comput (2020). https://doi.org/10.1007/s12190-020-01321-y

    Article  MathSciNet  Google Scholar 

  29. Zhang, X.B., Zhao, H.Y.: Bifurcation and optimal harvesting of a diffusive predator-prey system with delays and interval biological parameters. J. Theoret. Biol. 363, 390–403 (2014)

    Article  MathSciNet  Google Scholar 

  30. Wang, Q.L., Liu, Z.J., Zhang, X.A., Cheke, R.A.: Incorporating prey refuge into a predator-prey system with imprecise parameter estimates. Comput. Appl. Math. 36, 1067–1084 (2017)

    Article  MathSciNet  Google Scholar 

  31. Beddington, J.R., May, R.M.: Harvesting natural populations in a randomly fluctuating environment. Science 197, 463–465 (1977)

    Article  Google Scholar 

  32. Gard, T.C.: Persistence in stochastic food web models. Bull. Math. Biol. 46, 357–370 (1984)

    Article  MathSciNet  Google Scholar 

  33. Jiang, D.Q., Shi, N.Z.: A note on non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 303, 164–172 (2005)

    Article  MathSciNet  Google Scholar 

  34. Li, X.Y., Mao, X.R.: Population dynamical behavior of non-autonomous Lotka–Volterra competitive system with random perturbation. Discrete Contin. Dyn. Syst. B. 232, 427–448 (2009)

    MathSciNet  MATH  Google Scholar 

  35. Mao, X.R., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Process. Appl. 97, 95–110 (2002)

    Article  MathSciNet  Google Scholar 

  36. Wang, Z., Zhang, Q.M., Baese, A.M.: Imprecise parameters for near-optimal control of stochastic SIV epidemic model. Math. Methods Appl. Sci (2020). https://doi.org/10.1002/mma.6041

    Article  MathSciNet  Google Scholar 

  37. Mu, X.J., Zhang, Q.M., Rong, L.B.: Near-optimal control for a stochastic SIRS model with imprecise parameters. Asian J. Control. 1–16, (2019)

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

    Article  MathSciNet  Google Scholar 

  39. Ruan, S.G.: Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart. Appl. Math. 59, 159–173 (2001)

    Article  MathSciNet  Google Scholar 

  40. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)

    MATH  Google Scholar 

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Acknowledgements

The authors thank the editor and referees for their careful reading and valuable comments. The work is supported by the NNSF of China (No. 11871201), and NSF of Hubei Province, China (No. 2019CFB241).

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Authors and Affiliations

  1. Department of Mathematics, Hubei Minzu University, Enshi, 445000, Hubei, People’s Republic of China

    Siyu Chen, Zhijun Liu, Lianwen Wang & Jing Hu

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  1. Siyu Chen
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  2. Zhijun Liu
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  3. Lianwen Wang
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  4. Jing Hu
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Correspondence to Zhijun Liu.

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Chen, S., Liu, Z., Wang, L. et al. Stability of a delayed competitive model with saturation effect and interval biological parameters. J. Appl. Math. Comput. 64, 1–15 (2020). https://doi.org/10.1007/s12190-020-01341-8

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  • DOI: https://doi.org/10.1007/s12190-020-01341-8

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