AIMS Mathematics, 2020, 5(4): 3378-3390. doi: 10.3934/math.2020218.

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Asymptotic behavior for a class of population dynamics

1 School of Mathematics and Statistics, Changsha University of Science and Technology; Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha 410114, China
2 School of Mathematics, Southeast University, Nanjing, 211189, China

This paper investigates the asymptotic behavior for a class of n-dimensional population dynamics systems described by delay differential equations. With the help of technique of differential inequality, we show that each solution of the addressed systems tends to a constant vector as t → ∞, which includes many generalizations of Bernfeld-Haddock conjecture. By the way, our results extend some existing literatures.
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Keywords population dynamics; time-varying delay; asymptotic behavior; Bernfeld-Haddock conjecture

Citation: Chuangxia Huang, Luanshan Yang, Jinde Cao. Asymptotic behavior for a class of population dynamics. AIMS Mathematics, 2020, 5(4): 3378-3390. doi: 10.3934/math.2020218


  • 1. D. Yang, X. Li, J. Qiu, Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback, Nonlinear Anal. Hybrid Syst., 32 (2019), 294-305.    
  • 2. X. Yang, X. Li, Q. Xi, et al. Review of stability and stabilization for impulsive delayed systems, Math. Biosci. Eng., 15 (2018), 1495-1515.    
  • 3. X. Li, X. Yang, T. Huang, Persistence of delayed cooperative models: Impulsive control method, Appl. Math. Comput., 342 (2019), 130-146.
  • 4. Y. Tan, C. Huang, B. Sun, et al. Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition, J. Math. Anal. Appl., 458 (2018), 1115-1130.    
  • 5. C. Huang, X. Long, J. Cao, Stability of anti-periodic recurrent neural networks with multiproportional delays, Math. Method Appl. Sci., 2020.
  • 6. J. Zhang, C. Huang, Dynamics analysis on a class of delayed neural networks involving inertial terms, Adv. Differ. Equations, 120 (2020), 1-12.
  • 7. X. Long, S. Gong, New results on stability of Nicholson's blowflies equation with multiple pairs of time-varying delays, Appl. Math. Lett., 100 (2020), 106027.
  • 8. C. Huang, Y. Qiao, L. Huang, et al. Dynamical behaviors of a food-chain model with stage structure and time delays, Adv. Differ. Equations, 186 (2018).
  • 9. C. Huang, J. Cao, F. Wen, et al. Stability Analysis of SIR Model with Distributed Delay on Complex Networks, Plos One, 11 (2016), e0158813.
  • 10. H. Hu, X. Yuan, L. Huang, et al. Global dynamics of an SIRS model with demographics and transfer from infectious to susceptible on heterogeneous networks, Math. Biosci. Eng., 16 (2019), 5729-5749.    
  • 11. H. Hu, X. Zou, Existence of an extinction wave in the fisher equation with a shifting habitat, Proc. Am. Math. Soc., 145 (2017), 4763-4771.    
  • 12. H. Hu, T. Yi, X. Zou, On spatial-temporal dynamics of Fisher-KPP equation with a shifting environment, Proc. Amer. Math. Soc., 148 (2020), 213-221.
  • 13. J. Wang, C. Huang, L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst., 33 (2019), 162-178.    
  • 14. J. Wang, X. Chen, L. Huang, The number and stability of limit cycles for planar piecewise linear systems of nodeCsaddle type, J. Math. Anal. Appl., 469 (2019), 405-427.    
  • 15. C. Huang, Z. Yang, T. Yi, et al. On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differ. Equations, 256 (2014), 2101-2114.    
  • 16. C. Huang, H. Zhang, J. Cao, et al. Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator, Int. J. Bifurcation Chaos, 29 (2019), 1950091.
  • 17. C. Huang, H. Zhang, L. Huang, Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term, Commun. Pure Appl. Anal., 18 (2019), 3337-3349.    
  • 18. C. Qian, Y. Hu, Novel stability criteria on nonlinear density-dependent mortality Nicholson's blowflies systems in asymptotically almost periodic environments, J. Inequal. Appl., 13 (2020), 1-18.
  • 19. C. Huang, X. Long, L. Huang, et al. Stability of almost periodic Nicholson's blowflies model involving patch structure and mortality terms, Can. Math. Bull., (2019), 1-18.
  • 20. C. Huang, H. Yang, J. Cao, Weighted Pseudo Almost Periodicity of Multi-Proportional Delayed Shunting Inhibitory Cellular Neural Networks with D operator, Discrete Contin. Dyn. Syst. Ser. S, 2020.
  • 21. S. R. Bernfeld, J. R. A. Haddock, A variation of Razumikhin's method for retarded functional equations, In: Nonlinear systems and applications, An International Conference, New York: Academic Press, 1977, 561-566.
  • 22. C. Jehu, Comportement asymptotique des solutions de equation x'(t) = -f (t, x(t)) + f (t, x(t - 1)) + h(t) (in French), Ann. Soc. Sci. Brux. I, 92 (1979), 263-269.
  • 23. T. Ding, Asymptotic behavior of solutions of some retarded differential equations, Sci. China Ser. A-Math., 25 (1982), 363-371.
  • 24. T. Yi, L. Huang, Asymptotic behavior of solutions to a class of systems of delay differential equations, Acta Math. Sin. (Engl. Ser.), 23 (2007), 1375-1384.    
  • 25. M. Xu, W. Chen, X. Yi, New generalization of the two-dimensional Bernfeld-Haddock conjecture and its proof, Nonlinear Anal. Real World Appl., 11 (2010), 3413-3420.    
  • 26. Q. Zhou, W. Wang, Q. Fan, A generalization of the three-dimensional Bernfeld-Haddock conjecture and its proof, J. Comput. Appl. Math., 233 (2009), 473-481.    
  • 27. B. S. Chen, Asymptotic behavior of solutions of some infinite retarded differential equations(in Chinese), Acta Math. Sin. (Engl. Ser.), 3 (1990), 353-358.
  • 28. T. Ding, Applications of the qualitative methods in ordinary differential equations (in Chinese), Peking: China Higher Education Press, 2004, 155-163.
  • 29. T. Yi, L. Huang, Convergence of solution to a class of systems of delay differential equations, Nonlinear Dyn. Syst. Theory, 5 (2005), 189-200.
  • 30. Q. Zhou, Convergence for a two-neuron network with delays, Appl. Math. Lett., 22 (2009), 1181-1184.    
  • 31. S. Hu, L. Huang, T. Yi. Convergence of bounded solutions for a class of systems of delay differential equations, Nonlinear Anal., 61 (2005), 543-549.
  • 32. B. S. Chen, Asymptotic behavior of a class of nonautonomous retarded differential equations (in Chinese), Chinese Sci. Bull., 6 (1988), 413-415.
  • 33. T. Yi, L. Huang, Convergence for pseudo monotone semi-flows on product ordered topological spaces, J. Differ. Equations, 214 (2005), 429-456.    
  • 34. Q. Zhou, Asymptotic behavior of solutions to a first-order non-homogeneous delay differential equation, Electron. J. Differ. Equations, 103 (2011), 1-8.
  • 35. B. Liu, Asymptotic behavior of solutions to a class of non-autonomous delay differential equations, J. Math. Anal. Appl., 446 (2017), 580-590.    
  • 36. B. Liu, A generalization of the Bernfeld-Haddock conjecture, Appl. Math. Lett., 65 (2017), 7-13.    
  • 37. S. Xiao, Asymptotic behavior of solutions to a non-autonomous system of two-dimensional differential equations, Electron. J. Differ. Equations, 2017 (2017), 1-12.


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