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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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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