Positive periodic solutions of a class of non-autonomous single species population model with delays and feedback control

Acta Mathematica Sinica, English Series

Volumn 21, Issue 6, 2005, Pages 1319-1336

  De Chen, Feng ^a   Chen, Xiao Xin ^a   De Cao, Jin ^b   Chen, An Ping ^c  

^a FUZHOU UNIVERSITY   (China)

^b SOUTHEAST UNIVERSITY   (China)

^c Xiangnan College   (China)

Author keywords

Coincidence degree; Delays; Positive periodic solutions; Single species model

Indexed keywords

EID: 28144453816     PISSN: 14398516     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10114-005-0585-6     Document Type: Article

References (40)

1. Li, Y. K.: Existence and global attractivity of a positive periodic solution of a class of delay differential equation. Science in China (Series A), 41(3), 273-284 (1998)
2. Huo, H., Li, W.: Positive periodic solutions of a class of delay differential system with feedback control. Applied Mathematics and Computation, 148(1), 35-46 (2004)
3. Gopalsamy, K., Weng, P.: Feedback regulation of logistic growth. Internat. J. Maths and Math. Sci., 16(1), 177-192 (1993)
4. Yang, F., Jiang, D.: Existence and global attractivity of positive periodic solution of a logistic growth system with feedback control and deviating arguments. Ann. Differential Equations, 17(4), 377-384 (2001)
5. Cui, J., Chen, L.: The effect of dispersal on population growth with stage-structure. Computers Math. Applic., 39, 91-102 (2000)
6. Cui, J., Chen, L.: Permanence and extinction in Logistic and Lotka-Volterra systems with diffusion. J. Math. Anal. Appl., 258, 612-535 (2001)
7. Fan, M.: Periodicity, stability and some correlated problem on the nonlinear bio-dynamics, Ph.D degree thesis, Northeast Normal University, 2001
8. Fan, M., Wang, K.: Periodic solutions of single population model with hereditary effects. Mathematica Applicata, 13(2), 58-61 (2000)
9. Fan, M., Wang, K.: Periodicity in a food-limited population model with toxicant and time delays. Acta Mathematicae Applicatae Sinica, English Series, 18(2), 309-314 (2002)
10. Li, X., Fan, M., Wang, K.: Positive periodic solution of single specie model with feedback regulation and infinite delay. Appl. Math. J. Chinese Univ. Ser. A, 17(1), 13-21 (2002)(in Chinese)
11. Li, Y. K.: On a periodic Logistic equation with several delay. Advances in Mathematics, 28, 135-142 (1999)
12. Li, Y., Kuang, Y.: Periodic solutions in periodic state-dependent delay equations and population models. Proceedings of the American Mathematical Society, 130(5), 1345-1353 (2002)
13. Kuang, Y.: Delay Differential Equations with Application in Population Dynamics, 191, in the series of Mathematics in Science and Engineering, Academic Press, Boston, 1993
14. Yan, J., Feng, Q.: Global existence and oscillation in a nonlinear delay equation. Nonlinear Analysis, 43, 101-108 (2001)
15. Fan, G., Ouyang, Z.: Existence of positive periodic solution for a single species model with state dependent delay. Jouanal of Biomathematics, 17(2), 173-178 (2002)
16. Chen, Y.: Periodic solutions of a delayed periodic logistic equation. Applied Mathematics Letters, 16 1047-1051 (2003)
17. Chen, F. D., Shi, J. L.: Periodicity in a Logistic type system with several delays. Computer and Mathematics with Applications, 48(1-2), 35-44 (2004)
18. Chen, F. D. et al.: Periodicity in a food-limited population model with toxicants and state dependent delays. J. Math. Anal. Appl., 288(1), 132-142 (2003)
19. Chen, F. D., Lin, F. X., Chen, X. X.: Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control. Applied Mathematics and Computation, 158(1), 45-68 (2004)
20. Chen, F. D.: Positive periodic solutions of neutral Lotka-Volterra system with feedback control. Applied Mathematics and Computation, 162(3), 1279-1302 (2005)
21. Gui, Z., Chen, L.: Persistence and periodic solutions of a periodic Logistic equation with time delays. Journal of Mathematical Research and Exposition, 23(1), 109-114 (2003) (in Chinese)
22. Mallet-Pareta, J., Nussbaum, R. D.: Boundary layer phenomena for differential-delay equations withstate-dependent time lags: III. J. Differential Equations, 189, 640-692 (2003)
23. Magal, P., Arino, O.: Existence of periodic solutions for a state dependent delay differential equation. Journal of Differential Equations, 165, 61-95 (2000)
24. Arino, O., Hadeler, K. P.: Existence of periodic solutions for delay differential equations with state dependent delay. Journal of Differential Equations, 144, 263-301 (1998)
25. Li, Y. K., Liu, P., Zhu, L. F.: Positive periodic solutions of a class of functional differential systems with feedback controls. Nonlinear Analysis, 57(5-6), 655-666 (2004)
26. Mackey, M. C., Milton, J.: Feedback delays and the origin of blood cell dynamics. Comments Theoret Biol., 1, 299-327 (1990)
27. Longtin, A.: Nonlinear Oscillations, Noise and Chaos in Neural Delayed Feedback, Ph. D. Thesis, McGill University, Montréal, Canada, 1988
28. Longtin, A., Milton, J.: Complex oscillations in the human pupil light re.ex with mixed and delayed feedback. Math. Biosci., 90, 183-199 (1988)
29. Longtin, A., Milton, J.: Modelling autonomous oscillations in the human pupil light re.ex using nonlinear delay-differential equations. Bull. Math. Biol., 51, 605-624 (1989)
30. Longtin, A., Milton, J.: Insight into the transfer function, gain and oscillation onset for the pupil light re.ex using nonlinear delay-differential equations. Biol. Cybernet, 61, 51-58 (1989)
31. Li, Y., Kuang, Y.: Periodic solutions of periodic delay Lotka-Volterra equations and systems. J. Math. Anal. Appl., 255, 260-280 (2001)
32. Li, Y. K.: Periodic solutions for delay Lotka-Volterra competition systems. J. Math. Anal. Appl., 246(1), 230-244 (2000)
33. Fan, G. H., Li, Y. K.: Existence of positive periodic solutions for a periodic logistic equation. Applied Mathematics and Computation., 139(2-3), 311-321 (2003)
34. Weng, P., Jiang, D.: Existence and global stability of positive periodic solution of n-species periodic Lotka-Volterra competition system with feedback control and deviating arguments. Far East J. Math. Sci. (FJMS), 7(1), 45-65 (2002)
35. Fan, M., Wong, Patrica J. Y., Agarwal, Ravi P.: Periodicity and stability in Periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments. Acta Mathematics Sinica, English Series, 19(4), 801-822 (2003)
36. Weng, P.: Existence and global stability of positive periodic solution of periodic integro-differential systems with feedback controls. Comput. Math. Appl., 40(6-7),747-759 (2000)
37. Teng, Z.: On the permanence and extinction in non-autonomous Lotka-Volterra competition systems with delays. Acta Mathematica Sinica, Chinese Series, 44(2), 293-306 (2001)
38. Teng, Z., Chen, L.: Uniform persistence and existence of strictly positive solutions in non-autonomous Lotka-Volterra competition systems with delays. Computers and Mathematics with Applications, 37, 61-71 (1999)
39. Teng, Z.: Non-autonomous Lotka-Volterra competition systems with delays. Journal of Differential Equations, 179, 538-561 (2002)
40. Gaines, R. E., Mawhin, J. L.: Coincidence Degree and Nonlinear Differential Equations, Springer-Verlag, Berlin, 1977