Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control

Jiabo Xu; Zhidong Teng; Shujing Gao

Applications of Mathematics (2011)

  • Volume: 56, Issue: 2, page 207-225
  • ISSN: 0862-7940

Abstract

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A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.

How to cite

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Xu, Jiabo, Teng, Zhidong, and Gao, Shujing. "Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control." Applications of Mathematics 56.2 (2011): 207-225. <http://eudml.org/doc/116521>.

@article{Xu2011,
abstract = {A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.},
author = {Xu, Jiabo, Teng, Zhidong, Gao, Shujing},
journal = {Applications of Mathematics},
keywords = {discrete system; permanence; extinction; feedback control; time-varying delay},
language = {eng},
number = {2},
pages = {207-225},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control},
url = {http://eudml.org/doc/116521},
volume = {56},
year = {2011},
}

TY - JOUR
AU - Xu, Jiabo
AU - Teng, Zhidong
AU - Gao, Shujing
TI - Almost sufficient and necessary conditions for permanence and extinction of nonautonomous discrete logistic systems with time-varying delays and feedback control
JO - Applications of Mathematics
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 207
EP - 225
AB - A class of nonautonomous discrete logistic single-species systems with time-varying pure-delays and feedback control is studied. By introducing a new research method, almost sufficient and necessary conditions for the permanence and extinction of species are obtained. Particularly, when the system degenerates into a periodic system, sufficient and necessary conditions on the permanence and extinction of species are obtained. Moreover, a very important fact is found in our results, that is, the feedback control and delays are harmless for the permanence and extinction of species for discrete single-species systems. This shows that in a discrete single-species system introducing the feedback control to factitiously control the permanence and extinction of species is useless.
LA - eng
KW - discrete system; permanence; extinction; feedback control; time-varying delay
UR - http://eudml.org/doc/116521
ER -

References

top
  1. Agarwal, R. P., Difference Equations and Inequalities: Theory, Methods, and Applications, Marcel Dekker New York (1992). (1992) Zbl0925.39001MR1155840
  2. Agarwal, R. P., Li, W.-T., Pang, P. Y. H., 10.1080/1023619021000000735, J. Difference Equ. Appl. 8 (2002), 719-728. (2002) Zbl1010.39003MR1914599DOI10.1080/1023619021000000735
  3. Bohner, M., Fan, M., Zhang, J., 10.1016/j.nonrwa.2005.11.002, Nonlinear Anal., Real World Appl. 7 (2006), 1193-1204. (2006) Zbl1104.92057MR2260908DOI10.1016/j.nonrwa.2005.11.002
  4. Braverman, E., 10.1016/j.na.2004.12.015, Nonlinear Anal., Theory Methods Appl., Ser. A 63 (2005), 751-759 (Electronic only). (2005) Zbl1222.92060DOI10.1016/j.na.2004.12.015
  5. Braverman, E., Saker, S. H., 10.1016/j.na.2006.09.056, Nonlinear Anal., Theory Methods Appl. 67 (2007), 2955-2965. (2007) Zbl1125.39002MR2348007DOI10.1016/j.na.2006.09.056
  6. Chen, F., Permanence in a discrete Lotka-Volterra competition model with deviating arguments, Nonlinear Anal., Real World Appl. 9 (2008), 2150-2155. (2008) Zbl1156.39300MR2441771
  7. Chen, F., 10.1016/j.aml.2006.08.023, Appl. Math. Lett. 20 (2007), 729-733. (2007) Zbl1128.92029MR2314699DOI10.1016/j.aml.2006.08.023
  8. Chen, F., Wu, L., Li, Z., 10.1016/j.camwa.2006.12.015, Comput. Math. Appl. 53 (2007), 1214-1227. (2007) Zbl1127.92038MR2327675DOI10.1016/j.camwa.2006.12.015
  9. Chen, X., Chen, F., 10.1016/j.amc.2006.02.039, Appl. Math. Comput. 181 (2006), 1446-1454. (2006) Zbl1106.39003MR2270776DOI10.1016/j.amc.2006.02.039
  10. Chen, Y., Zhou, Z., 10.1016/S0022-247X(02)00611-X, J. Math. Anal. Appl. 277 (2003), 358-366. (2003) Zbl1019.39004MR1954481DOI10.1016/S0022-247X(02)00611-X
  11. Douraki, M. J., Mashreghi, J., 10.1080/10236190701466504, J. Difference Equ. Appl. 14 (2008), 231-257. (2008) Zbl1135.92026MR2397323DOI10.1080/10236190701466504
  12. Emmert, K. E., Allen, L. J. S., 10.1080/10236190410001654151, J. Difference Equ. Appl. 10 (2004), 1177-1199. (2004) Zbl1067.92052MR2100721DOI10.1080/10236190410001654151
  13. Fan, M., Wang, Q., 10.3934/dcdsb.2004.4.563, Disc. Cont. Dyn. Syst., Ser. B 4 (2004), 563-574. (2004) Zbl1100.92064MR2073960DOI10.3934/dcdsb.2004.4.563
  14. Fan, Y.-H., Li, W.-T., 10.1016/j.jmaa.2004.02.061, J. Math. Anal. Appl. 299 (2004), 357-374. (2004) Zbl1063.39013MR2098248DOI10.1016/j.jmaa.2004.02.061
  15. Giang, D. V., Huong, D. C., 10.1016/j.jmaa.2004.11.035, J. Math. Anal. Appl. 305 (2005), 291-295. (2005) Zbl1096.92034MR2128128DOI10.1016/j.jmaa.2004.11.035
  16. Győri, I., Trofimchuk, S. I., 10.1080/10236190008808250, J. Difference Equ. Appl. 6 (2000), 647-665. (2000) MR1813210DOI10.1080/10236190008808250
  17. Huo, H.-F., Li, W.-T., 10.1016/j.aml.2004.07.002, Appl. Math. Letters 17 (2004), 1007-1013. (2004) Zbl1067.39009MR2087748DOI10.1016/j.aml.2004.07.002
  18. Kocic, V. L., Ladas, G., Global Behavior of Nonlinear Difference Equations of Higher Order with Application, Kluwer Academic Publishers Dordrecht (1993). (1993) MR1247956
  19. Kon, R., 10.1007/s00285-003-0224-8, J. Math. Biol. 48 (2004), 57-81. (2004) Zbl1050.92045MR2035520DOI10.1007/s00285-003-0224-8
  20. Li, Y. K., Zhu, L. F., 10.1016/j.aml.2004.09.002, Appl. Math. Lett. 18 (2005), 61-67. (2005) Zbl1085.39009MR2121555DOI10.1016/j.aml.2004.09.002
  21. Liao, X., Ouyang, Z., Zhou, S., 10.1016/j.cam.2006.10.084, J. Comput. Appl. Math. 211 (2008), 1-10. (2008) Zbl1143.39005MR2386823DOI10.1016/j.cam.2006.10.084
  22. Liao, X., Zhou, S., Chen, Y., Permanence and global stability in a discrete n -species competition system with feedback controls, Nonlinear Anal., Real World Appl. 9 (2008), 1661-1671. (2008) Zbl1154.34352MR2422571
  23. Liao, L., Yu, J., Wang, L., 10.1016/S0898-1221(02)00265-1, Comput. Math. Appl. 44 (2002), 1403-1411. (2002) Zbl1043.93053MR1938776DOI10.1016/S0898-1221(02)00265-1
  24. Liu, Z., Chen, L., 10.1016/j.cam.2005.09.023, J. Comput. Appl. Math. 197 (2006), 446-456. (2006) Zbl1098.92066MR2260418DOI10.1016/j.cam.2005.09.023
  25. Liz, E., 10.1016/j.jmaa.2006.06.030, J. Math. Anal. Appl. 330 (2007), 740-743. (2007) Zbl1108.92033MR2302956DOI10.1016/j.jmaa.2006.06.030
  26. Liz, E., Tkachenko, V., Trofimchuk, S., 10.1016/j.mbs.2005.03.016, Math. Biosci. 199 (2006), 26-37. (2006) Zbl1086.92045MR2205557DOI10.1016/j.mbs.2005.03.016
  27. Lu, Z., Wang, W., 10.1007/s002850050171, J. Math. Biol. 39 (1999), 269-282. (1999) Zbl0945.92022MR1716315DOI10.1007/s002850050171
  28. Merdan, H., Duman, O., 10.1016/j.chaos.2007.08.081, Chaos Solitons Fractals 40 (2009), 1169-1175. (2009) Zbl1197.39009MR2526104DOI10.1016/j.chaos.2007.08.081
  29. Muroya, Y., 10.1016/j.jmaa.2006.07.070, J. Math. Anal. Appl. 330 (2007), 24-33. (2007) Zbl1124.39011MR2298155DOI10.1016/j.jmaa.2006.07.070
  30. Muroya, Y., 10.1016/j.jmaa.2003.12.033, J. Math. Anal. Appl. 293 (2004), 446-461. (2004) Zbl1059.39007MR2053890DOI10.1016/j.jmaa.2003.12.033
  31. Papaschinopoulos, G., Schinas, C. J., 10.1080/10236199808808146, J. Difference Equ. Appl. 4 (1998), 315-323. (1998) Zbl0913.39009MR1657224DOI10.1080/10236199808808146
  32. Sadhukhan, D., Mondal, B., Maiti, M., 10.1016/j.amc.2007.12.063, Appl. Math. Comput. 201 (2008), 631-639. (2008) Zbl1143.92041MR2431960DOI10.1016/j.amc.2007.12.063
  33. Saito, Y., Ma, W., Hara, T., 10.1006/jmaa.2000.7303, J. Math. Anal. Appl. 256 (2001), 162-174. (2001) Zbl0976.92031MR1820074DOI10.1006/jmaa.2000.7303
  34. Saker, S. H., 10.1016/j.mcm.2007.04.007, Math. Comput. Modelling 47 (2008), 278-297. (2008) Zbl1130.92044MR2378835DOI10.1016/j.mcm.2007.04.007
  35. Teng, Z., 10.1080/14689360110102312, Dyn. Syst. 17 (2002), 187-202. (2002) Zbl1035.34086MR1927807DOI10.1080/14689360110102312
  36. Wang, W., Mulone, G., Salemi, F., Salone, V., 10.1006/jmaa.2001.7666, J. Math. Anal. Appl. 264 (2001), 147-167. (2001) Zbl1006.92025MR1868334DOI10.1006/jmaa.2001.7666
  37. Xia, Y., Cao, J., Lin, M., Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Anal., Real World Appl. 8 (2007), 1079-1095. (2007) Zbl1127.39038MR2331428
  38. Xiong, X., Zhang, Z., 10.1016/j.mcm.2007.10.004, Math. Comput. Modelling 48 (2008), 333-343. (2008) Zbl1145.34334MR2431471DOI10.1016/j.mcm.2007.10.004
  39. Yang, X., 10.1016/j.jmaa.2005.04.036, J. Math. Anal. Appl. 316 (2006), 161-177. (2006) Zbl1107.39017MR2201755DOI10.1016/j.jmaa.2005.04.036

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