Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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PERMANENCE OF A TWO SPECIES DELAYED COMPETITIVE MODEL WITH STAGE STRUCTURE AND HARVESTING

  • XU, CHANGJIN (GUIZHOU KEY LABORATORY OF ECONOMICS SYSTEM SIMULATION SCHOOL OF MATHEMATICS AND STATISTICS GUIZHOU UNIVERSITY OF FINANCE AND ECONOMICS) ;
  • ZU, YUSEN (SCHOOL OF MATHEMATICS AND STATISTICS HENAN UNIVERSITY OF SCIENCE AND TECHNOLOGY)
  • Received : 2013.07.29
  • Published : 2015.07.31
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Abstract

In this paper, a two species competitive model with stage structure and harvesting is investigated. By using the differential inequality theory, some new sufficient conditions which ensure the permanence of the system are established. Our result supplements the main results of Song and Chen [Global asymptotic stability of a two species competitive system with stage structure and harvesting, Commun. Nonlinear Sci. Numer. Simul. 19 (2001), 81-87].

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References

  1. J. R. Bence and R. M. Nisbet, Space-limited recruitment in open systems: the importance of time delays, Ecology 70 (1989), no. 5, 1434-1441. https://doi.org/10.2307/1938202
  2. A. A. Berryman, The origns and evolution of predator-prey theory, Ecology 73 (1992), no. 5, 1530-1535. https://doi.org/10.2307/1940005
  3. L. M. Cai and X. Y. Song, Permanence and stability of a predator-prey system with stage structure for predator, J. Comput. Appl. Math. 201 (2007), no. 2, 356-366. https://doi.org/10.1016/j.cam.2005.12.035
  4. F. D. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback control, Nonlinear Anal. Real World Appl. 7 (2006), no. 1, 133-143. https://doi.org/10.1016/j.nonrwa.2005.01.006
  5. F. D. Chen, Permanence of a discrete n-species food-chain system with time delays, Appl. Math. Comput. 185 (2007), no. 1, 719-726. https://doi.org/10.1016/j.amc.2006.07.079
  6. F. D. Chen, Partial survival and extinction of a delayed predator-prey model with stage structure, Appl. Math. Comput. 219 (2012), no. 8, 4157-4162. https://doi.org/10.1016/j.amc.2012.10.055
  7. F. D. Chen, W. L. Chen, Y. M. Wu, and Z. Z. Ma, Permanence of a stage-structured predator-prey system, Appl. Math. Comput. 219 (2013), no. 17, 8856-8862. https://doi.org/10.1016/j.amc.2013.03.055
  8. F. D. Chen and M. S. You, Permanence for an integrodifferential model of mutualism, Appl. Math. Comput. 186 (2007), no. 1, 30-34. https://doi.org/10.1016/j.amc.2006.07.085
  9. K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcial. Ekvac. 29 (1986), no. 1, 77-90.
  10. J. M. Cushing, Periodic time-dependent predator-prey system, SIAM J. Appl. Math. 32 (1977), no. 1, 82-95. https://doi.org/10.1137/0132006
  11. J. Dhar and K. S. Jatav, Mathematical analysis of a delayed stage-structured predator-prey model with impulsive diffusion between two predators territories, Ecol. Complex. in Press.
  12. Y. H. Fan and W. T. Li, Permanence for a delayed discrete ratio-dependent predator-prey model with Holling type functional response, J. Math. Anal. Appl. 299 (2004), no. 2, 357-374. https://doi.org/10.1016/j.jmaa.2004.02.061
  13. Z. Y. Hou, On permanence of all subsystems of competitive Lotka-Volterra systems with delays, Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 4285-4301. https://doi.org/10.1016/j.nonrwa.2010.05.015
  14. Z. Y. Hou, On permanence of Lotka-Volterra systems with delays and variable intrinsic growth rates, Nonlinear Anal. Real World Appl. 14 (2013), no. 2, 960-975. https://doi.org/10.1016/j.nonrwa.2012.08.010
  15. H. X. Hu, Z. D. Teng, and H. J. Jiang, On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls, Nonlinear Anal. Real World Appl. 10 (2009), no. 3, 1803-1815. https://doi.org/10.1016/j.nonrwa.2008.02.017
  16. V. A. A. Jansen, R. M. Nisbet, and W. S. C Gurney, Generation cycles in stage structured populations, Bull. Math. Biol. 52 (1990), no. 3, 375-396. https://doi.org/10.1007/BF02458578
  17. X. Y. Liao, S. F. Zhou, and Y. M. Chen, Permanence and global stability in a discrete n-species competition system with feedback controls, Nonlinear Anal. Real World Appl. 9 (2008), no. 4, 1661-1671. https://doi.org/10.1016/j.nonrwa.2007.05.001
  18. S. Q. Liu and L. S. Chen, Necessary-sufficient conditions for permanence and extinction in lotka-volterra system with distributed delay, Appl. Math. Lett. 16 (2003), no. 6, 911-917. https://doi.org/10.1016/S0893-9659(03)90016-4
  19. S. Q. Liu, L. S. Chen, and Z. J. Liu, Extinction and permanence in nonautonomous competitive system with stage structure, J. Math. Anal. Appl. 274 (2002), no. 2, 667-684. https://doi.org/10.1016/S0022-247X(02)00329-3
  20. X. Lv, S. P. Lu, and P. Lu, Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments, Nonlinear Anal. Real World Appl. 11 (2010), no. 1, 574-583. https://doi.org/10.1016/j.nonrwa.2009.09.004
  21. Z. H. Ma, Z. Z. Li, S. F. Wang, T. Li, and F. P. Zhang, Permanence of a predator-prey system with stage structure and time delay, Appl. Math. Comput. 201 (2008), no. 1-2, 65-71. https://doi.org/10.1016/j.amc.2007.11.050
  22. F. Montes de Oca and M. Vivas, Extinction in a two dimensional Lotka-Volterra sysstem with infinite delay, Nonlinear Anal. Real World Appl. 7 (2006), no. 5, 1042-1047. https://doi.org/10.1016/j.nonrwa.2005.09.005
  23. Y. Muroya, Permanence and global stability in a Lotka-Volterra predator-prey system with delays, Appl. Math. Lett. 16 (2003), no. 8, 1245-1250. https://doi.org/10.1016/S0893-9659(03)90124-8
  24. R. M. Nisbet, W. S. C. Gurney, and J. A. J. Metz, Stage structure models applied in evolutionary ecology, in Applied Mathematical Ecology, Biomathematics, pp. 428-449, Springer, Berlin, Germany, 1989.
  25. X. Y. Song and L. S. Chen, Global asymptotic stability of a two species competitive system with stage structure and harvesting, Commun. Nonlinear Sci. Numer. Simul. 6 (2001), no. 2, 81-87. https://doi.org/10.1016/S1007-5704(01)90006-1
  26. Z. D. Teng, Y. Zhang, and S. J. Gao, Permanence criteria for general delayed discrete nonautonomous-species Kolmogorov systems and its applications, Comput. Math. Appl. 59 (2010), no. 2, 812-828. https://doi.org/10.1016/j.camwa.2009.10.011
  27. J. Y.Wang, Q. S. Lu, and Z. S. Feng, A nonautonomous predator-prey system with stage structure and double time delays, J. Comput. Appl. Math. 230 (2009), no. 1, 283-299. https://doi.org/10.1016/j.cam.2008.11.014
  28. R. Xu, L. S. Chen, and F. L. Hao, Periodic solutions of an n-species Lotka-Volterra type food-chain model with time delays, Appl. Math. Comput. 171 (2005), no. 1, 511-530. https://doi.org/10.1016/j.amc.2005.01.067
  29. J. D. Zhao and J. F. Jiang, Average conditons for permanence and extinction in nonantonomous Lotka-Volterra system, J. Math. Anal. Appl. 299 (2004), no. 2, 663-675. https://doi.org/10.1016/j.jmaa.2004.06.019

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