The existence of eight positive periodic solutions for a generalized prey-predator system with delay and stocking
Authors:
Zhengqiu Zhang and Xinsheng Xiong
Journal:
Quart. Appl. Math. 65 (2007), 317-337
MSC (2000):
Primary 34K13, 34K60
DOI:
https://doi.org/10.1090/S0033-569X-07-01056-6
Published electronically:
March 7, 2007
MathSciNet review:
2330560
Full-text PDF Free Access
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Additional Information
Abstract: By means of using Mawhin’s continuation theorem of coincidence degree theory, we derive criteria for the existence of eight positive periodic solutions of a generalized prey-predator system with time delay and stocking.
References
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- H. I. Freedman and Paul Waltman, Persistence in models of three interacting predator-prey populations, Math. Biosci. 68 (1984), no. 2, 213–231. MR 738903, DOI https://doi.org/10.1016/0025-5564%2884%2990032-4
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- Zhengqiu Zhang and Zhicheng Wang, The existence of a periodic solution for a generalized prey-predator system with delay, Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 2, 475–486. MR 2092072, DOI https://doi.org/10.1017/S0305004103007527
References
- L. Chen and J. Chen, Nonlinear Dynamical System in Biology (Science Press, Beijing, 1993).
- M. Farkas and H. J. Freedman, The stable coexistence of competing species an an renewable resource, J. Math. Anal. Appl. 138(1989), 401-472.
- H. I. Freedman and S. Ruan, Uniform persistence in functional differential equations, J. Differential. Equations 115(1995), 173-192. MR 1308612 (96e:34115)
- H. I. Freedman and P. Waltman, Persistence in three interacting predator-prey populations, Math. Biosci. 88(1984), 213-231. MR 738903 (85h:92037)
- R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differetial Equation (Springer-Verlag, 1977). MR 0637067 (58:30551)
- J. L. Mawhin, Topological Degree Methods in Nonlinear Boundary Value problems. CBMS Reg. Conf. Ser. Math. 40 (Amer, Math. Soc., Providence, 1979). MR 525202 (80c:47055)
- D. Mukherjee and A. B. Roy, Uniform persistence and global attractivity theorem for generalized prey-predator system with time delay, J. Nonlin Anal. TMA 38 (1999), 59-74. MR 1693012 (2000e:92031)
- Z. Q. Zhang, Z. C. Wang, The existence of a periodic solution for a generalized prey-predator system with delay, Math. Proc Camb Phil. Soc. 137 (2004), 475. MR 2092072 (2005e:34221)
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Additional Information
Zhengqiu Zhang
Affiliation:
School of Mathematics and Computing Technology, Central South University, Changsha, 410083, People’s Republic of China
Email:
z_q_zhang@sina.com.cn
Xinsheng Xiong
Affiliation:
Department of Applied Mathematics, Hunan University, Changsha, 410082, People’s Republic of China
Email:
xxs82@163.com
Keywords:
Eight positive periodic solutions,
a generalized prey-predator system,
delay and stocking,
the continuation theorem of coincidence degree theory,
topological degree theory.
Received by editor(s):
February 3, 2006
Published electronically:
March 7, 2007
Additional Notes:
This project was supported by NNSF of China (10271044) and by Chinese postdoctoral science fund (20060400267).
Article copyright:
© Copyright 2007
Brown University
