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On the existence of positive periodic solutions to a Lotka-Volterra cooperative population model with multiple delays. (English) Zbl 1139.34317
Summary: The author studies the existence of positive periodic solutions to a Lotka-Volterra cooperative population model with multiple delays as follows $$\multline x_i'(t)=x_i(t)\left[r_i(t)-p_{ii}(t)x_i(t)+\sum^n_{k\ne i}p_{ik}(t)x_k(t)-\sum^n_{j=1}q_{ij}(t)x_i(t-\tau_{ij}(t))+\right.\\ \left.\sum^n_{k\ne i}\sum^n_{j=1}c_{ikj}(t)x_k(t-\gamma_{ikj}(t))\right],\quad (i=1,2,\dots,n).\endmultline$$ By using Mawhin’s continuation theorem of coincidence degree principle, a new result is obtained.

##### MSC:
 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 34K60 Qualitative investigation and simulation of models
##### Keywords:
Mawhin’s continuation theorem
Full Text:
##### References:
 [1] Fang, Meng; Wang, Ke; Jiang, Daqing: Existence and global attractivity of positive periodic solutions of periodic n-species Lotka--Volterra competition systems with several deviating arguments, Math. biosci. 160, 47-61 (1999) · Zbl 0964.34059 · doi:10.1016/S0025-5564(99)00022-X [2] Chen, Fengde; Xie, Xiangdong; Shi, Jinlin: Existence, uniqueness and stability of positive periodic solution for a nonlinear prey-competition model with delays, J. comput. Appl. math. 194, No. 2, 368-387 (2006) · Zbl 1104.34050 · doi:10.1016/j.cam.2005.08.005 [3] Li, Yongkun: Periodic solutions for delay Lotka--Volterra competition systems, J. math. Anal. appl. 246, 230-244 (2000) · Zbl 0972.34057 · doi:10.1006/jmaa.2000.6784 [4] Gopalsamy, K.: Global asymptotic stability in a periodic Lotka--Volterra system, J. aust. Math. soc. Ser B. 24, 160-170 (1982) · Zbl 0498.92016 [5] Tang, Xianhua; Zou, Xingfu: On positive periodic solutions of Lotka--Volterra competition systems with deviating arguments, Proc. amer. Math. soc. 134, 2967-2974 (2006) · Zbl 1101.34056 · doi:10.1090/S0002-9939-06-08320-1 [6] Chen, F. D.; Shi, J. L.; Chen, X. X.: Periodicity in Lotka--Volterra facultative mutualism system with several delays, J. engrg. Math. 21, No. 3, 403-409 (2004) [7] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002 [8] Gaines, R. E.; Mawhin, J. L.: Lecture notes in mathematics, Lecture notes in mathematics 568 (1977) [9] Lu, Shiping: On the existence of positive periodic solutions for neutral functional differential equation with multiple deviating arguments, J. math. Anal. appl. 280, No. 2, 321-333 (2003) · Zbl 1034.34084 · doi:10.1016/S0022-247X(03)00049-0