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Existence and global attractivity of positive periodic solutions of Lotka-Volterra predator-prey systems with deviating arguments. (English) Zbl 1186.34118

Summary: The model discussed in this paper is described by the following periodic 3-species Lotka-Volterra predator-prey system with several deviating arguments:

x 1 ' (t)=x 1 (t)(r 1 (t)-a 11 (t)x 1 (t-τ 11 (t))-a 12 (t)x 2 (t-τ 12 (t))-a 13 (t)x 3 (t-τ 13 (t)))x 2 ' (t)=x 2 (t)(r 2 (t)-a 21 (t)x 1 (t-τ 21 (t))-a 22 (t)x 2 (t-τ 22 (t))-a 23 (t)x 3 (t-τ 23 (t)))x 3 ' (t)=x 3 (t)(r 2 (t)-a 31 (t)x 1 (t-τ 31 (t))-a 32 (t)x 2 (t-τ 32 (t))-a 33 (t)x 3 (t-τ 33 (t)))(*)

where x 1 (t) denotes the density of prey species at time t, x 2 (t) and x 3 (t) denote the density of predator species at time t, r i ,a ij C(,[0,)) and τ ij C(,) are w-periodic functions with

r ¯ i =1 w 0 w r i (s)ds>0;a ¯ ij =1 w 0 w a ij (s)>0,I,j=1,2,3·

By using Krasnoselskii’s fixed point theorem and the construction of Lyapunov function, a set of easily verifiable sufficient conditions are derived for the existence and global attractivity of positive periodic solutions of (*).

MSC:
34K60Qualitative investigation and simulation of models
34K20Stability theory of functional-differential equations
34K13Periodic solutions of functional differential equations
92D25Population dynamics (general)
References:
[1]Alvarez, C.; Lazer, A. C.: An application of topological degree to the periodic competing species model, J. aust. Math. soc. Ser. B 28, 202-219 (1986) · Zbl 0625.92018 · doi:10.1017/S0334270000005300
[2]Ahmad, S.: On the nonautoomous Lotka–Volterra competition equations, Proc. amer. Math. soc. 117, 199-204 (1993) · Zbl 0848.34033 · doi:10.2307/2159717
[3]Battaaz, A.; Zanolin, F.: Coexistence states for periodic competition Kolmogorov systems, J. math. Anal. appl. 219, 179-199 (1998) · Zbl 0911.34037 · doi:10.1006/jmaa.1997.5726
[4]Chen, Y.; Zhou, Z.: Stable periodic solution of a discrete periodic Lotka–Volterra competition system, J. math. Anal. appl. 277, 358-366 (2003) · Zbl 1019.39004 · doi:10.1016/S0022-247X(02)00611-X
[5]Cushing, J. M.: Two species competition in a periodic environment, J. math. Biol. 10, 385-400 (1980) · Zbl 0455.92012 · doi:10.1007/BF00276097
[6]Fan, M.; Wang, K.: Global periodic solutions of a generalized n-species gilpn–ayala competition model, Comput. math. Appl. 40, 1141-1151 (2000) · Zbl 0954.92027 · doi:10.1016/S0898-1221(00)00228-5
[7]Fan, M.; Wang, K.; Jiang, D. Q.: Existence and global attractivity of positive peridic solutions of 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
[8]Gopalsamy, K.: Global asymptotical stability in a periodic Lotka–Volterra system, J. aust. Math. soc. Ser. B 29, 66-72 (1985) · Zbl 0588.92019 · doi:10.1017/S0334270000004768
[9]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[10]Li, Y. K.: Periodic solutions of N-species competition system with delays, J. biomath. 12, 1-12 (1997) · Zbl 0891.92027
[11]Li, Y. K.: On a periodic delay logistic type population model, Ann. differential equations 14, 29-36 (1998) · Zbl 0966.34067
[12]Li, Y. K.: Periodic solutions for delay Lotka–Volterra competition systems, J. math. Anal. appl. 255, 260-280 (2001)
[13]Li, Y. K.; Kuang, Y.: Periodic solutions of periodic delay Lotka–Volterra equations and systems, J. math. Anal. appl. 255, 260-280 (2001) · Zbl 1024.34062 · doi:10.1006/jmaa.2000.7248
[14]Tang, H. X.; Zou, F. X.: 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
[15]Tang, X. H.; Zou, X. F.: Global attractivity of positive periodic solution to periodic Lotka–Volterra competition systems with pure delay, J. differential equations 228, 580-610 (2006) · Zbl 1113.34052 · doi:10.1016/j.jde.2006.06.007
[16]Krasnoselskii, M. A.: Positive solutions of operator equations, noordhoff, Groningen, (1964) · Zbl 0121.10604
[17]Roydin, H. L.: Real analysis, (1998)