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Global attractivity of the periodic Lotka-Volterra system. (English) Zbl 0973.92039
Summary: A Lotka-Volterra periodic model with $m$-predators and $n$-preys is studied. A set of easily verifiable sufficient conditions that guarantee the existence, uniqueness and global attractivity of the positive periodic solutions is obtained. Finally, a suitable example is given to illustrate that the conditions of the main theorem are feasible.

##### MSC:
 92D40 Ecology 34C25 Periodic solutions of ODE 37N25 Dynamical systems in biology
##### Keywords:
Lotka-Volterra system; global attractivity
Full Text:
##### References:
 [1] Lansun, C.: Mathematical models and methods in ecology. (1988) [2] De Mottoni, P.; Schiaffino, A.: Competition system with periodic coefficients: A geometric approach. J. math. Biol. 11, 319-335 (1981) · Zbl 0474.92015 [3] Cushing, J. M.: Two species competition in a periodic environment. J. math. Biol. 10, 385-400 (1980) · Zbl 0455.92012 [4] Cushing, J. M.: Periodic Lotka--Volterra competition equations. J. math. Biol. 24, 381-403 (1986) · Zbl 0608.92019 [5] Ahmad, S.: Convergence and ultimate bounds of solutions of the nonautonomous Volterra--Lotka competition equations. J. math. Anal. appl. 127, 377-387 (1987) · Zbl 0648.34037 [6] Ahmad, S.: On almost periodic solutions of the competing species problems. Proc. amer. Math. soc. 102, 855-865 (1988) · Zbl 0668.34042 [7] Ahmad, S.: On the nonautonomous Volterra--Lotka competition equations. Proc. amer. Math. soc. 177, 199-204 (1993) · Zbl 0848.34033 [8] Gopalsamy, K.: Exchange of equilibria in two species Lotka--Volterra competition models. J. austral. Math. soc. Ser. B 24, 160-170 (1982) · Zbl 0498.92016 [9] Gopalsamy, K.: Global asymptotic stability in a periodic Lotka--Volterra system. J. austral. Math. soc. Ser. B 27, 66-72 (1985) · Zbl 0588.92019 [10] Gopalsamy, K.: Global asymptotic stability in an almost periodic Lotka--Volterra system. J. austral. Math. soc. Ser. B 27, 346-360 (1986) · Zbl 0591.92022 [11] Alvarz, C.; Lazer, A. C.: An application of topological degree to the periodic competing species problem. J. austral. Math. soc. Ser. B 28, 202-219 (1986) · Zbl 0625.92018 [12] Tineo, A.; Alvarez, C.: A defferent consideration about the globally asymptotically stable solution of the periodic n-competing species problem. J. math. Anal. appl. 159, 44-50 (1991) · Zbl 0729.92025 [13] Zhien, M.; Wendi, W.: Asymptotic behavior of predator--prey system with time dependent coefficients. Appl. anal. 34, 79-90 (1989) · Zbl 0658.34044 [14] Zhonghua, L.; Lansu, C.: Global asymptotic stability of the periodic Lotka--Volterra system with two-predator and one-prey. Appl. math. J. chinese univ. Ser. B. 10, 267-274 (1995) · Zbl 0840.34036