# zbMATH — the first resource for mathematics

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 to ordinary differential equations 37N25 Dynamical systems in biology
##### Keywords:
Lotka-Volterra system; global attractivity
Full Text:
##### References:
 [1] Lansun, C., Mathematical models and methods in ecology, (1988), Science Press Beijing [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.