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Average conditions for permanence and extinction in nonautonomous Gilpin-Ayala competition model. (English) Zbl 1119.34038
Let the functions $$b_{i}(t)$$ and $$a_{ij}(t)$$ be continuous on $$[c,\infty )$$ and bounded above and below by strictly positive constants and let $$d_{ij}$$ be positive numbers; $$i,j=1,2,\ldots ,n$$. The author studies the $$n$$-species M. E. Gilpin, F. J. Ayala competitive system [Proc. Natl. Acad. Sci. USA 70, 3590-3593 (1973; Zbl 0272.92016)] $x'_{i}(t) = x_{i}(t) \left\{ b_{i}(t)-\sum ^{n}_{j=1} a_{ij}(t) (x_{j}(t)) ^{d_{ij}} \right\} . \eqno{(1)}$ The system (1) is called permanent if for any positive solution $$X(t)$$ with components $$x_{i}(t)$$, there exist positive constants $$\lambda _{i}$$, $$k_{i}$$, $$T$$ such that for $$t\geq T$$ one has $$\lambda _{i}\leq x_{i}(t)\leq k_{i}$$. It is called globally attractive if the components of any two positive solutions $$X(t)$$, $$Y(t)$$ satisfy $\lim _{t\rightarrow \infty } | x_{i}(t)-y_{i}(t)| = 0 .$ Under some “average conditions” as introduced by S. Ahmad and A. C. Lazer [Nonlinear Anal., Theory Methods Appl. 40, No. 1–8(A), 37–49 (2000; Zbl 0955.34041)] the author proves the permanency and global attractiveness of the system (1). Further, similar results are obtained for the subsets $$x_{i}(t)$$, $$i=1,2,\ldots ,r$$, with $$r<n$$, to the effect that $$r$$ of the species in the system are permanent whereas the remaining $$n-r$$ ones are exposed to extinction. Inasmuch as system (1) is for $$d_{ij} = 1$$ reduced to the Lotka-Volterra one, the results obtained here generalize those of several authors for the later system.

MSC:
 34D05 Asymptotic properties of solutions to ordinary differential equations 92D25 Population dynamics (general) 34C60 Qualitative investigation and simulation of ordinary differential equation models
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References:
 [1] Ahmad, S., On the nonautonomous volterra – lotka competition equations, Proc. amer. math. soc., 117, 199-204, (1993) · Zbl 0848.34033 [2] Ahmad, S., Extinction of species in nonautonomous lotka – volterra systems, Proc. amer. math. soc., 127, 2905-2910, (1999) · Zbl 0924.34040 [3] Ahmad, S.; Lazer, A.C., Average conditions for global asymptotic stability in a nonautonomous lotka – volterra system, Nonlinear anal., 40, 37-49, (2000) · Zbl 0955.34041 [4] Ahmad, S.; Montes de Oca, F., Extinction in nonautonomous T-periodic competitive lotka – volterra system, Appl. math. comput., 90, 155-166, (1998) · Zbl 0906.92024 [5] Ayala, F.J.; Gilpin, M.E.; Eherenfeld, J.G., Competition between species: theoretical models and experimental tests, Theor. popul. biol., 4, 331-356, (1973) [6] Chen, Y., New results on positive periodic solutions of a periodic integro-differential competition system, Appl. math. comput., 153, 2, 557-565, (2004) · Zbl 1051.45004 [7] Chen, F.D., Persistence and periodic orbits for two-species non-autonomous diffusion lotka – volterra models, Appl. math. J. Chinese univ. ser. B., 19, 4, 359-366, (2004) · Zbl 1074.34053 [8] Chen, F.D., On a nonlinear non-autonomous predator – prey model with diffusion and distributed delay, J. comput. appl. math., 80, 1, 33-49, (2005) · Zbl 1061.92058 [9] Chen, F.D., Periodicity in a nonlinear predator – prey system with state dependent delays, Acta mathematicae applicatae sinica, English series, 21, 1, 49-60, (2005) · Zbl 1096.34050 [10] Chen, F.D., Positive periodic solutions of neutral lotka – volterra system with feedback control, Appl. math. comput., 162, 3, 1279-1302, (2005) · Zbl 1125.93031 [11] F.D. Chen, The permanence and global attractivity of Lotka-Volterra competition system with feedback controls, Nonlinear Anal.: Real World Appl., in press. · Zbl 1103.34038 [12] F.D. Chen, On a periodic multi-species ecological model, Appl. Math. Comput., in press. · Zbl 1080.92059 [13] Chen, F.D.; Lin, S.J., Periodicity in a logistic type system with several delays, Comput. math. appl., 48, 1-2, 35-44, (2004) · Zbl 1061.34050 [14] Chen, F.D.; Lin, F.X.; Chen, X.X., Sufficient conditions for the existence of positive periodic solutions of a class of neutral delay models with feedback control, Appl. math. comput., 158, 1, 45-68, (2004) · Zbl 1096.93017 [15] Chen, F.D.; Sun, D.X.; Lin, F.X., Periodicity in a food-limited population model with toxicants and state dependent delays, J. math. anal. appl., 288, 1, 132-142, (2003) [16] Fan, M.; Wang, K., Global periodic solutions of a generalized n-species gilpin – ayala competition model, Comput. math. appl., 40, 1141-1151, (2000) · Zbl 0954.92027 [17] Gilpin, M.E.; Ayala, F.J., Global models of growth and competition, Proc. nat. acad. sci. USA, 70, 3590-3593, (1973) · Zbl 0272.92016 [18] Goh, B.S.; Agnew, T.T., Stability in gilpin and Ayala’s model of competition, J. math. biol., 4, 275-279, (1977) · Zbl 0379.92017 [19] Huo, H.F.; Li, W.T., Periodic solutions of a periodic lotka – volterra system with delay, Appl. math. comput., 156, 3, 787-803, (2004) · Zbl 1069.34099 [20] Li, Y.K.; Kuang, Y., Periodic solutions of periodic delay lotka – volterra equations and systems, J. math. anal. appl., 255, 1, 260-280, (2001) · Zbl 1024.34062 [21] Liao, X.X.; Li, J., Stability in gilpin – ayala competition models with diffusion, Nonlinear anal. TMA, 28, 1751-1758, (1997) · Zbl 0872.35054 [22] Lisena, B., Global stability in periodic competitive systems, Nonlinear anal.: real world appl., 5, 4, 613-627, (2004) · Zbl 1089.34044 [23] Montes de Oca, F.; Zeeman, M.L., Balancing survival and extinction in nonautonomous competitive lotka – volterra systems, J. math. anal. appl., 192, 360-370, (1995) · Zbl 0830.34039 [24] Montes de Oca, F.; Zeeman, M.L., Extinction in nonautonomous competitive lotka – volterra systems, Proc. amer. math. soc., 124, 3677-3687, (1996) · Zbl 0866.34029 [25] Muroya, Y., Boundedness and partial survival of species in nonautonomous lotka – volterra systems, Nonlinear anal.: real world appl., 6, 2, 263-272, (2005) · Zbl 1077.34077 [26] Teng, Z., On the nonautonomous lotka – volterra N-species competing systems, Appl. math. comput., 114, 175-185, (2000) · Zbl 1016.92045 [27] Teng, Z., The almost periodic komogorov competitive systems, Nonlinear anal., 42, 1221-1230, (2000) · Zbl 1135.34319 [28] Tineo, A., On the asymptotic behavior of some population models, J. math. anal. appl., 167, 516-529, (1992) · Zbl 0778.92018 [29] Tineo, A., An iterative scheme for the N-competing species problem, J. differential equations, 116, 1-15, (1995) · Zbl 0823.34048 [30] Tineo, A.; Alvarz, C., A different consideration about the globally asymptotically stable solution of the periodic n-competing species problem, J. math. anal. appl., 159, 44-55, (1991) · Zbl 0729.92025 [31] Zhao, J.D.; Chen, W.C., The qualitative analysis of N-species nonlinear prey-competition systems, Appl. math. comput., 149, 567-576, (2004) · Zbl 1045.92038 [32] Zhao, J.D.; Jiang, J.F., Permanence in nonautonomous lotka – volterra system with predator – prey, Appl. math. comput., 152, 99-109, (2004) · Zbl 1047.92050 [33] Zhao, J.D.; Jiang, J.F., Average conditions for permanence and extinction in nonautonomous lotka – volterra system, J. math. anal. appl., 299, 663-675, (2004) · Zbl 1066.34050 [34] Zhao, J.D.; Jiang, J.F.; Lazer, A.C., The permanence and global attractivity in a nonautonomous lotka – volterra system, Nonlinear anal.: real world appl., 5, 4, 265-276, (2004) · Zbl 1085.34040
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