Permanence and asymptotic behavior of the \(N\)-species nonautonomous Lotka-Volterra competitive systems. (English) Zbl 0959.34039

Teng, Zhidong; Li, Zhiming

Comput. Math. Appl. 39, No. 7-8, 107-116 (2000).

The authors study the permanence and global asymptotic behavior for the \(n\)-species Lotka-Volterra competitive systems \[ {dx_i\over dt}=x_i \left(b_i(t)- \sum^n_{j=1} a_{ij}(t)x_j\right), \quad i=1,2, \dots,n, \] where all parameters are time dependent and asymptotically approximate periodic functions, respectively. The authors obtain sufficient conditions for the permanence and global asymptotic stability of the system. They extend related results for \(n=2\) by Q. Peng and L.-S. Chen [Comput. Math. Appl. 27, No. 12, 53-60 (1994; Zbl 0798.92023)].

Reviewer: Chen Lan Sun (Beijing)

MSC:

34D05	Asymptotic properties of solutions to ordinary differential equations
92D25	Population dynamics (general)
34D23	Global stability of solutions to ordinary differential equations

Keywords:

nonautonomous system; Lotka-Volterra competitive system; global asymptotic system

Citations:

Zbl 0798.92023

References:

[1]	Peng, Q.L.; Chen, L.S., Asymptotic behavior of the nonautonomous two-species Lotka-Volterra competition models, Computers math. applic., 27, 12, 53-60, (1994) · Zbl 0798.92023
[2]	Ahmad, S., On the nonautonomous Volterra-Lotka competition equations, (), 199-205 · Zbl 0848.34033
[3]	Montes de Oca, F.; Zeeman, M.L., Extinction in nonautonomous competitive Lotka-Volterra systems, (), 3677-3687 · Zbl 0866.34029
[4]	Tineo, A., An iterative scheme for the N-competing species problem, J. diff. equs., 116, 1-15, (1995) · Zbl 0823.34048
[5]	Ahmad, S.; Lazer, A.C., On the nonautonomous N-competing species problems, Appl. anal., 57, 309-323, (1995) · Zbl 0859.34033
[6]	Zhao, X.Q., The qualitative analysis of N-species Lotka-Volterra periodic competition systems, Mathl. comput. modelling, 15, 11, 3-8, (1991) · Zbl 0756.34048
[7]	Gopalsamy, K., Global asymptotic stability in a periodic Lotka-Volterra system, J. austral. math. soc., 27, 66-72, (1985), Ser. B · Zbl 0588.92019
[8]	Tineo, A., On the asymptotic behavior of some population models, J. math. anal. appl., 167, 516-529, (1992) · Zbl 0778.92018
[9]	Tineo, A.; Alvarez, C., A different consideration about the globally asymptotically stable solution of the periodic n-competing species problems, J. math. anal. appl., 159, 44-50, (1991) · Zbl 0729.92025
[10]	Ahmad, S., On almost periodic solutions of the competing species problems, (), 855-861 · Zbl 0668.34042
[11]	Ahmad, S.; Lazer, A.C., Necessary and sufficient average growth in a Lotka-Volterra system, Nonlinear analysis, 34, 191-228, (1998) · Zbl 0934.34037
[12]	Coleman, S.D., Nonautonomous logistic equations as models of the adjustment of populations to environmental change, Math. biosci., 45, 159-173, (1979) · Zbl 0425.92013