zbMATH — the first resource for mathematics

Permanence and global stability for nonautonomous \(N\)-species Lotka-Volterra competitive system with impulses. (English) Zbl 1200.34051
The authors consider the following \(N\)-species Lotka-Volterra competitive system with impulsive effects
\[ \begin{aligned} & \dot x_i(t)= x_i(t)\Biggl[a_i(t)- \displaystyle\sum^n_{j=1} b_{ij}(t) x_j(t)\Biggr],\quad t\neq t_k,\\ & x_i(t^+_k)= h_{i\alpha}x_i(t_k),\quad i= 1,2,\dots, n,\;k= 1,2,\dots,\end{aligned}\tag{1} \]
where \(a_i\) and \(b_{ij}\) \((i,j= 1,2,\dots)\) are bounded and continuous. The authors find sufficient conditions for the permanence, persistance and extinction of (1).

34C60 Qualitative investigation and simulation of ordinary differential equation models
34A37 Ordinary differential equations with impulses
34D05 Asymptotic properties of solutions to ordinary differential equations
92D25 Population dynamics (general)
Full Text: DOI
[1] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, (1993), Longman Harlow · Zbl 0815.34001
[2] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Press Singapore · Zbl 0719.34002
[3] E. Liz, Boundary value problems for new types of differential equations, Ph.D. Thesis, University of Vigo Spain, 1994 (in Spanish)
[4] Samoilenko, A.M.; Perestyuk, N.A., Impulsive differential equations, (1995), World Scientific Press Singapore · Zbl 0837.34003
[5] Ahmad, S.; Stamova, I.M., Almost periodic solutions of \(n\)-dimensional impulsive systems, Nonlinear anal. RWA, 9, 4, 1721-1740, (2008)
[6] Ahmad, S.; Stamova, I.M., Asymptotic stability of competitive systems with delays and impulsive perturbations, J. math. anal. appl., 334, 686-700, (2007) · Zbl 1153.34044
[7] Ahmad, S.; Stamova, I.M., Asymptotic stability of an \(N\)-dimensional impulsive competitive system, Nonlinear anal. RWA, 8, 654-663, (2007) · Zbl 1152.34342
[8] Akhmet, M.U.; Beklioglu, M.; Ergenc, T.; Tkachenko, V.I., An impulsive ratio-dependent predator-prey system with diffusion, Nonlinear anal. RWA, 7, 1255-1267, (2006) · Zbl 1114.35097
[9] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. comput. modelling, 26, 59-72, (1997) · Zbl 1185.34014
[10] Liu, X.; Ballinger, G., Boundedness for impulsive delay differential equations and applications to population growth models, Nonlinear anal., 53, 1041-1062, (2003) · Zbl 1037.34061
[11] Liu, X.; Chen, L., Global dynamics of the periodic logistic system with periodi impulsive perturbations, J. math. anal. appl., 289, 279-2913, (2004)
[12] Men, X.; Chen, L.; Li, Q., The dynamics of an impulsive delay predator-prey model with variable coefficients, Appl. math. comput., 198, 361-374, (2008) · Zbl 1133.92029
[13] Saker, S.H.; Alzabut, J.O., Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear anal. RWA, 8, 1029-1039, (2007) · Zbl 1124.34054
[14] Wang, W.; Shen, J.; Luo, Z., Partial survival and extinction in two competing species with impulses, Nonlinear anal. RWA, 10, 3, 1243-1254, (2009) · Zbl 1162.34308
[15] Wen, X.; Wang, Z., Persistence and extinction in two species models with impulse, Acta math. sin. (engl. ser.), 2, 447-454, (2006) · Zbl 1115.34046
[16] Zhang, H.; Georgescu, P.; Chen, L., On the impulsive controllability and bifurcation of a predator-prey model of IPM, Biosystems, 93, 151-171, (2008)
[17] Jin, Z.; Han, M.; Li, G., The persistence in a Lotka-Volterra competition systems with impulsive, Chaos solitons fractals, 24, 1105-1117, (2005) · Zbl 1081.34045
[18] Ahmad, S.; Lazer, A.C., Necessary and sufficient average growth in a lotka – volterra system, Nonlinear anal., 34, 191-228, (1998) · Zbl 0934.34037
[19] 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
[20] Ahmad, S.; Lazer, A.C., Average growth and extinction in a competitive lotka – volterra system, Nonlinear anal., 62, 545-557, (2005) · Zbl 1088.34047
[21] Teng, Z.; Li, Z., Permanence and asymptotic behavior of the N-species nonautonomous lotka – volterra competitive systems, Comput. math. appl., 39, 107-116, (2000) · Zbl 0959.34039
[22] Berman, J.; Plemmons, R.J., Nonnegative matrices in the mathematical sciences, (1979), Academic Press New York · Zbl 0484.15016
[23] Teng, Z.; Chen, L., The positive periodic solutions in periodic Kolmogorov type systems with delays, Acta math. appl. sin., 22, 446-456, (1999), (in Chinese) · Zbl 0976.34063
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.