Zhen, Jin; Ma, Zhien Stability for a competitive Lotka-Volterra system with delays. (English) Zbl 1015.34060 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51, No. 7, 1131-1142 (2002). The authors consider the following Lotka-Volterra type competitive system with discrete delays \[ \dot x(t) =x(t)[b_1-a_{11}x(t-\tau_{11})-a_{12}y(t-\tau_{12})],\quad \dot y(t) =y(t)[b_2-a_{21}x(t-\tau_{21})-a_{22}y(t-\tau_{22})], \] with the initial conditions \[ \begin{aligned} x(t)&=\phi_1(t)\geq 0, \quad t\in[-\tau,0],\quad\phi_1(0)>0,\\ y(t)&=\phi_2(t)\geq 0,\quad t\in[-\tau,0],\quad\phi_2(0)>0,\\ \tau&=\max\{\tau_{ij}\},\end{aligned} \] where \(x(t), y(t)\) stand for densities of both the population at time \(t\), respectively, \(b_i, a_{ij}\) are all positive constants and \(\tau_{ij}\) are nonnegative. Conditions for local and global asymptotic stability of the positive equilibrium \(Z^*=(x^*,y^*)\) are obtained. Reviewer: Alexander Olegovich Ignatyev (Donetsk) Cited in 1 ReviewCited in 33 Documents MSC: 34K20 Stability theory of functional-differential equations 34K60 Qualitative investigation and simulation of models involving functional-differential equations 92D25 Population dynamics (general) Keywords:delay differential equations; stability PDFBibTeX XMLCite \textit{J. Zhen} and \textit{Z. Ma}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 51, No. 7, 1131--1142 (2002; Zbl 1015.34060) Full Text: DOI References: [1] Beretta, E.; Kuang, Y., Convergence results in a well-known predator-prey system, J. Math. Anal. Appl., 204, 840-853 (1996) · Zbl 0876.92021 [2] Cooke, K. L.; Gpossman, Z., Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl., 86, 592-627 (1982) · Zbl 0492.34064 [3] Freedman, H. I.; Rao, V. S.H., Stability criteria for a system involving two time delays, SIAM J. Appl. Math., 46, 552-560 (1986) · Zbl 0624.34066 [4] Gopalsamy, K., Stability and Oscillations in Delay Differential Equations of Population Dynamics (1992), Kluwer Academic, Dordrecht: Kluwer Academic, Dordrecht The Netherlands · Zbl 0752.34039 [5] He, X.-Z., Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198, 355-370 (1996) · Zbl 0873.34062 [6] He, X.-Z., The Lyapunov functionals for delay Lotka-Volterra-type models, SIAM J. Appl. Math., 58, 1222-1236 (1998) · Zbl 0917.34047 [7] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic press: Academic press New York · Zbl 0777.34002 [8] Lu, Z.; Takeuchi, Y., Permanence and global attractivity for competitive Lotka-Volterra systems with delay, Nonlinear Anal., 22, 847-856 (1994) · Zbl 0809.92025 [9] May, R. M., Time delay versus stability in population models with two or three tropics levels, Ecology, 54, 2, 315-325 (1973) [10] Wendi, W., Uniform persistence in competition models, J. Biomath., 6, 2, 164-169 (1991) · Zbl 0825.92108 [11] Yongkun, Li., Periodic solutions of N-species competition system with time delays, J. Math. Biol., 12, 1, 1-7 (1997) · Zbl 0891.92027 [12] Zhien, M., Stability of predation models with time delays, Appl. Anal., 22, 169-192 (1986) · Zbl 0592.92020 [13] M. Zhien, Mathematical Modeling and Research on the Population Ecology, AnHui educational press, Hefei, 1996.; M. Zhien, Mathematical Modeling and Research on the Population Ecology, AnHui educational press, Hefei, 1996. 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.