Liu, Shengqiang; Chen, Lansun Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delays. (English) Zbl 1058.34101 Appl. Math. Lett. 16, No. 6, 911-917 (2003). The author considers the following autonomous two-species Lotka-Volterra competitive system with distributed delays \[ \frac{dx_{1}(t)}{dt} =x_{1}(t)\left( b_{1}-\sum_{j=1}^{2}\sum_{k=1}^{n}\int_{-\tau _{1j}^{(k)}}^{0}a_{1j}^{(k)}x_{j}(t+\theta )d\mu _{1j}^{(k)}(\theta )\right) , \]\[ \frac{dx_{2}(t)}{dt} =x_{\text{2}}(t)\left( b_{2}-\sum_{j=1}^{2}\sum_{k=1}^{n}\int_{-\tau _{2j}^{(k)}}^{0}a_{2j}^{(k)}x_{j}(t+\theta )d\mu _{2j}^{(k)}(\theta )\right) . \]Necessary and sufficient conditions for the system to be permanent are given and it is shown that \(x_{1}\) and \(x_{2}\) extinct if and only if the coefficients \(b_{1},b_{2}\) and \(A_{ij}=\Sigma _{k=1}^{n}a_{ij}^{(k)}\), \(i,j=1,2,\) in the system satisfy some inequalities, respectively. Reviewer: Takeshi Taniguchi (Kurume) Cited in 16 Documents MSC: 34K25 Asymptotic theory of functional-differential equations 92D25 Population dynamics (general) Keywords:Necessary and sufficient conditions; Lotka-Volterra system; Distributed delays; Extinction; Permanence PDFBibTeX XMLCite \textit{S. Liu} and \textit{L. Chen}, Appl. Math. Lett. 16, No. 6, 911--917 (2003; Zbl 1058.34101) Full Text: DOI References: [1] Lu, Z. Y.; Takeuchi, Y., Permanence and global attractivity for competitive Lotka-Volterra systems with delay, Nonli. Anal., 22, 847-856 (1994) · Zbl 0809.92025 [2] Gopalsamy, K., Stability criteria for the linear system ξ(t) +A(t)x(t-T) = 0 and an application to non-linear system, Int. J. Systems. Sci., 21, 1841-1853 (1990) · Zbl 0708.93071 [3] Hale, J. K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J. Math. Anal., 20, 388-395 (1989) · Zbl 0692.34053 [4] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press · Zbl 0777.34002 [5] Waltman, P., A brief survey of persistence in infinite-dimensional system, SIAM J. Math. Anal., 20, 388-395 (1989) · Zbl 0692.34053 [6] S.Q. Liu and L.S. Chen, Permanence, extinction and balancing survival in nonautonomous LotkalVolterra system with delays, Appl. Math. Comput.; S.Q. Liu and L.S. Chen, Permanence, extinction and balancing survival in nonautonomous LotkalVolterra system with delays, Appl. Math. Comput. [7] Wang, W. D.; Ma, Z. E., Harmless delays for uniform persistence, J. Math. Anal. Applic., 158, 256-268 (1991) · Zbl 0731.34085 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.