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Necessary-sufficient conditions for permanence and extinction in Lotka-Volterra system with distributed delays. (English) Zbl 1058.34101
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.

34K25Asymptotic theory of functional-differential equations
92D25Population dynamics (general)
Full Text: DOI
[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 ${\xi}(t) +A(t)x(t-T) = 0$ 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) · 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. (to appear).
[7] Wang, W. D.; Ma, Z. E.: Harmless delays for uniform persistence. J. math. Anal. applic. 158, 256-268 (1991) · Zbl 0731.34085