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Persistence and global stability in a delayed Gause-type predator-prey system without dominating instantaneous negative feedbacks. (English) Zbl 1013.34074
For the following predator-prey system without dominating instantaneous negative feedbacks \[ \begin{aligned} \dot{x}_1 &= x_1(t)\left(a_1-a_{11}x_1(t-\tau_1)-a_{12}\frac{x_2(t)}{m+x_1(t)}\right),\\ \dot{x}_2 &= x_2(t)\left(-a_2+a_{21}\frac{x_1(t-\tau_2)}{m+x_1(t-\tau_2)}-a_{22}x_2(t-\tau_3)\right), \end{aligned} \] the authors obtain sufficient conditions for uniform persistence, and local and global asymptotic stability of a positive equilibrium.

34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
Full Text: DOI
[1] Hofbauer, J.; Sigmund, K., The theory of evolution and dynamical systems, (1988), Cambridge Univ. Press New York
[2] Kuang, Y.; Smith, H.L., Global stability for infinite delay lotka – volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077
[3] Kuang, Y., Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. differential equations, 119, 503-532, (1995) · Zbl 0828.34066
[4] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064
[5] Wang, W.; Ma, Z., Harmless delays for uniform persistence, J. math. anal. appl., 158, 256-268, (1991) · Zbl 0731.34085
[6] Wright, E.M., A nonlinear differential difference equation, J. reine angew. math., 194, 66-87, (1955) · Zbl 0064.34203
[7] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002
[8] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Academic Dordrecht/Norwell · Zbl 0752.34039
[9] He, X.Z., Stability and delays in a predator – prey system, J. math. anal. appl., 198, 355-370, (1996) · Zbl 0873.34062
[10] Xu, R.; Yang, P.H., Persistence and stability in a three species food-chain system with time delays, (), 277-283 · Zbl 0985.92038
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