×

zbMATH — the first resource for mathematics

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.

MSC:
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[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
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.