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Stability analysis on a predator-prey system with distributed delays. (English) Zbl 0897.34062
A Lotka-Volterra predator-prey system with distributed delays is considered and local and global dynamical properties of two possible equilibria $$P_1= (x_0,0)$$ and $$P_2= (x^*, y^*)$$ are discussed. It is shown that when the delays are sufficiently small, if $$P_2$$ does not exist, then $$P_1$$ is globally asymptotically stable or globally attractive; otherwise, $$P_2$$ is locally asymptotically stable. Furthermore, a region of explicit asymptotic stability is obtained for $$P_2$$ based on a Lyapunov functional.

##### MSC:
 34K20 Stability theory of functional-differential equations
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##### References:
 [1] Beretta, E.; Kuang, Y., Convergence results in a well known delayed predator-prey system, J. math. anal. appl., 204, 840-853, (1996) · Zbl 0876.92021 [2] Freedman, H.I., Deterministic mathematical models in population ecology, (1980), Marcel Dekker New York · Zbl 0448.92023 [3] Freedman, H.I.; Ruan, S., Uniform persistence in functional differential equations, J. differential equations, 115, 173-192, (1995) · Zbl 0814.34064 [4] Gopalsamy, K., Stability and oscillations in delay differential equations of population dynamics, (1992), Kluwer Dordrecht · Zbl 0752.34039 [5] Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048 [6] He, X., Stability and delays in a prey-predator system, J. math. anal. appl., 198, 355-370, (1996) · Zbl 0873.34062 [7] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · Zbl 0777.34002 [8] Kuang, Y., Global stability in delay differential systems without dominating instantaneous negative feedbacks, J. differential equations, 119, 503-532, (1995) · Zbl 0828.34066 [9] Kuang, Y.; Smith, H.L., Global stability for infinite delay Lotka-Volterra type systems, J. differential equations, 103, 221-246, (1993) · Zbl 0786.34077 [10] Lu, Z.; Takeuchi, Y., Persistence and global attractivity for competitive Lotka-Volterra systems with delay, Nonlinear anal. TMA, 22, 847-856, (1994) · Zbl 0809.92025 [11] Takeuchi, Y., Global dynamical properties of Lotka-Volterra systems, (1996), World Scientific Singapore · Zbl 0844.34006 [12] Takeuchi, Y.; Ma, W., Global attractivity of mixed Lotka-Volterra differential systems with delays, (1996), preprint · Zbl 0926.34060 [13] Waltman, P., A second course in elementary differential equations, (1986), Academic Press Orlando · Zbl 0689.34001 [14] Zhao, T.; Kuang, Y.; Smith, H.L., Global existence of periodic solutions in a class of delayed gause-type predator-prey systems, Nonlinear anal. TMA, 28, 1373-1394, (1997) · Zbl 0872.34047
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