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Global qualitative analysis for a predator-prey system with delay. (English) Zbl 1145.34042
This paper concerns the following predator-prey system with delay, $$ \cases \frac {dx(t)}{dt} = x(t)[r_1-a_{11}x(t)-a_{12}y(t)], \\ \frac{dy(t)}{dt} = y(t)[-r_2+a_{21}x(t)-a_{22}y(t-\tau)]. \endcases\tag$*$ $$ Under the assumption that $$ r_1a_{21}-r_2a_{11}>0, \tag{\text{H}} $$ system ($*$) has a unique positive equilibrium $E_{*}=(\frac {r_1a_{22}+r_2a_{12}}{a_{11}a_{22}+a_{12}a_{21}}, \frac {r_1a_{21}-r_2a_{11}}{a_{11}a_{22}+a_{12}a_{21}})$. When $\tau=0$, it is known that $E_{*}$ is globally asymptotically stable. Also, {\it Wendi Wang} and {\it Zhien Ma} showed that solutions of ($*$) are bounded, uniformly persistent and the delay is harmless for the uniform persistence [J. Math. Anal. Appl. 158, 256--268 (1991; Zbl 0731.34085)]. First, by constructing suitable Lyapunov functionals, the authors provide a set of sufficient conditions on the global stability of $E_{*}$. Notice that the conditions can not be reduced to those for the case where $\tau=0$. Then, it was shown that local Hopf bifurcation can occur. Moreover, the direction, stability and period of the periodic solution bifurcating from $E_{*}$ was investigates through the normal form theorem and center manifold argument [{\it R. D. Nussbaum}, Ann. Mat. Pura Appl. IV. Ser. 101, 263--306 (1974; Zbl 0323.34061)]. Finally, using the global Hopf bifurcation theorem due to {\it J. Wu} [Trans. Am. Math. Soc. 350, 4799--4838 (1998; Zbl 0905.34034)], the authors also investigates the global existence of a periodic solution of ($*$). This may be the most important contribution of this paper as results on global existence of periodic solutions are scarce in the literature.

MSC:
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
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Full Text: DOI
References:
[1] Xiao, Y.; Cheng, D.; Tang, S.: Dynamic complexities in predator-prey ecosystem models with age-structure for predator. Chaos, solitons & fractals 14, 1403-1411 (2002) · Zbl 1032.92033
[2] Krise, S.; Choudhury, S. R.: Bifurcations and chaos in a predator-prey model with delay and a laser-diode system with self-sustained pulsations. Chaos, solitons & fractals 16, 59-77 (2003) · Zbl 1033.37048
[3] Gopalsamy, K.: Time lags and global stability in two species competition. Bull math biol 42, 728-737 (1980) · Zbl 0453.92014
[4] Kuang, Y.: Delay differential equations with applications in population dynamics. (1993) · Zbl 0777.34002
[5] Zhou, L.; Tang, Y.: Stability and Hopf bifurcation for a delay competition diffusion system. Chaos, solitons & fractals 14, 1201-1225 (2002) · Zbl 1038.35147
[6] Ruan, S.: Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays. Quart appl math 59, 159-173 (2001) · Zbl 1035.34084
[7] Chen, L.: Mathematical models and methods in ecology. (1988)
[8] Wang, W.; Ma, Z.: Harmless delays for uniform persistence. J math anal appl 158, 256-268 (1991) · Zbl 0731.34085
[9] Nussbaum, R. D.: Periodic solutions of some nonlinear autonomous functional equations. Ann mat pura appl 10, 263-306 (1974) · Zbl 0323.34061
[10] Leung, A.: Periodic solutions for a prey-predator differential delay equation. J differential equat 26, 391-403 (1977) · Zbl 0365.34078
[11] Nussbaum, R. D.: A Hopf bifurcation theorem for retarded functional differential equations. Trans am math soc 238, 139-163 (1978)
[12] Táboas, P.: Periodic solutions of a planar delay equation. Proc roy soc Edinburgh sect A 116, 85-101 (1990) · Zbl 0719.34125
[13] Erbe, L. H.; Geba, K.; Krawcewicz, W.; Wu, J.: S1-degree and global Hopf bifurcations. J differential equat 98, 277-298 (1992) · Zbl 0765.34023
[14] Krawcewicz, W.; Wu, J.; Xia, H.: Global Hopf bifurcation theory for considering fields and neural equations with applications to lossless transmission problems. Canad appl math quart 1, 167-219 (1993) · Zbl 0801.34069
[15] Hassard, B.; Kazarino, D.; Wan, Y.: Theory and applications of Hopf bifurcation. (1981) · Zbl 0474.34002
[16] Araki, M.; Kondo, B.: Stability and transient behavior of composite nonlinear system. IEEE trans automat control 17, 537-541 (1972) · Zbl 0275.93031
[17] Hahn, W.: Theory and applications of Lyapunov’s direct method. (1963) · Zbl 0111.28403
[18] Song, Y.; Han, M.; Peng, Y.: Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays. Chaos, solitons & fractals 22, 1139-1148 (2004) · Zbl 1067.34075
[19] Dieuonné: Foundations of modern analysis. (1960)
[20] Hale, J.; Lunel, S.: Introduction to functional differential equations. (1993) · Zbl 0787.34002
[21] Wu, J.: Symmetric functional differential equations and neural networks with memory. Trans am math soc 350, 4799-4838 (1998) · Zbl 0905.34034