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The stability and Hopf bifurcation for a predator-prey system with time delay. (English) Zbl 1152.34059
The author incorporates delay into an ODE considered by {\it S.-R. Zhou} {et al.} [Theor. Popul. Biol. 67, 23--31 (2005; Zbl 1072.92060)] to obtain $$ \align \frac {dN(t)}{dt} & = r_1N(t)-\varepsilon P(t)N(t), \\ \frac{dP(t)}{dt} & = P(t)\left(r_2-\theta \frac {P(t-\tau)}{N(t)}\right), \endalign $$ which describes the dynamics of a ratio-dependent predator ($P$)-prey ($N$) system. First, the local stability of the positive equilibrium point $E^\ast=(\frac {r_1\theta}{r_2\varepsilon}, \frac{r_1}{\varepsilon})$ is studied. $E^ast$ is stable for $\tau\in [0,\tau_0)$ and Hopf bifurcation occurs for $\tau=\tau_k$, where $\tau_k=\frac {(2k+1)\pi}{r_2+\sqrt{r_2^2+4r_1r_2}}$ for $k=0,1,\dots$. Then the stability and direction of bifurcating periodic solutions is discussed using the normal form theory and center manifold theorem due to [{\it B. D. Hassard} and {\it N. D. Kazarinoff}, Theory and applications of Hopf bifurcation. Moskva: Mir (1985; Zbl 0662.34001)].

MSC:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K20Stability theory of functional-differential equations
92D25Population dynamics (general)
34K19Invariant manifolds (functional-differential equations)
34K17Transformation and reduction of functional-differential equations and systems; normal forms
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Full Text: DOI
References:
[1] Chen, X.: Periodicity in a nonlinear discrete predator -- prey system with state dependent delays, Nonlinear anal RWA 8, 435-446 (2007) · Zbl 1152.34367 · doi:10.1016/j.nonrwa.2005.12.005
[2] Çelik C, Duman O. Allee effect in a discrete-time predator -- prey system. Chaos, Solitons & Fractals, in press. doi:10.1016/j.chaos.2007.09.077.
[3] Fowler, M. S.; Ruxton, G. D.: Population dynamic consequences of allee effects, J theor biol 215, 39-46 (2002)
[4] Gopalsamy, K.: Time lags and global stability in two species competition, Bull math biol 42, 728-737 (1980) · Zbl 0453.92014
[5] Hadjiavgousti, D.; Ichtiaroglou, S.: Allee effect in a predator -- prey system, Chaos, solitons & fractals 36, 334-342 (2008) · Zbl 1128.92045
[6] Hassard, N. D.; Kazarinoff, Y. H.: Theory and applications of Hopf bifurcation, (1981) · Zbl 0474.34002
[7] He, X.: Stability and delays in a predator -- prey system, J math anal appl 198, 355-370 (1996) · Zbl 0873.34062 · doi:10.1006/jmaa.1996.0087
[8] Huo, H. -F.; Li, W. -T.: Existence and global stability of periodic solutions of a discrete predator -- prey system with delays, Appl math comput 153, 337-351 (2004) · Zbl 1043.92038 · doi:10.1016/S0096-3003(03)00635-0
[9] Jang, S. R. -J.: Allee effects in a discrete-time host-parasitoid model, J diff equat appl 12, 165-181 (2006) · Zbl 1088.92058 · doi:10.1080/10236190500539238
[10] Jiang, G.; Lu, Q.: Impulsive state feedback of a predator -- prey model, J comput appl math 200, 193-207 (2007) · Zbl 1134.49024 · doi:10.1016/j.cam.2005.12.013
[11] 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 · doi:10.1016/S0960-0779(02)00199-6
[12] Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[13] Leung, A.: Periodic solutions for a prey -- predator differential delay equation, J diff equat 26, 391-403 (1977) · Zbl 0365.34078 · doi:10.1016/0022-0396(77)90087-0
[14] Liu, Z.; Yuan, R.: Stability and bifurcation in a harvested one-predator -- two-prey model with delays, Chaos, solitons & fractals 27, No. 5, 1395-1407 (2006) · Zbl 1097.34051 · doi:10.1016/j.chaos.2005.05.014
[15] Liu, B.; Teng, Z.; Chen, L.: Analysis of a predator -- prey model with Holling II functional response concerning impulsive control strategy, J comput appl math 193, 347-362 (2006) · Zbl 1089.92060 · doi:10.1016/j.cam.2005.06.023
[16] Liu, X.; Xiao, D.: Complex dynamic behaviors of a discrete-time predator -- prey system, Chaos, solitons & fractals 32, 80-94 (2007) · Zbl 1130.92056 · doi:10.1016/j.chaos.2005.10.081
[17] Ma, W.; Takeuchi, Y.: Stability analysis on a predator -- prey system with distributed delays, J comput appl math 88, 79-94 (1998) · Zbl 0897.34062 · doi:10.1016/S0377-0427(97)00203-3
[18] Mccarthy, M. A.: The allee effect finding mates and theoretical models, Ecol model 103, 99-102 (1997)
[19] Murray, J. D.: Mathematical biology, (1993) · Zbl 0779.92001
[20] 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
[21] Ruan, S.; Wei, J.: Periodic solutions of planar systems with two delays, Proc roy soc Edinburgh sect A 129, 1017-1032 (1999) · Zbl 0946.34062 · doi:10.1017/S0308210500031061
[22] Scheuring, I.: Allee effect increases the dynamical stability of populations, J theor biol 199, 407-414 (1999)
[23] Sun, C.; Han, M.; Lin, Y.; Chen, Y.: Global qualitative analysis for a predator -- prey system with delay, Chaos, solitons & fractals 32, 1582-1596 (2007) · Zbl 1145.34042 · doi:10.1016/j.chaos.2005.11.038
[24] Teng, Z.; Rehim, M.: Persistence in nonautonomous predator -- prey systems with infinite delays, J comput appl math 197, 302-321 (2006) · Zbl 1110.34054 · doi:10.1016/j.cam.2005.11.006
[25] Wang, L. -L.; Li, W. -T.; Zhao, P. -H.: Existence and global stability of positive periodic solutions of a discrete predator -- prey system with delays, Adv diff equat 4, 321-336 (2004) · Zbl 1081.39007 · doi:10.1155/S1687183904401058
[26] Wang, F.; Zeng, G.: Chaos in Lotka -- Volterra predator -- prey system with periodically impulsive ratio-harvesting the prey and time delays, Chaos, solitons & fractals 32, 1499-1512 (2007) · Zbl 1130.37042 · doi:10.1016/j.chaos.2005.11.102
[27] Wen, X.; Wang, Z.: The existence of periodic solutions for some models with delay, Nonlinear anal RWA 3, 567-581 (2002) · Zbl 1095.34549 · doi:10.1016/S1468-1218(01)00049-9
[28] Xu, R.; Wang, Z.: Periodic solutions of a nonautonomous predator -- prey system with stage structure and time delays, J comput appl math 196, 70-86 (2006) · Zbl 1110.34051 · doi:10.1016/j.cam.2005.08.017
[29] Yan, X. P.; Chu, Y. D.: Stability and bifurcation analysis for a delayed Lotka -- Volterra predator -- prey system, J comput appl math 196, 198-210 (2006) · Zbl 1095.92071 · doi:10.1016/j.cam.2005.09.001
[30] Zhou, S. R.; Liu, Y. F.; Wang, G.: The stability of predator -- prey systems subject to the allee effects, Theor populat biol 67, 23-31 (2005) · Zbl 1072.92060 · doi:10.1016/j.tpb.2004.06.007
[31] Zhou, L.; Tang, Y.: Stability and Hopf bifurcation for a delay competition diffusion system, Chaos, solitons & fractals 14, 1201-1225 (2002) · Zbl 1038.35147 · doi:10.1016/S0960-0779(02)00068-1