# zbMATH — the first resource for mathematics

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 S.-R. Zhou et al. [Theor. Popul. Biol. 67, 23–31 (2005; Zbl 1072.92060)] to obtain
\begin{aligned} \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), \end{aligned} 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 [B. D. Hassard and N. D. Kazarinoff, Theory and applications of Hopf bifurcation. Moskva: Mir (1985; Zbl 0662.34001)].

##### MSC:
 34K60 Qualitative investigation and simulation of models involving functional-differential equations 34K18 Bifurcation theory of functional-differential equations 34K20 Stability theory of functional-differential equations 92D25 Population dynamics (general) 34K19 Invariant manifolds of functional-differential equations 34K17 Transformation and reduction of functional-differential equations and systems, normal forms
##### Keywords:
Predator-prey system; delay; stability; Hopf bifurcation; normal form
Full Text:
##### 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 [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), Cambridge University Press Cambridge · 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 [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 [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 [10] Jiang, G.; Lu, Q., Impulsive state feedback of a predator – prey model, J comput appl math, 200, 193-207, (2007) · Zbl 1134.49024 [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 [12] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press Boston · 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 [14] Liu, Z.; Yuan, R., Stability and bifurcation in a harvested one-predator – two-prey model with delays, Chaos, solitons & fractals, 27, 5, 1395-1407, (2006), [march] · Zbl 1097.34051 [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 [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 [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 [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), Springer-Verlag New York · 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 [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 [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 [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 [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 [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 [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 [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 [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 [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
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.