×
Meng, Xin-You; Huo, Hai-Feng

Bifurcation analysis of a Lotka-Volterra mutualistic system with multiple delays. (English) Zbl 1474.34485

Abstr. Appl. Anal. 2014, Article ID 958140, 18 p. (2014).
Summary: A class of Lotka-Volterra mutualistic system with time delays of benefit and feedback delays is introduced. By analyzing the associated characteristic equation, the local stability of the positive equilibrium and existence of Hopf bifurcation are obtained under all possible combinations of two or three delays selecting from multiple delays. Not only explicit formulas to determine the properties of the Hopf bifurcation are shown by using the normal form method and center manifold theorem, but also the global continuation of Hopf bifurcation is investigated by applying a global Hopf bifurcation result due to J. Wu [Trans. Am. Math. Soc. 350, No. 12, 4799–4838 (1998; Zbl 0905.34034)]. Numerical simulations are given to support the theoretical results.

Cited in 1 Document

MSC:

34K18 Bifurcation theory of functional-differential equations
34K60 Qualitative investigation and simulation of models involving functional-differential equations
92D25 Population dynamics (general)
37N25 Dynamical systems in biology

Citations:

Zbl 0905.34034
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Lotka, A. J., Elements of Physical Biology (1925), New York, NY, USA: Williams and Wilkins, New York, NY, USA · JFM 51.0416.06
[2] Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature, 118, 2972, 558-560 (1926) · JFM 52.0453.03 · doi:10.1038/118558a0
[3] Jin, Z.; Ma, Z. E., Stability for a competitive Lotka-Volterra system with delays, Nonlinear Analysis: Theory, Methods and Applications, 51, 7, 1131-1142 (2002) · Zbl 1015.34060 · doi:10.1016/S0362-546X(01)00881-1
[4] Song, Y.; Han, M.; Peng, Y., Stability and Hopf bifurcations in a competitive Lotka-Volterra system with two delays, Chaos, Solitons & Fractals, 22, 5, 1139-1148 (2004) · Zbl 1067.34075 · doi:10.1016/j.chaos.2004.03.026
[5] Zhang, J. Z.; Jin, Z.; Yan, J. R.; Sun, G. Q., Stability and Hopf bifurcation in a delayed competition system, Nonlinear Analysis: Theory, Methods & Applications, 70, 2, 658-670 (2009) · Zbl 1166.34049 · doi:10.1016/j.na.2008.01.002
[6] Zhang, J. F., Stability and bifurcation periodic solutions in a Lotka-Volterra competition system with multiple delays, Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, 70, 1, 849-860 (2012) · Zbl 1267.34128 · doi:10.1007/s11071-012-0501-4
[7] Faria, T., Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, Journal of Mathematical Analysis and Applications, 254, 2, 433-463 (2001) · Zbl 0973.35034 · doi:10.1006/jmaa.2000.7182
[8] Song, Y. L.; Peng, Y. H.; Wei, J. J., Bifurcations for a predator-prey system with two delays, Journal of Mathematical Analysis and Applications, 337, 1, 466-479 (2008) · Zbl 1132.34053 · doi:10.1016/j.jmaa.2007.04.001
[9] Yan, X. P.; Li, W. T., Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Applied Mathematics and Computation, 177, 1, 427-445 (2006) · Zbl 1090.92052 · doi:10.1016/j.amc.2005.11.020
[10] Xu, C. Q.; Yuan, S. L., Stability and Hopf bifurcation in a delayed predator-prey system with herd behavior, Abstract and Applied Analysis, 2014 (2014) · Zbl 1449.92041 · doi:10.1155/2014/568943
[11] Meng, X. Y.; Huo, H. F.; Zhang, X. B.; Xiang, H., Stability and Hopf bifurcation in a three-species system with feedback delays, Nonlinear Dynamics, 64, 4, 349-364 (2011) · doi:10.1007/s11071-010-9866-4
[12] Song, Y. L.; Wei, J. J., Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Journal of Mathematical Analysis and Applications, 301, 1, 1-21 (2005) · Zbl 1067.34076 · doi:10.1016/j.jmaa.2004.06.056
[13] Chen, L. S.; Lu, Z. Y.; Wang, W. D., The effect of delays on the permanence for Lotka-Volterra systems, Applied Mathematics Letters, 8, 4, 71-73 (1995) · Zbl 0833.34071 · doi:10.1016/0893-9659(95)00050-Z
[14] Hale, J. K., Theory of Functional Differential Equations (1977), New York, NY, USA: Springer, New York, NY, USA · Zbl 0352.34001
[15] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0777.34002
[16] Gause, J. D., Mathematical Biology (1989), New York, NY, USA: Springer, New York, NY, USA · Zbl 0682.92001
[17] He, X. Z.; Gopalsamy, K., Persistence, attractivity, and delay in facultative mutualism, Journal of Mathematical Analysis and Applications, 215, 1, 154-173 (1997) · Zbl 0893.34036 · doi:10.1006/jmaa.1997.5632
[18] Meng, X. Z.; Wei, J. J., Stability and bifurcation of mutual system with time delay, Chaos, Solitons & Fractals, 21, 3, 729-740 (2004) · Zbl 1048.34122 · doi:10.1016/j.chaos.2003.12.050
[19] Zhang, F. Q.; Zhang, Y. J., Hopf bifrucation for Lotka-Volterra mutualistic systems with three delays, Jouranl of Biomathematics, 26, 2, 223-233 (2011) · Zbl 1249.34245
[20] Hassard, B. D.; Kazarinoff, N. D.; Wan, Y. H., Theory and Applications of Hopf Bifurcation (1981), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0474.34002
[21] Wu, J. H., Symmetric functional-differential equations and neural networks with memory, Transactions of the American Mathematical Society, 350, 12, 4799-4838 (1998) · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2
[22] Ruan, S. G.; Wei, J. J., On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete & Impulsive Systems A: Mathematical Analysis, 10, 6, 863-874 (2003) · Zbl 1068.34072
[23] Cushing, J., Periodic time-dependent predator-prey systems, SIAM Journal on Applied Mathematics, 32, 1, 82-95 (1977) · Zbl 0348.34031 · doi:10.1137/0132006
[24] He, X. Z., Stability and delays in a predator-prey system, Journal of Mathematical Analysis and Applications, 198, 2, 355-370 (1996) · Zbl 0873.34062 · doi:10.1006/jmaa.1996.0087
[25] Clark, C. W., Bioeconomic Modelling and Fisheries Management (1985), New York, NY, USA: John Wiley & Sons, New York, NY, USA
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.