The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays

Changjin Xu; Maoxin Liao; Xiaofei He

International Journal of Applied Mathematics and Computer Science (2011)

  • Volume: 21, Issue: 1, page 97-107
  • ISSN: 1641-876X

Abstract

top
In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.

How to cite

top

Changjin Xu, Maoxin Liao, and Xiaofei He. "Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays." International Journal of Applied Mathematics and Computer Science 21.1 (2011): 97-107. <http://eudml.org/doc/208040>.

@article{ChangjinXu2011,
abstract = {In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.},
author = {Changjin Xu, Maoxin Liao, Xiaofei He},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {predator-prey model; delay; stability; Hopf bifurcation},
language = {eng},
number = {1},
pages = {97-107},
title = {Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays},
url = {http://eudml.org/doc/208040},
volume = {21},
year = {2011},
}

TY - JOUR
AU - Changjin Xu
AU - Maoxin Liao
AU - Xiaofei He
TI - Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays
JO - International Journal of Applied Mathematics and Computer Science
PY - 2011
VL - 21
IS - 1
SP - 97
EP - 107
AB - In this paper, a two-species Lotka-Volterra predator-prey model with two delays is considered. By analyzing the associated characteristic transcendental equation, the linear stability of the positive equilibrium is investigated and Hopf bifurcation is demonstrated. Some explicit formulae for determining the stability and direction of Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using normal form theory and center manifold theory. Some numerical simulations for supporting the theoretical results are also included.
LA - eng
KW - predator-prey model; delay; stability; Hopf bifurcation
UR - http://eudml.org/doc/208040
ER -

References

top
  1. Bhattacharyya, R. and Mukhopadhyay, B. (2006). Spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, Ecological Complexity 3(2): 160-169. 
  2. Faria, T. (2001). Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, Journal of Mathematical Analysis and Applications 254(2): 433-463. Zbl0973.35034
  3. Gao, S.J., Chen, L.S. and Teng, Z.D. (2008). Hopf bifurcation and global stability for a delayed predator-prey system with stage structure for predator, Applied Mathematics and Computation 202(2): 721-729. Zbl1151.34067
  4. Hale, J. (1977). Theory of Functional Differential Equations, Springer-Verlag, Berlin. Zbl0352.34001
  5. Hassard, B., Kazarino, D. and Wan, Y. (1981). Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge. 
  6. Kar, T. and Pahari, U. (2007). Modelling and analysis of a preypredator system stage-structure and harvesting, Nonlinear Analysis: Real World Applications 8(2): 601-609. Zbl1152.34374
  7. Klamka, J. (1991). Controllability of Dynamical Systems, Kluwer, Dordrecht. Zbl0732.93008
  8. Kuang, Y. (1993). Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, MA. Zbl0777.34002
  9. Kuang, Y. and Takeuchi, Y. (1994). Predator-prey dynamics in models of prey dispersal in two-patch environments, Mathematical Biosciences 120(1): 77-98. Zbl0793.92014
  10. Li, K. and Wei, J. (2009). Stability and Hopf bifurcation analysis of a prey-predator system with two delays, Chaos, Solitons & Fractals 42(5): 2603-2613. Zbl1198.34144
  11. May, R.M. (1973). Time delay versus stability in population models with two and three trophic levels, Ecology 4(2): 315-325. 
  12. Prajneshu Holgate, P. (1987). A prey-predator model with switching effect, Journal of Theoretical Biology 125(1): 61-66. 
  13. Ruan, S. and Wei, J. (2003). On the zero of some transcendential functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous, Discrete and Impulsive Systems Series A 10(1): 863-874. Zbl1068.34072
  14. Song, Y.L. and Wei, J. (2005). Local Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Journal of Mathematical Analysis and Applications 301(1): 1-21. Zbl1067.34076
  15. Teramoto, E.I., Kawasaki, K. and Shigesada, N. (1979). Switching effects of predaption on competitive prey species, Journal of Theoretical Biology 79(3): 303-315. 
  16. Xu, R., Chaplain, M.A.J. and Davidson F.A. (2004). Periodic solutions for a delayed predator-prey model of prey dispersal in two-patch environments, Nonlinear Analysis: Real World Applications 5(1): 183-206. Zbl1066.92059
  17. Xu, R. and Ma, Z.E. (2008). Stability and Hopf bifurcation in a ratio-dependent predator-prey system with stage structure, Chaos, Solitons & Fractals 38(3): 669-684. Zbl1146.34323
  18. Yan, X.P. and Li, W.T. (2006). Hopf bifurcation and global periodic solutions in a delayed predator-prey system, Applied Mathematics and Computation 177(1): 427-445. Zbl1090.92052
  19. Yan, X.P. and Zhang, C.H. (2008). Hopf bifurcation in a delayed Lokta-Volterra predator-prey system, Nonlinear Analysis: Real World Applications 9(1): 114-127. Zbl1149.34048
  20. Zhou, X.Y., Shi, X.Y. and Song, X.Y. (2008). Analysis of nonautonomous predator-prey model with nonlinear diffusion and time delay, Applied Mathematics and Computation 196(1): 129-136. Zbl1147.34355

Citations in EuDML Documents

top
  1. Kun-Yi Yang, Ling-Li Zhang, Jie Zhang, Stability analysis of a three-dimensional energy demand-supply system under delayed feedback control
  2. Dariusz Myszor, Krzysztof A. Cyran, Mathematical modelling of molecule evolution in protocells
  3. Qiaoling Chen, Zhidong Teng, Zengyun Hu, Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response
  4. Hasim A. Obaid, Rachid Ouifki, Kailash C. Patidar, An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.