Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay

ABSTRACT

A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.

A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.

Cite this paper

X. Wang and H. Liu, "Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 652-661. doi: 10.4236/am.2012.36099.

X. Wang and H. Liu, "Hopf Bifurcations in a Predator-Prey System of Population Allelopathy with Discrete Delay,"

References

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[2] V. Volterra, “Variazionie Fluttuazioni del Numero d’Individui in Specie Animali Conviventi,” Mem. Acad. Licei, Vol. 2, 1926, pp. 31-113.

[3] M. Brelot, “Sur le probleme biologique hereditaire de deux especes devorante et devore,” Annali di Matematica Pura ed Applicata, Vol. 9, No. 1, 1931, pp. 58-74. doi:10.1007/BF02414092

[4] L. Chen, “Mathematical Models and Methods in Ecology,” Science Press, Beijing, 1988.

[5] Y. Song and S. Yuan, “Bifurcation Analysis in a Predator-Prey System with Delay,” Nonlinear Analysis: Real World Applications, Vol. 7, 2006, pp. 265-284. doi:10.1016/j.nonrwa.2005.03.002

[6] T. Faria, “Stability and Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion,” Journal of Mathematical Analysis and Applications, Vol. 254, No. 2, 2001, pp. 433-463.

[7] S. Ruan, “Absolute Stability, Conditional Stability and Bifurcation in Kolmogorov-Type Predator-Prey System with Discrete Delays,” Quarterly of Applied Mathematics, Vol. 59, 2001, pp. 159-172.

[8] X. P. Yan and Y. D. Chu, “Stability and Bifurcation Analysis for a Delayed Lotka-Volterra Predator-Prey System,” Journal of Computational and Applied Mathematics, Vol. 196, No. 1, 2006, pp. 198-210.

[9] X. P. Yan and C. H. Zhang, “Hopf Bifurcation in a Delayed Lokta-Volterra Predator-Prey System,” Nonlinear Analysis, Vol. RWA 9, 2008, pp. 114-127. doi:10.1016/j.nonrwa.2006.09.007

[10] Y. Song and J. Wei, “Local Hopf Bifurcation and Global Periodic Solutions in a Delayed Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 301, 2005, pp. 1-21. doi:10.1016/j.jmaa.2004.06.056

[11] X. P. Yan and W. T. Li, “Hopf Bifurcation and Global Periodic Solutions in a Delayed Predatorprey System,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 427-445.

[12] J. Zhang and B. Feng, “Geometric Theory and Bifurcation Problems of Ordinary Differential Equations,” Beijing University Press, Beijing, 2000.

[13] S. A. Gourley, “Travelling Fronts in the Diffusive Nicholsons Blowflies Equation with Distributed Delays,” Mathematical and Computer Modelling, Vol. 32, 2000, pp. 843-853. doi:10.1016/S0895-7177(00)00175-8

[14] J. K. Hale, “Theory of Functional Differential Equations,” Spring-Verlag, New York, 1977. doi:10.1007/978-1-4612-9892-2

[15] J. Wu, “Theory and Applications of Partial Functional Differential Equations,” Springer-Verlag, New York, 1996. doi:10.1007/978-1-4612-4050-1

[16] C. H. Zhang, X. P. Yan and G. H. Cui, “Hopf Bifurcations in a Predator-Prey System with a Discrete Delay and a Distributed Delay,” Nonlinear Analysis, Vol. RWA 11, 2010, pp. 4141-4153. doi:10.1016/j.nonrwa.2010.05.001

[17] X. Z. He, “Stability and Delays in a Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 198, No. 2, 1996, pp. 355-370.

[18] Z. Lu and W. Wang, “Global Stability for Two-Species Lotka-Volterra Systems with Delay,” Journal of Mathematical Analysis and Applications, Vol. 208, No. 1, 1997, pp. 277-280.

[19] E. L. Rice, “Allelopathy,” 2nd Edition, Academic Press, New York, 1984.

[20] J. M. Smith, “Models in Ecology,” Cambridge University, Cambridge, 1974.

[21] J. Chattopadhyay, “Effects of toxic Substance on a Two-Species Competitive System,” Ecological Modelling, Vol. 84, 1996, pp. 287-289. doi:10.1016/0304-3800(94)00134-0

[22] A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, “A Delay Differential Equations Model of Plankton Allelopathy,” Mathematical Biosciences, Vol. 149, No. 2, 1998, pp. 167-189.

[23] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, “Theory and Applications of Hopf Bifurcation,” Cambridge University Press, Cambridge, 1981.

[24] Y. Song, M. Han and J. Wei, “Stability and Hopf Bifurcation on a Simplified BAM Neural Network with Delays,” Physica D, Vol. 200, 2005, pp. 185-204. doi:10.1016/j.physd.2004.10.010

[1] A. J. Lotka, “Elements of Physical Biology,” Williams and Wilkins, New York, 1925.

[2] V. Volterra, “Variazionie Fluttuazioni del Numero d’Individui in Specie Animali Conviventi,” Mem. Acad. Licei, Vol. 2, 1926, pp. 31-113.

[3] M. Brelot, “Sur le probleme biologique hereditaire de deux especes devorante et devore,” Annali di Matematica Pura ed Applicata, Vol. 9, No. 1, 1931, pp. 58-74. doi:10.1007/BF02414092

[4] L. Chen, “Mathematical Models and Methods in Ecology,” Science Press, Beijing, 1988.

[5] Y. Song and S. Yuan, “Bifurcation Analysis in a Predator-Prey System with Delay,” Nonlinear Analysis: Real World Applications, Vol. 7, 2006, pp. 265-284. doi:10.1016/j.nonrwa.2005.03.002

[6] T. Faria, “Stability and Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion,” Journal of Mathematical Analysis and Applications, Vol. 254, No. 2, 2001, pp. 433-463.

[7] S. Ruan, “Absolute Stability, Conditional Stability and Bifurcation in Kolmogorov-Type Predator-Prey System with Discrete Delays,” Quarterly of Applied Mathematics, Vol. 59, 2001, pp. 159-172.

[8] X. P. Yan and Y. D. Chu, “Stability and Bifurcation Analysis for a Delayed Lotka-Volterra Predator-Prey System,” Journal of Computational and Applied Mathematics, Vol. 196, No. 1, 2006, pp. 198-210.

[9] X. P. Yan and C. H. Zhang, “Hopf Bifurcation in a Delayed Lokta-Volterra Predator-Prey System,” Nonlinear Analysis, Vol. RWA 9, 2008, pp. 114-127. doi:10.1016/j.nonrwa.2006.09.007

[10] Y. Song and J. Wei, “Local Hopf Bifurcation and Global Periodic Solutions in a Delayed Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 301, 2005, pp. 1-21. doi:10.1016/j.jmaa.2004.06.056

[11] X. P. Yan and W. T. Li, “Hopf Bifurcation and Global Periodic Solutions in a Delayed Predatorprey System,” Applied Mathematics and Computation, Vol. 177, No. 1, 2006, pp. 427-445.

[12] J. Zhang and B. Feng, “Geometric Theory and Bifurcation Problems of Ordinary Differential Equations,” Beijing University Press, Beijing, 2000.

[13] S. A. Gourley, “Travelling Fronts in the Diffusive Nicholsons Blowflies Equation with Distributed Delays,” Mathematical and Computer Modelling, Vol. 32, 2000, pp. 843-853. doi:10.1016/S0895-7177(00)00175-8

[14] J. K. Hale, “Theory of Functional Differential Equations,” Spring-Verlag, New York, 1977. doi:10.1007/978-1-4612-9892-2

[15] J. Wu, “Theory and Applications of Partial Functional Differential Equations,” Springer-Verlag, New York, 1996. doi:10.1007/978-1-4612-4050-1

[16] C. H. Zhang, X. P. Yan and G. H. Cui, “Hopf Bifurcations in a Predator-Prey System with a Discrete Delay and a Distributed Delay,” Nonlinear Analysis, Vol. RWA 11, 2010, pp. 4141-4153. doi:10.1016/j.nonrwa.2010.05.001

[17] X. Z. He, “Stability and Delays in a Predator-Prey System,” Journal of Mathematical Analysis and Applications, Vol. 198, No. 2, 1996, pp. 355-370.

[18] Z. Lu and W. Wang, “Global Stability for Two-Species Lotka-Volterra Systems with Delay,” Journal of Mathematical Analysis and Applications, Vol. 208, No. 1, 1997, pp. 277-280.

[19] E. L. Rice, “Allelopathy,” 2nd Edition, Academic Press, New York, 1984.

[20] J. M. Smith, “Models in Ecology,” Cambridge University, Cambridge, 1974.

[21] J. Chattopadhyay, “Effects of toxic Substance on a Two-Species Competitive System,” Ecological Modelling, Vol. 84, 1996, pp. 287-289. doi:10.1016/0304-3800(94)00134-0

[22] A. Mukhopadhyay, J. Chattopadhyay and P. K. Tapaswi, “A Delay Differential Equations Model of Plankton Allelopathy,” Mathematical Biosciences, Vol. 149, No. 2, 1998, pp. 167-189.

[23] B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, “Theory and Applications of Hopf Bifurcation,” Cambridge University Press, Cambridge, 1981.

[24] Y. Song, M. Han and J. Wei, “Stability and Hopf Bifurcation on a Simplified BAM Neural Network with Delays,” Physica D, Vol. 200, 2005, pp. 185-204. doi:10.1016/j.physd.2004.10.010