**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**30172

##### Bifurcation Analysis of a Delayed Predator-prey Fishery Model with Prey Reserve in Frequency Domain

**Authors:**
Changjin Xu

**Abstract:**

In this paper, applying frequency domain approach, a delayed predator-prey ﬁshery model with prey reserve is investigated. By choosing the delay τ as a bifurcation parameter, It is found that Hopf bifurcation occurs as the bifurcation parameter τ passes a sequence of critical values. That is, a family of periodic solutions bifurcate from the equilibrium when the bifurcation parameter exceeds a critical value. The length of delay which preserves the stability of the positive equilibrium is calculated. Some numerical simulations are included to justify the theoretical analysis results. Finally, main conclusions are given.

**Keywords:**
Predator-prey model,
stability,
Hopf bifurcation,
frequency domain,
Nyquist criterion.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1081928

**References:**

[1] D. J. Allwright, Harmonic balance and the Hopf bifurcation theorem, Math. Proc. Cambridge Phil. Soc. 82 (3) (1977) 453-467.

[2] Y. Y. Chen and C. M. Song, Stability and bifurcation analysis in a prey-predator system with stage-structure for prey and time delay, Chaos, Solitons and Fractals 38 (4) (2008) 1104-1114.

[3] H. Freedman and V. S. H. Rao, The trade-off between mutual interference and time lags in predator-prey systems, Bull. Math. Biol., 45 (6) (1983) 991-1004.

[4] X. F. Liao and S. W. Li, Hopf bifurcation on a two-neuron system with distributed delays: a frequency domain approach, Nonlinear Dyn. 31 (3) (2003) 299-326.

[5] X. F. Liao, S. W. Li and G.R. Chen, Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain, Neural Netw. 17 (4) (2004) 545-561.

[6] A. I. Mees and L. 0. Chua, The Hopf bifurcation theorem and its applications to nonlinear oscillations in circuits and systems, IEEE Trans. Circuits Syst. 26 (4) (1979) 235-254.

[7] J. L. Moiola and G. R. Chen, Hopf bifurcation analysis: a frequency domain approach, Singapore: World Scientific, 1996.

[8] J. L. Moiola and G. R. Chen, Frequency domain approach to computa-tional analysis of bifurcations and limit cycles: a tutorial, Int. J. Bifur. Chaos 3 (4) (1993) 843-867.

[9] Y. L. Song, S. L. Yuan and J.M. Zhang, Bifurcation analysis in the delayed Leslie-Gower predator-prey system, Appl. Math. Modelling 33 (11) (2009) 4049-4061.

[10] W. Y. Wang and L. J. Pei, Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system, Acta Mech. Sin. 27 (2) (2011) 285¬296.

[11] C. J. Xu, M. X. Liao and X. F. He, Stability and Hopf bifurcation analysis for a Lokta-Volterra predator-prey model with two delays, Int. J. Appl. Math. Comput. Sci. 21 (1) (2011) 97-107.

[12] C. J. Xu, X. H. Tang and M. X. Liao, Frequency domain analysis for bifurcation in a simplified tri-neuron BAM network model with two delays, Neural Netw. 23 (7) 2010 872-880.

[13] C. J. Xu, X. H. Tang and M. X. Liao, Stability and bifurcation analysis of a delayed predator-prey model of prey dispersal in two-patch environments, Appl. Math. Comput. 216 (10) (2010) 2920-2936.

[14] C. J. Xu, X. H. Tang and M. X. Liao, Stability and bifurcation analysis of a six-neuron BAM neural network model with discrete delays, Neurocomputing, 74 (5) (2011) 689-707.

[15] C. J. Xu, X. H. Tang, M. X. Liao and X. F. He, Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays, Nonlinear Dyn. doi: 10.1007/s11071-010-9919-8.

[16] X. P. Yan and C.H. Zhang, Asymptotic stability of positive equilibrium solution for a delayed preyCpredator diffusion system, Appl. Math. Modelling 34 (1) (2010) 184-199.

[17] R. Zhang, J. F. Sun and H. X. Yang, Analysis of a prey-predator fishery model with prey reserve, Appl. Math. Sci. 50 (1) (2007) 2481-2492.

[18] X. Y. Zhou, X. Y. Shi and X. Y. Song, Analysis of non-autonomous predator-prey model with nonlinear diffusion and time delay, Appl. Math. Comput. 196 (1) (2008) 129-136.