Stability and bifurcation analysis of a discrete predator-prey model with Holling type III functional response

Volume 9, Issue 12, pp 6228--6243 http://dx.doi.org/10.22436/jnsa.009.12.27

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2313 Downloads
• 3055 Views

Authors

Jiangang Zhang - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Tian Deng - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Yandong Chu - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Shuang Qin - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Wenju Du - School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. Hongwei Luo - Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu, 730070, China. - Department of Information Engineering, Gansu forestry technological College, Tianshui, Gansu 741020, China.

Abstract

The paper studies the dynamical behaviors of a discrete predator-prey system with Holling type III functional response. More precisely, we investigate the local stability of equilibriums, flip bifurcation and Neimark-Sacker bifurcation of the model by using the center manifold theorem and the bifurcation theory. And analyze the dynamic characteristics of the system in two-dimensional parameter-spaces, one can observe the ''cluster'' phenomenon. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the model. The results show that we can more clearly and directly observe the chaotic phenomenon, period-adding and Neimark-Sacker bifurcation from two-dimensional parameter-spaces and the optimal parameters matching interval can also be found easily.

Share and Cite

Keywords

• Predator-prey system
• stability
• flip bifurcation
• Neimark-Sacker bifurcation
• cluster
• parameter-space.

MSC

•  74H60
•  42C05

References

• [1] S. Banerjee, A. Tsygvintsev, Stability and bifurcations of equilibria in a delayed Kirschner-Panetta model, Appl. Math. Lett., 40 (2015), 65--71


• [2] J. Carr, Applications of centre manifold theory, Applied Mathematical Sciences, Springer-Verlag, New York--Berlin (1981)


• [3] B.-S. Chen, J.-J. Chen, Bifurcation and chaotic behavior of a discrete singular biological economic system, Appl. Math. Comput., 219 (2012), 2371--2386


• [4] M. Danca, S. Codreanu, B. Bako, Detailed analysis of a nonlinear prey-predator model, J. Biol. Phys., 23 (1997), 11--20


• [5] E. M. Elabbasy, A. A. Elabbasy, Y. Zhang, Bifurcation analysis and chaos in a discrete reduced Lorenz system, Appl. Math. Comput., 228 (2014), 184--194


• [6] R. K. Ghaziani, W. Govaerts, C. Sonck, Resonance and bifurcation in a discrete-time predator-prey system with Holling functional response, Nonlinear Anal. Real World Appl., 13 (2012), 1451--1465


• [7] A. Gkana, L. Zachilas, Incorporating prey refuge in a preypredator model with a Holling type I functional response: random dynamics and population outbreaks, J. Biol. Phys., 39 (2013), 587--606


• [8] A. Gkana, L. Zachilas, Non-overlapping generation species: complex prey-predator interactions, Int. J. Nonlinear Sci. Numer. Simul., 16 (2015), 207--219


• [9] J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, Springer-Verlag, New York (1983)


• [10] K. P. Hadeler, I. Gerstmann, The discrete Rosenzweig model, Math. Biosci., 98 (1990), 49--72


• [11] Z. He, X. Lai, Bifurcation and chaotic behavior of a discrete-time predator-prey system,, Nonlinear Anal. Real World Appl., 12 (2011), 403--417


• [12] D.-P. Hu, H.-J. Cao, Bifurcation and chaos in a discrete-time predator-prey system of Holling and Leslie type, Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 702--715


• [13] Z.-Y. Hu, Z.-D. Teng, L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, Math. Comput. Simulation, 97 (2014), 80--93


• [14] D. Jana, Chaotic dynamics of a discrete predator-prey system with prey refuge, Appl. Math. Comput., 224 (2013), 848--865


• [15] Z.-J. Jing, J.-P. Yang, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals, 27 (2006), 259--277


• [16] Y. A. Kuznetsov, Elements of applied bifurcation theory, Second edition, Applied Mathematical Sciences, Springer-Verlag, New York (1998)


• [17] S.-Y. Li, Z.-L. Xiong, Bifurcation analysis of a predator-prey system with sex-structure and sexual favoritism, Adv. Difference Equ., 2013 (2013), 24 pages


• [18] X.-L. Liu, D.-M. Xiao, Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32 (2007), 80--94


• [19] O. P. Misra, P. Sinha, C. Singh, Stability and bifurcation analysis of a prey-predator model with age based predation, Appl. Math. Model., 37 (2013), 6519--6529


• [20] S. L. Qiu, C. Q. Wu, J. B. Chen, Qualitative analysis for a predator-prey model of Holling II-function response with nonlinear density dependence, Nat. Sci. ED, 34 (2006), 14--18


• [21] C. Robinson, Dynamical systems, Stability, symbolic dynamics, and chaos, Studies in Advanced Mathematics, CRC Press, Boca Raton (1995)


• [22] C. Robinson, Dynamical systems, Stability, symbolic dynamics, and chaos, Second edition, Studies in Advanced Mathematics, CRC Press, Boca Raton (1999)


• [23] P. Sangapate, New sufficient conditions for the asymptotic stability of discrete time-delay systems, Adv. Difference Equ., 2012 (2012), 8 pages


• [24] C. Wang, X.-Y. Li, Further investigations into the stability and bifurcation of a discrete predator-prey model, J. Math. Anal. Appl., 422 (2015), 920--939


• [25] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York (2003)


• [26] G.-D. Zhang, Y. Shen, B.-S. Chen, Bifurcation analysis in a discrete differential-algebraic predator-prey system, Appl. Math. Model., 38 (2014), 4835--4848


• [27] G.-D. Zhang, Y. Shen, B.-S. Chen, Bifurcation analysis in a discrete differential-algebraic predator-prey system, Appl. Math. Model., 38 (2014), 4835--4848