Dynamics and bifurcations of a ratio-dependent predator-prey model
DOI:
https://doi.org/10.5269/bspm.41174Abstract
In this paper, we study a ratio-dependent predator-prey model with modied Holling-Tanner formalism, by using dynamical techniques and numerical continuation algorithms implemented in Matcont. We determine codim-1 and 2 bifurcation points and their corresponding normal form coecients. We also compute a curve of limit cycles of the system emanating from a Hopf point.
References
1. Banerjee M , Petrovskii S., Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system, Theor Ecol 2011;4:37-53.
2. Baurmann M , Gross T , Feudel U , Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighbourhood of turing-hopf bifurcations, J Theor Biol 2007;245:220-9.
3. Huffaker CB, Experimental studies predators: dispersion factors and predator–prey oscillations, Hilgardia 1958;27:343–83.
4. Luckinbill LS, Coexistence of laboratory populations of paramecium aurelia and its predator didinium nautum., Ecology 1973;54:1320-7 .
5. Okubo A Diffusion and ecological problems: mathematical models, Berlin: Springer; 1980.
6. Segel LA , Jackson JL , Dissipative structure: an explanation and an ecological example, J. Math. Biol. 36 (1998), 389-406.
7. Shi HB , Ruan SG , Su Y , Zhang JF, Spatiotemporal dynamics of a diffusive leslie–gower predator–prey model with ratio-dependent functional response, Int J Bifurcat Chaos 2015; 25:1530014 .
8. Turing AM, The chemical basis of morphogenesis , Philos Trans R Soc Lond B 1952;237:37–7
9. Wang W , Liu QX , Jin Z, Spatiotemporal complexity of a ratio-dependent predator-prey system, Phys Rev E 2007;75:051913
10. Lai Zhang a , Jia Liu b , Malay Banerjee, Hopf and steady state bifurcation analysis in a ratio-dependent predator–prey model, 44 (2017) 52–73.
11. E. L. Allgower and K. Georg. Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin, (1990).
12. Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed, Springer-Verlag, New York, (2004).
13. S. Wiggins, Introduction to Applied Non-linear Dynamical Systems and Chaos, 3rd ed, Springer-Verlag, University of Bristol (2000)
14. J.D. Murray, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996.
15. L. Perko, Mathematical Biology, Springer, Berlin (1989).
16. Song YL , Zou XF , Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a hopf–turing bifurcation point, Comput Math Appl 2014;67:1978–97
17. Y. Tang, W. Zhang, Heteroclinic bifurcations in a ratio-dependent predator-prey system, J. Math. Biol. 50, (2005) 699-712.
18. D. Xiao, S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Institute Communications, 21, (1999) 493-506.
19. D. Xiao, K.F. Zhang, Multiple bifurcations of a predator-prey system, Discrete and Continuous Dynamical Systems Series B, 8, (2007) 417-433.
20. D. Xiao, H. Zhu, Multiple focus and hopf bifurcations in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 66, (2006) 802-819.
21. D. M. Xiao, W. X. Li, Dynamics in ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl. 324(1), (2006) 14-29.
22. H. Zhu, S. A. Campbell, G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63(2), (2002) 636-682.
2. Baurmann M , Gross T , Feudel U , Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighbourhood of turing-hopf bifurcations, J Theor Biol 2007;245:220-9.
3. Huffaker CB, Experimental studies predators: dispersion factors and predator–prey oscillations, Hilgardia 1958;27:343–83.
4. Luckinbill LS, Coexistence of laboratory populations of paramecium aurelia and its predator didinium nautum., Ecology 1973;54:1320-7 .
5. Okubo A Diffusion and ecological problems: mathematical models, Berlin: Springer; 1980.
6. Segel LA , Jackson JL , Dissipative structure: an explanation and an ecological example, J. Math. Biol. 36 (1998), 389-406.
7. Shi HB , Ruan SG , Su Y , Zhang JF, Spatiotemporal dynamics of a diffusive leslie–gower predator–prey model with ratio-dependent functional response, Int J Bifurcat Chaos 2015; 25:1530014 .
8. Turing AM, The chemical basis of morphogenesis , Philos Trans R Soc Lond B 1952;237:37–7
9. Wang W , Liu QX , Jin Z, Spatiotemporal complexity of a ratio-dependent predator-prey system, Phys Rev E 2007;75:051913
10. Lai Zhang a , Jia Liu b , Malay Banerjee, Hopf and steady state bifurcation analysis in a ratio-dependent predator–prey model, 44 (2017) 52–73.
11. E. L. Allgower and K. Georg. Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin, (1990).
12. Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed, Springer-Verlag, New York, (2004).
13. S. Wiggins, Introduction to Applied Non-linear Dynamical Systems and Chaos, 3rd ed, Springer-Verlag, University of Bristol (2000)
14. J.D. Murray, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996.
15. L. Perko, Mathematical Biology, Springer, Berlin (1989).
16. Song YL , Zou XF , Spatiotemporal dynamics in a diffusive ratio-dependent predator-prey model near a hopf–turing bifurcation point, Comput Math Appl 2014;67:1978–97
17. Y. Tang, W. Zhang, Heteroclinic bifurcations in a ratio-dependent predator-prey system, J. Math. Biol. 50, (2005) 699-712.
18. D. Xiao, S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Institute Communications, 21, (1999) 493-506.
19. D. Xiao, K.F. Zhang, Multiple bifurcations of a predator-prey system, Discrete and Continuous Dynamical Systems Series B, 8, (2007) 417-433.
20. D. Xiao, H. Zhu, Multiple focus and hopf bifurcations in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 66, (2006) 802-819.
21. D. M. Xiao, W. X. Li, Dynamics in ratio-dependent predator-prey model with predator harvesting, J. Math. Anal. Appl. 324(1), (2006) 14-29.
22. H. Zhu, S. A. Campbell, G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math. 63(2), (2002) 636-682.
Downloads
Published
2021-12-16
Issue
Section
Research Articles
License
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
