Dynamics and bifurcations of a ratio-dependent predator-prey model
Resumen
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.
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Banerjee M , Petrovskii S., Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system, Theor Ecol 2011;4:37-53. DOI: https://doi.org/10.1007/s12080-010-0073-1
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. DOI: https://doi.org/10.1016/j.jtbi.2006.09.036
Huffaker CB, Experimental studies predators: dispersion factors and predator–prey oscillations, Hilgardia 1958;27:343–83. DOI: https://doi.org/10.3733/hilg.v27n14p343
Luckinbill LS, Coexistence of laboratory populations of paramecium aurelia and its predator didinium nautum., Ecology 1973;54:1320-7 . DOI: https://doi.org/10.2307/1934194
Okubo A Diffusion and ecological problems: mathematical models, Berlin: Springer; 1980.
Segel LA , Jackson JL , Dissipative structure: an explanation and an ecological example, J. Math. Biol. 36 (1998), 389-406. DOI: https://doi.org/10.1007/s002850050105
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 . DOI: https://doi.org/10.1142/S0218127415300141
Turing AM, The chemical basis of morphogenesis , Philos Trans R Soc Lond B 1952;237:37–7 DOI: https://doi.org/10.1098/rstb.1952.0012
Wang W , Liu QX , Jin Z, Spatiotemporal complexity of a ratio-dependent predator-prey system, Phys Rev E 2007;75:051913 DOI: https://doi.org/10.1103/PhysRevE.75.051913
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. DOI: https://doi.org/10.1016/j.cnsns.2016.07.027
E. L. Allgower and K. Georg. Numerical Continuation Methods: An Introduction, Springer-Verlag, Berlin, (1990). DOI: https://doi.org/10.1007/978-3-642-61257-2
Yu. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3rd ed, Springer-Verlag, New York, (2004). DOI: https://doi.org/10.1007/978-1-4757-3978-7
S. Wiggins, Introduction to Applied Non-linear Dynamical Systems and Chaos, 3rd ed, Springer-Verlag, University of Bristol (2000)
J.D. Murray, Differential Equations and Dynamical Systems, Springer-Verlag, New York, 1996.
L. Perko, Mathematical Biology, Springer, Berlin (1989).
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 DOI: https://doi.org/10.1016/j.camwa.2014.04.015
Y. Tang, W. Zhang, Heteroclinic bifurcations in a ratio-dependent predator-prey system, J. Math. Biol. 50, (2005) 699-712. DOI: https://doi.org/10.1007/s00285-004-0307-1
D. Xiao, S. Ruan, Bogdanov-Takens bifurcations in harvested predator-prey systems, Fields Institute Communications, 21, (1999) 493-506. DOI: https://doi.org/10.1090/fic/021/41
D. Xiao, K.F. Zhang, Multiple bifurcations of a predator-prey system, Discrete and Continuous Dynamical Systems Series B, 8, (2007) 417-433. DOI: https://doi.org/10.3934/dcdsb.2007.8.417
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. DOI: https://doi.org/10.1137/050623449
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. DOI: https://doi.org/10.1016/j.jmaa.2005.11.048
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. DOI: https://doi.org/10.1137/S0036139901397285
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