In predator-prey population model, the density capacity restricts the growth rate of prey population. Prey species is assumed with
logistic growth rate, which is
exponent dependent. The generalized Hopf point is found, hence the Bautin bifurcation with limit cycle hysteresis phenomena is observed, which brings forth the limit point cycle appearing on Bautin bifurcation line. The homoclinc bifurcation line starts from Bogdanov-Takens point and tangents to saddle-node bifurcation line. The homoclinic orbits and near dynamics are simulated, which manifests the complex dynamics of system.
References
[1]
Cai, L., Chen, G. and Xiao, D. (2012) Multiparametric Bifurcations of an Epidemiological Model with Strong Allee Effect. JournalofMathematicalBiology, 67, 185-215. https://doi.org/10.1007/s00285-012-0546-5
[2]
Hsu, S.B. and Huang, T.W. (1999) Hopf Bifurcation Analysis for a Predator-Prey System of Holling and Leslie Type. TaiwaneseJournalofMathematics, 3, 35-53. https://doi.org/10.11650/twjm/1500407053
[3]
Gasull, A., Kooij, R.E. and Torregrosa, J. (1997) Limit Cycles in the Holling-Tanner Model. PublicacionsMatemàtiques, 41, 149-167. https://doi.org/10.5565/publmat_41197_09
[4]
Lamontagne, Y., Coutu, C. and Rousseau, C. (2008) Bifurcation Analysis of a Predator-Prey System with Generalised Holling Type III Functional Response. Journal of DynamicsandDifferentialEquations, 20, 535-571. https://doi.org/10.1007/s10884-008-9102-9
[5]
Xiang, C., Huang, J. and Wang, H. (2022) Linking Bifurcation Analysis of Holling-Tanner Model with Generalist Predator to a Changing Environment. Studies in AppliedMathematics, 149, 124-163. https://doi.org/10.1111/sapm.12492
[6]
Xiang, C., Huang, J., Ruan, S. and Xiao, D. (2020) Bifurcation Analysis in a Host-Generalist Parasitoid Model with Holling II Functional Response. JournalofDifferentialEquations, 268, 4618-4662. https://doi.org/10.1016/j.jde.2019.10.036
[7]
Beddington, J.R. and May, R.M. (1982) The Harvesting of Interacting Species in a Natural Ecosystem. ScientificAmerican, 247, 62-69. https://doi.org/10.1038/scientificamerican1182-62
[8]
Brauer, F. and Soudack, A.C. (1979) Stability Regions in Predator-Prey Systems with Constant-Rate Prey Harvesting. JournalofMathematicalBiology, 8, 55-71. https://doi.org/10.1007/bf00280586
[9]
Brauer, F. and Soudack, A.C. (1982) Coexistence Properties of Some Predator-Prey Systems under Constant Rate Harvesting and Stocking. JournalofMathematicalBiology, 12, 101-114. https://doi.org/10.1007/bf00275206
[10]
Chowdhury, J., Al Basir, F., Mukherjee, A. and Roy, P.K. (2025) A Theta Logistic Model for the Dynamics of Whitefly Borne Mosaic Disease in Cassava: Impact of Roguing and Insecticide Spraying. JournalofAppliedMathematicsandComputing, 71, 4897-4914. https://doi.org/10.1007/s12190-025-02419-x
[11]
Chow, S.N., Li, C. and Wang, D. (1994) Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press. https://doi.org/10.1017/cbo9780511665639
[12]
Kuznetsov, Yu.A. (1998) Elements of Applied Bifrucation Theory. 2nd Edition, Springer-Verlag.
[13]
Kuznetsov, Yu.A. (2011) Practical Computation of Normal Forms on Center Manifolds at Degenerate Bogdanov-Takens Bifurcations. InternationalJournalofBifurcation&Chaos, 15, 3535-3546.
[14]
Dhooge, A., Govaerts, W. and Kuznetsov, Y.A. (2003) MATCONT: A MATLAB Package for Numerical Bifrucation Analysis of ODEs. ACM Transactions on Mathematical Software, 29, 141-164. https://doi.org/10.1145/779359.779362
[15]
Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Yu.A., Sandstede, B. and Wang, X.J. (2000) Auto97-Auto2000: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont), User’s Guide. Concordia University. http://indy.cs.concordia.ca
[16]
Govaerts, W.J.F. (2000) Numerical Methods for Bifurcations of Dynamical Equilibria. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9780898719543
