November  2020, 13(11): 3253-3269. doi: 10.3934/dcdss.2020130

Bogdanov-Takens bifurcation in predator-prey systems

a. 

School of Mathematics and Statistics, Lingnan Normal University, Zhanjiang, Guandong 524048, China

b. 

Department of Applied Mathematics, Western University, London, Ontario, N6A 5B7, Canada

c. 

School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China

* Corresponding author: Pei Yu

Received  November 2018 Revised  February 2019 Published  November 2020 Early access  November 2019

In this paper, we consider Bogdanov-Takens bifurcation in two predator-prey systems. It is shown that in the full parameter space, Bogdanov-Talens bifurcation can be codimension $ 2 $, $ 3 $ or $ 4 $. First, the simplest normal form theory is applied to determine the codimension of the systems as well as the unfolding terms. Then, bifurcation analysis is carried out to describe the dynamical behaviour and bifurcation property of the systems around the critical point.

Citation: Bing Zeng, Shengfu Deng, Pei Yu. Bogdanov-Takens bifurcation in predator-prey systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3253-3269. doi: 10.3934/dcdss.2020130
References:
[1]

A. D. Bazykin, Nonlinear Dynamics of Interaction Populations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 11. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798725.

[2]

R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funktsional. Anal. i Priložen, 9 (1975), 63.

[3]

K. S. ChengS. B. Hsu and S. S. Lin, Some results on global stability of a predator-prey system, J. Math. Biology, 12 (1981), 115-126.  doi: 10.1007/BF00275207.

[4] S. N. ChowC. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
[5]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987), 375-413.  doi: 10.1017/S0143385700004119.

[6]

H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980.

[7]

M. Gazor and P. Yu, Formal decomposition method and parametric normal form, Int. J. Bifurcation and Chaos Appl. Sci. Engrg., 20 (2010), 3487-3515.  doi: 10.1142/S0218127410027830.

[8]

M. Gazor and P. Yu, Spectral sequences and parametric normal forms, J. Differential Equations, 252 (2012), 1003-1031.  doi: 10.1016/j.jde.2011.09.043.

[9]

M. Gazor and M. Moazeni, Parametric normal forms for Bogdanov-Takens singularity; The generalized saddle-node case, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 205-224.  doi: 10.3934/dcds.2015.35.205.

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[11]

M. A. Han, J. Llibre and J. M. Yang, On uniqueness of limit cycles in general Bogdanov-Takens bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850115, 12 pp. doi: 10.1142/S0218127418501158.

[12]

M. A. Han, Bifurcation of limit cycles and the cusp of order n, Acta Math. Sinica, New Ser., 13 (1997), 64-75.  doi: 10.1007/BF02560525.

[13]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.

[14]

J. Jiang and P. Yu, Multistable phenomena involving equilibria and periodic motions in predator-prey systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750043, 28 pp. doi: 10.1142/S0218127417500432.

[15]

J. Jiang, W. Zhang and P. Yu, Tristable phenomenon in a predator-prey system arsing from multiple limit cycles bifurcation, submitted for publication, 2018.

[16]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.

[17]

C. Z. LiJ. Q. Li and Z. E. Ma, Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116.  doi: 10.3934/dcdsb.2015.20.1107.

[18]

A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. U.S., 6 (1920), 410-415.  doi: 10.1073/pnas.6.7.410.

[19]

P. Mardešić, The number of limit cycles of polynomial deformations of a Hamiltonian vector field, Ergodic Theory Dynam. Systems, 10 (1990), 523-529.  doi: 10.1017/S0143385700005721.

[20]

S. Q. Ruan and D. M. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896.

[21]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100. 

[22]

V. Volterra, Variazionie fluttuazioni del numero d'individui in specie animali, Mem. Acad. Lincei Roma., 2 (1926), 31-113. 

[23]

D. M. XiaoW. X. Li and M. A. Han, Dynamics in a ratio-dependent predator prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048.

[24]

P. Yu, Simplest normal forms of Hopf and generalized Hopf bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1917-1939.  doi: 10.1142/S0218127499001401.

[25]

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211 (1998), 19-38.  doi: 10.1006/jsvi.1997.1347.

[26]

P. Yu and A. Y. T. Leung, The simplest normal form of Hopf bifurcation, Nonlinearity, 16 (2003), 277-300.  doi: 10.1088/0951-7715/16/1/317.

show all references

References:
[1]

A. D. Bazykin, Nonlinear Dynamics of Interaction Populations, World Scientific Series on Nonlinear Science, Series A: Monographs and Treatises, 11. World Scientific Publishing Co., Inc., River Edge, NJ, 1998. doi: 10.1142/9789812798725.

[2]

R. I. Bogdanov, Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues, Funktsional. Anal. i Priložen, 9 (1975), 63.

[3]

K. S. ChengS. B. Hsu and S. S. Lin, Some results on global stability of a predator-prey system, J. Math. Biology, 12 (1981), 115-126.  doi: 10.1007/BF00275207.

[4] S. N. ChowC. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
[5]

F. DumortierR. Roussarie and J. Sotomayor, Generic 3-parameter families of vector fields on the plane, unfolding a singularity with nilpotent linear part. The cusp case of codimension 3, Ergodic Theory Dynam. Systems, 7 (1987), 375-413.  doi: 10.1017/S0143385700004119.

[6]

H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, 57. Marcel Dekker, Inc., New York, 1980.

[7]

M. Gazor and P. Yu, Formal decomposition method and parametric normal form, Int. J. Bifurcation and Chaos Appl. Sci. Engrg., 20 (2010), 3487-3515.  doi: 10.1142/S0218127410027830.

[8]

M. Gazor and P. Yu, Spectral sequences and parametric normal forms, J. Differential Equations, 252 (2012), 1003-1031.  doi: 10.1016/j.jde.2011.09.043.

[9]

M. Gazor and M. Moazeni, Parametric normal forms for Bogdanov-Takens singularity; The generalized saddle-node case, Discrete Contin. Dyn. Syst. Ser. A, 35 (2015), 205-224.  doi: 10.3934/dcds.2015.35.205.

[10]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.

[11]

M. A. Han, J. Llibre and J. M. Yang, On uniqueness of limit cycles in general Bogdanov-Takens bifurcation, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 28 (2018), 1850115, 12 pp. doi: 10.1142/S0218127418501158.

[12]

M. A. Han, Bifurcation of limit cycles and the cusp of order n, Acta Math. Sinica, New Ser., 13 (1997), 64-75.  doi: 10.1007/BF02560525.

[13]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.

[14]

J. Jiang and P. Yu, Multistable phenomena involving equilibria and periodic motions in predator-prey systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 27 (2017), 1750043, 28 pp. doi: 10.1142/S0218127417500432.

[15]

J. Jiang, W. Zhang and P. Yu, Tristable phenomenon in a predator-prey system arsing from multiple limit cycles bifurcation, submitted for publication, 2018.

[16]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition, Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.

[17]

C. Z. LiJ. Q. Li and Z. E. Ma, Codimension 3 B-T bifurcations in an epidemic model with a nonlinear incidence, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1107-1116.  doi: 10.3934/dcdsb.2015.20.1107.

[18]

A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Natl. Acad. Sci. U.S., 6 (1920), 410-415.  doi: 10.1073/pnas.6.7.410.

[19]

P. Mardešić, The number of limit cycles of polynomial deformations of a Hamiltonian vector field, Ergodic Theory Dynam. Systems, 10 (1990), 523-529.  doi: 10.1017/S0143385700005721.

[20]

S. Q. Ruan and D. M. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896.

[21]

F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., 43 (1974), 47-100. 

[22]

V. Volterra, Variazionie fluttuazioni del numero d'individui in specie animali, Mem. Acad. Lincei Roma., 2 (1926), 31-113. 

[23]

D. M. XiaoW. X. Li and M. A. Han, Dynamics in a ratio-dependent predator prey model with predator harvesting, J. Math. Anal. Appl., 324 (2006), 14-29.  doi: 10.1016/j.jmaa.2005.11.048.

[24]

P. Yu, Simplest normal forms of Hopf and generalized Hopf bifurcations, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 9 (1999), 1917-1939.  doi: 10.1142/S0218127499001401.

[25]

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vib., 211 (1998), 19-38.  doi: 10.1006/jsvi.1997.1347.

[26]

P. Yu and A. Y. T. Leung, The simplest normal form of Hopf bifurcation, Nonlinearity, 16 (2003), 277-300.  doi: 10.1088/0951-7715/16/1/317.

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