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More degeneracy but fewer bifurcations in a predator–prey system having fully null linear part

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Abstract

Although the Poincaré normal form theory is not applicable, a predator–prey system having fully null linear part was proved to be degenerate of codimension 2 within the class of the GLV vector fields and unfolded versally within the GLV class. In this paper, we study the case that the nondegeneracy condition no longer holds, i.e., a quadratic term vanishes. We prove that the quadratic terms in the GLV normal form cannot be eliminated any more, showing that the vanished quadratic term substantially contributes to the degeneracy. We give its versal unfolding of codimension 3 within the GLV class, display all its bifurcations near the equilibrium, and see that the Hopf bifurcation and the heteroclinic bifurcation, which occur in the codimension 2 case, do not happen but two transcritical bifurcations at different equilibria may occur simultaneously, which is impossible in the codimension 2 case.

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Acknowledgements

The second author is partly supported by the National Natural Science Foundation of China (Nos. 11871041, 11931016), Science and Technology Innovation Action Plan of Science and Technology Commission of Shanghai Municipality (STCSM, No. 20JC1413200) and the Innovation Program of Shanghai Municipal Education Commission (No. 2021-01-07-00-02-E00087). The third author is partly supported by the National Natural Science Foundation of China (Nos. 11771307, 11831012, 11821001, 12171336).

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  1. School of Mathematics, Sichuan University, Chengdu, 610064, Sichuan, People’s Republic of China

    Zhifei Guo & Weinian Zhang

  2. School of Mathematical Sciences, Institute of Modern Analysis-A Frontier Research Center of Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, People’s Republic of China

    Yilei Tang

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  1. Zhifei Guo
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  2. Yilei Tang
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  3. Weinian Zhang
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Correspondence to Weinian Zhang.

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Guo, Z., Tang, Y. & Zhang, W. More degeneracy but fewer bifurcations in a predator–prey system having fully null linear part. Z. Angew. Math. Phys. 73, 122 (2022). https://doi.org/10.1007/s00033-022-01763-3

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  • DOI: https://doi.org/10.1007/s00033-022-01763-3

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