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Imperfect and Bogdanov–Takens bifurcations in biological models: from harvesting of species to isolation of infectives

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Abstract

A natural biological system under human interventions may exhibit complex dynamical behaviors which could lead to either the collapse or stabilization of the system. The bifurcation theory plays an important role in understanding this evolution process by modeling and analyzing the biological system. In this paper, we examine two types of biological models that Fred Brauer made pioneer contributions: predator–prey models with stocking/harvesting and epidemic models with importation/isolation. First we consider the predator–prey model with Holling type II functional response whose dynamics and bifurcations are well-understood. By considering human interventions such as constant harvesting or stocking of predators, we show that the system under human interventions undergoes imperfect bifurcation and Bogdanov-Takens bifurcation, which induces much richer dynamical behaviors such as the existence of limit cycles or homoclinic loops. Then we consider an epidemic model with constant importation/isolation of infective individuals and observe similar imperfect and Bogdanov-Takens bifurcations when the constant importation/isolation rate varies.

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Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Shigui Ruan

  2. School of Mathematical Sciences, CMA-Shanghai, Shanghai Jiao Tong University, Shanghai, 200240, China

    Dongmei Xiao

Authors
  1. Shigui Ruan
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  2. Dongmei Xiao
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Correspondence to Shigui Ruan.

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The authors declare no conflict of interest.

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In memory of Professor Fred Brauer.

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Research was partially supported by NSF (DMS-1853622 and DMS-2052648).

Research was partially supported by NSFC Grants (Nos. 12271353, 11931016 and 12161131001).

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Ruan, S., Xiao, D. Imperfect and Bogdanov–Takens bifurcations in biological models: from harvesting of species to isolation of infectives. J. Math. Biol. 87, 17 (2023). https://doi.org/10.1007/s00285-023-01951-3

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  • DOI: https://doi.org/10.1007/s00285-023-01951-3

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