Abstract
In this paper we completely study bifurcations of an epidemic model with five parameters introduced by Hilker et al. (Am Nat 173:72–88, 2009), which describes the joint interplay of a strong Allee effect and infectious diseases in a single population. Existence of multiple positive equilibria and all kinds of bifurcation are examined as well as related dynamical behavior. It is shown that the model undergoes a series of bifurcations such as saddle-node bifurcation, pitchfork bifurcation, Bogdanov–Takens bifurcation, degenerate Hopf bifurcation of codimension two and degenerate elliptic type Bogdanov–Takens bifurcation of codimension three. Respective bifurcation surfaces in five-dimensional parameter spaces and related dynamical behavior are obtained. These theoretical conclusions confirm their numerical simulations and conjectures by Hilker et al., and reveal some new bifurcation phenomena which are not observed in Hilker et al. (Am Nat 173:72–88, 2009). The rich and complicated dynamics exhibit that the model is very sensitive to parameter perturbations, which has important implications for disease control of endangered species.
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This research was partially supported by the National Natural Science Foundations of China (No. 10831003 and No. 10925102), and the Program of Shanghai Subject Chief Scientists (No. 10XD1406200). The third author also thank Université de Lille 1 for financial support when she visited it and finished this paper.
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Cai, L., Chen, G. & Xiao, D. Multiparametric bifurcations of an epidemiological model with strong Allee effect. J. Math. Biol. 67, 185–215 (2013). https://doi.org/10.1007/s00285-012-0546-5
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DOI: https://doi.org/10.1007/s00285-012-0546-5
Keywords
- Epidemic model
- Allee effect
- Degenerate elliptic type
- Bogdanov–Takens bifurcation
- Codimension three
- Degenerate Hopf bifurcation