### Mathematical Biosciences and Engineering

2012, Issue 4:767-784. doi: 10.3934/mbe.2012.9.767

# Global analysis of a simple parasite-host model with homoclinic orbits

• Received: 01 January 2012 Accepted: 29 June 2018 Published: 01 October 2012
• MSC : 92D30, 34C37, 37G35.

• In this paper, a simple parasite-host model proposed by Ebert et al.(2000) is reconsidered. The basic epidemiological reproduction number of parasite infection ($R_0$) and the basic demographic reproduction number of infected hosts ($R_1$) are given. The global dynamics of the model is completely investigated, and the existence of heteroclinic and homoclinic orbits is theoretically proved, which implies that the outbreak of parasite infection may happen. The thresholds determining the host extinction in the presence of parasite infection and variation in the equilibrium level of the infected hosts with $R_0$ are found. The effects of $R_0$ and $R_1$ on dynamics of the model are considered and we show that the equilibrium level of the infected host may not be monotone with respect to $R_0$. In particular, it is found that full loss of fecundity of infected hosts may lead to appearance of the singular case.

Citation: Jianquan Li, Yanni Xiao, Yali Yang. Global analysis of a simple parasite-host model withhomoclinic orbits[J]. Mathematical Biosciences and Engineering, 2012, 9(4): 767-784. doi: 10.3934/mbe.2012.9.767

### Related Papers:

• In this paper, a simple parasite-host model proposed by Ebert et al.(2000) is reconsidered. The basic epidemiological reproduction number of parasite infection ($R_0$) and the basic demographic reproduction number of infected hosts ($R_1$) are given. The global dynamics of the model is completely investigated, and the existence of heteroclinic and homoclinic orbits is theoretically proved, which implies that the outbreak of parasite infection may happen. The thresholds determining the host extinction in the presence of parasite infection and variation in the equilibrium level of the infected hosts with $R_0$ are found. The effects of $R_0$ and $R_1$ on dynamics of the model are considered and we show that the equilibrium level of the infected host may not be monotone with respect to $R_0$. In particular, it is found that full loss of fecundity of infected hosts may lead to appearance of the singular case.

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沈阳化工大学材料科学与工程学院 沈阳 110142

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