Global analysis of a simple parasite-host model with
homoclinic orbits
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1.
Faculty of Science, Air Force Engineering University, Xi'an 710051
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2.
Department of Applied Mathematics, Xi'an Jiaotong University, Xi'an 710049
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Received:
01 January 2012
Accepted:
29 June 2018
Published:
01 October 2012
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MSC :
92D30, 34C37, 37G35.
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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
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Abstract
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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