# American Institute of Mathematical Sciences

November  2018, 23(9): 3817-3836. doi: 10.3934/dcdsb.2018082

## A non-autonomous predator-prey model with infected prey

 1 College of Mathematics and Statistics, Northeast Petroleum University, Daqing, 163318, China 2 College of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, China 3 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

1Corresponding author

Received  September 2016 Revised  December 2017 Published  March 2018

A non-constant eco-epidemiological model with SIS-type infectious disease in prey is formulated and investigated, it is assumed that the disease is endemic in prey before the invasion of predator and that predation is more likely on infected prey than on the uninfected. Sufficient conditions for both permanence and extinction of the infected prey, and the necessary conditions for the permanence of the infected prey are established. It is shown that the predation preference to infected prey may even increase the possibility of disease endemic, and that the introduction of new resource for predator could be helpful for it to eradicate the infected prey. Numerical simulations have been performed to verify/extend our analytical results.

Citation: Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082
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##### References:
Solutions of system (2) with different hunting rate $\eta_1(t)$ on susceptible prey $s$. Here $(a)$: $\eta_1(t) = 0$; $(b)$: $\eta_1(t) = 1.5+\cos t$; $(c)$: $\eta_1(t) = 3+\cos t$; $(d)$: $\eta_1(t) = 4.5+\cos t$
Solutions of system (2) with different hunting rate $\eta_1(t)$ on susceptible prey $s$. $(a)$: $\eta_1(t) = 0$; $(b)$: $\eta_1(t) = 1.5+\cos(\sqrt{t})$; $(c)$: $\eta_1(t) = 3+\cos(\sqrt{t})$; $(d)$: $\eta_1(t) = 4.5+\cos(\sqrt{t})$
Basic behavior of solutions of model (49) with different intrinsic growth rate $r(t)$ for predator $y$. $(a)$: $r(t)\equiv 0$; $(b)$: $r(t) = \sin t+13$; $(c)$: $r(t) = \sin t+19$.Here we set $\eta_1(t) = \cos t+1$, and all the other parameters are same as those for FIG. 1
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