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June  2017, 10(3): 445-461. doi: 10.3934/dcdss.2017021

## A periodic and diffusive predator-prey model with disease in the prey

 1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China 2 School of Mathematical and Statistical Sciences, University of Texas-Rio Grande Valley, Edinburg, TX 78539, USA 3 College of Atmospheric Science, Nanjing University of Information Science and Technology, Nanjing 210044, China

* Corresponding author: Guangping Hu

Received  December 2015 Revised  October 2016 Published  February 2017

In this paper, we are concerned with a time periodic and diffusivepredator-prey model with disease transmission in the prey. Firstwe consider a $SI$ model when the predator species is absent. Byintroducing the basic reproduction number for the $SI$ model, weshow the sufficient conditions for the persistence and extinctionof the disease. When the presence of the predator is taken intoaccount, a number of sufficient conditions for the co-existence ofthe prey and predator species, the global extinction of predatorspecies and the global extinction of both the prey and predatorspecies are given.

Citation: Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021
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