# American Institute of Mathematical Sciences

October  2018, 23(8): 3087-3107. doi: 10.3934/dcdsb.2017209

## Positive steady states of a density-dependent predator-prey model with diffusion

 1 School of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, China 2 School of Mathematical Science, Huaiyin Normal University, Huaian, 223300, China 3 School of Physical and Mathematical Sciences, Nanjing Tech University, Nanjing, Jiangsu 211816, China

* Corresponding author

Received  March 2017 Revised  July 2017 Published  September 2017

Fund Project: This research was supported by the National Science Foundation of China (Grant number 61672013,11601179,11601226,11361055 and 61373005) and the Natural Science Foundation of Jiangsu Province of China (BK20140927)

In this paper, we investigate the rich dynamics of a diffusive Holling type-Ⅱ predator-prey model with density-dependent death rate for the predator under homogeneous Neumann boundary condition. The value of this study lies in two-aspects. Mathematically, we show the stability of the constant positive steady state solution, the existence and nonexistence, the local and global structure of nonconstant positive steady state solutions. And biologically, we find that Turing instability is induced by the density-dependent death rate, and both the general stationary pattern and Turing pattern can be observed as a result of diffusion.

Citation: Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209
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##### References:
Numerical simulations of the long time behavior of solution $(N(x, t), P(x, t))$ of model (5) with different values of $d_2$. (a) $d_2=0.6$; (b) $d_2=0.25$;

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