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

Kaigang Huang; Yongli Cai; Feng Rao; Shengmao Fu; Weiming Wang

doi:10.3934/dcdsb.2017209

Article Contents

2018, Volume 23, Issue 8: 3087-3107. Doi: 10.3934/dcdsb.2017209

This issue Previous Article Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions Next Article Continuous and discrete one dimensional autonomous fractional ODEs

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
* Corresponding author

Received: February 28, 2017

Revised: June 30, 2017

Early access: August 2017

Published: August 2018

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 35J65; 35B32; Secondary: 92D30.

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Figure 1. 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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