Combined effects of the spatial heterogeneity and the functional response

Yu-Xia Wang; Wan-Tong Li

doi:10.3934/dcds.2019002

Article Contents

2019, Volume 39, Issue 1: 19-39. Doi: 10.3934/dcds.2019002

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Combined effects of the spatial heterogeneity and the functional response

Yu-Xia Wang^1, , and
Wan-Tong Li^2,

1.
School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

E-mail address: wangyux10@163.com (Y.-X. Wang, Corresponding author)
E-mail address: wangyux10@163.com (Y.-X. Wang, Corresponding author)

Received: June 30, 2017

Revised: April 30, 2018

Published: December 2018

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Abstract

This paper deals with a predator-prey model with Beddington-DeAngelis functional response, in which a protection zone is created for the prey species. Whether the combination of the protection zone and the Beddington-DeAngelis functional response can yield new results or not is of interest. The result reveals that they jointly produce a new critical value, which is smaller than that determined by either the protection zone or the functional response singly. As a result, rather different stationary patterns can be found and the combined effects are very prominent. Then the effect of the parameter $k$ in the Beddington-DeAngelis functional response is studied. The result deduces that as $k$ is large enough, there exists a unique positive stationary solution and it is linearly stable except a special case. Actually, we can obtain that the positive stationary solution is globally asymptotically stable.

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Mathematics Subject Classification: Primary: 35J60, 92D25; Secondary: 35B32.

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