# American Institute of Mathematical Sciences

January  2019, 39(1): 19-39. doi: 10.3934/dcds.2019002

## Combined effects of the spatial heterogeneity and the functional response

 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)

Received  July 2017 Revised  May 2018 Published  October 2018

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.

Citation: Yu-Xia Wang, Wan-Tong Li. Combined effects of the spatial heterogeneity and the functional response. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 19-39. doi: 10.3934/dcds.2019002
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##### References:
 [1] Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214 [2] Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507 [3] Ming Liu, Dongpo Hu, Fanwei Meng. Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020259 [4] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [5] Liang Zhang, Zhi-Cheng Wang. Spatial dynamics of a diffusive predator-prey model with stage structure. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1831-1853. doi: 10.3934/dcdsb.2015.20.1831 [6] Kexin Wang. Influence of feedback controls on the global stability of a stochastic predator-prey model with Holling type Ⅱ response and infinite delays. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1699-1714. doi: 10.3934/dcdsb.2019247 [7] Shanshan Chen. Nonexistence of nonconstant positive steady states of a diffusive predator-prey model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 477-485. doi: 10.3934/cpaa.2018026 [8] Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189 [9] Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173 [10] Xiaoyuan Chang, Junjie Wei. Stability and Hopf bifurcation in a diffusive predator-prey system incorporating a prey refuge. Mathematical Biosciences & Engineering, 2013, 10 (4) : 979-996. doi: 10.3934/mbe.2013.10.979 [11] Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701 [12] Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005 [13] Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041 [14] Jicai Huang, Yijun Gong, Shigui Ruan. Bifurcation analysis in a predator-prey model with constant-yield predator harvesting. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2101-2121. doi: 10.3934/dcdsb.2013.18.2101 [15] Qing Zhu, Huaqin Peng, Xiaoxiao Zheng, Huafeng Xiao. Bifurcation analysis of a stage-structured predator-prey model with prey refuge. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2195-2209. doi: 10.3934/dcdss.2019141 [16] Fasma Diele, Carmela Marangi. Positive symplectic integrators for predator-prey dynamics. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2661-2678. doi: 10.3934/dcdsb.2017185 [17] Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026 [18] Arnaud Ducrot, Vincent Guyonne, Michel Langlais. Some remarks on the qualitative properties of solutions to a predator-prey model posed on non coincident spatial domains. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 67-82. doi: 10.3934/dcdss.2011.4.67 [19] Li Zu, Daqing Jiang, Donal O'Regan. Persistence and stationary distribution of a stochastic predator-prey model under regime switching. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2881-2897. doi: 10.3934/dcds.2017124 [20] Eric Avila-Vales, Gerardo García-Almeida, Erika Rivero-Esquivel. Bifurcation and spatiotemporal patterns in a Bazykin predator-prey model with self and cross diffusion and Beddington-DeAngelis response. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 717-740. doi: 10.3934/dcdsb.2017035

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