# American Institute of Mathematical Sciences

April  2022, 27(4): 2189-2219. doi: 10.3934/dcdsb.2021129

## A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term

 1 School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China 2 School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China

* Corresponding author: Hongmei Cheng

Received  January 2021 Published  April 2022 Early access  April 2021

Fund Project: The second author is supported by NSFC grant 11701341. The third author is supported by NSFC grant 11771044

This paper is mainly considered a Leslie-Gower predator-prey model with nonlocal diffusion term and a free boundary condition. The model describes the evolution of the two species when they initially occupy the bounded region $[0,h_0]$. We first show that the problem has a unique solution defined for all $t>0$. Then, we establish the long-time dynamical behavior, including Spreading-vanishing dichotomy and Spreading-vanishing criteria.

Citation: Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2189-2219. doi: 10.3934/dcdsb.2021129
##### References:

show all references

##### References:
 [1] Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240 [2] Jianping Wang, Mingxin Wang. Free boundary problems with nonlocal and local diffusions Ⅱ: Spreading-vanishing and long-time behavior. Discrete and Continuous Dynamical Systems - B, 2020, 25 (12) : 4721-4736. doi: 10.3934/dcdsb.2020121 [3] Yunfeng Liu, Zhiming Guo, Mohammad El Smaily, Lin Wang. A Leslie-Gower predator-prey model with a free boundary. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2063-2084. doi: 10.3934/dcdss.2019133 [4] Siyu Liu, Haomin Huang, Mingxin Wang. A free boundary problem for a prey-predator model with degenerate diffusion and predator-stage structure. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1649-1670. doi: 10.3934/dcdsb.2019245 [5] Jia-Feng Cao, Wan-Tong Li, Meng Zhao. On a free boundary problem for a nonlocal reaction-diffusion model. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4117-4139. doi: 10.3934/dcdsb.2018128 [6] Yu-Xia Hao, Wan-Tong Li, Fei-Ying Yang. Traveling waves in a nonlocal dispersal predator-prey model. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3113-3139. doi: 10.3934/dcdss.2020340 [7] Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete and Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057 [8] Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189 [9] Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations and Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005 [10] Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501 [11] Fang Li, Xing Liang, Wenxian Shen. Diffusive KPP equations with free boundaries in time almost periodic environments: I. Spreading and vanishing dichotomy. Discrete and Continuous Dynamical Systems, 2016, 36 (6) : 3317-3338. doi: 10.3934/dcds.2016.36.3317 [12] Christian Kuehn, Thilo Gross. Nonlocal generalized models of predator-prey systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (3) : 693-720. doi: 10.3934/dcdsb.2013.18.693 [13] Xun Cao, Weihua Jiang. Double zero singularity and spatiotemporal patterns in a diffusive predator-prey model with nonlocal prey competition. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3461-3489. doi: 10.3934/dcdsb.2020069 [14] Feiying Yang, Wantong Li, Renhu Wang. Invasion waves for a nonlocal dispersal predator-prey model with two predators and one prey. Communications on Pure and Applied Analysis, 2021, 20 (12) : 4083-4105. doi: 10.3934/cpaa.2021146 [15] Xinjian Wang, Guo Lin. Asymptotic spreading for a time-periodic predator-prey system. Communications on Pure and Applied Analysis, 2019, 18 (6) : 2983-2999. doi: 10.3934/cpaa.2019133 [16] Hongmei Cheng, Rong Yuan. Existence and stability of traveling waves for Leslie-Gower predator-prey system with nonlocal diffusion. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5433-5454. doi: 10.3934/dcds.2017236 [17] Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002 [18] Yanfei Du, Ben Niu, Junjie Wei. A predator-prey model with cooperative hunting in the predator and group defense in the prey. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021298 [19] Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807 [20] Julián López-Gómez, Eduardo Muñoz-Hernández. A spatially heterogeneous predator-prey model. Discrete and Continuous Dynamical Systems - B, 2021, 26 (4) : 2085-2113. doi: 10.3934/dcdsb.2020081

2021 Impact Factor: 1.497

## Metrics

• HTML views (387)
• Cited by (0)

• on AIMS