Turing instability and coexistence in an extended Klausmeier model with nonlocal grazing |
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| Institution: | 1. Department of Computer Science and Technology, North University of China, Taiyuan, Shanxi 030051, China;2. Data Science and Technology, North University of China, Taiyuan, Shanxi 030051, China;3. Department of Mathematics, North University of China, Taiyuan, Shanxi 030051, China;4. Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, China;5. Center for Ecology and Environmental Sciences, Northwestern Polytechnical University, Xi’an 710072, China;6. School of Computer and Information Technology, Shanxi University, Taiyuan 030006, China;7. Science and Technology on Electronic Test and Measurement Laboratory, North University of China, Taiyuan 030051, China;8. School of Mechanical Engineering and Center for OPTical IMagery Analysis and Learning (OPTIMAL),Northwestern Polytechnical University, Xi’an 710072, China;1. School of Ecology and Environment, Northwestern Polytechnical University, Xi’an 710072, China;2. School of Computer and Information Technology, Shanxi University, Taiyuan 030006, China;3. School of Mechanical Engineering and Center for OPTical IMagery Analysis and Learning, Northwestern Polytechnical University, Xi’an 710072, China |
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| Abstract: | In this paper, we study the coexistence of an extended Klausmeier model with cross-diffusion and nonlocal sustained grazing. First, we analyze a saddle–node bifurcation of spatially homogeneous system. Second, we focus on the reaction–diffusion system with nonlocal sustained grazing. Our main result is that nonlocal terms promote linear stability, and the system may produce pattern under the influences of self-diffusion and cross-diffusion. Moreover, both the grazing parameter and rainfall rate can induce transitions among bare soil state, vegetation pattern state and homogeneous vegetation state. Finally, we address the nonlocal reaction–diffusion system as a bifurcation problem, and analyze the existence and stability of bifurcation solutions. Furthermore, numerical simulations have been illustrated to verify our theoretical findings. |
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| Keywords: | Vegetation pattern Cross-diffusion Nonlocal grazing Bifurcation |
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