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Numerical simulations for the predator-prey model on surfaces with lumped mass method

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Abstract

The predator-prey model is powerful mathematical tool to describe the dynamics of biological systems and promote research on biological populations. In this paper, we present a lumped mass finite element method for solving the predator-prey models on surfaces. The main purpose of the proposed method is to overcome the difficulty of the positivity preservation of the solutions. Based on positivity preservation results, we investigate the stabilities of semi-discrete and fully discrete approximations. Besides, numerical simulations are considered to illustrate the feasibility of the numerical method by convergence tests. Two classical phenomena of the predator-prey model are simulated on three different implicit surfaces.

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Acknowledgements

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this article. This work is in parts supported by the NSF of China (No. U19A2079, No. 11671345, No. 61962056, No. 11971377), the Xinjiang Provincial University Research Foundation and the NSF of Shanxi Province (No. 2019JM-367).

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  1. College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China

    Meng Zhang, Xufeng Xiao & Xinlong Feng

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  1. Meng Zhang
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  2. Xufeng Xiao
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  3. Xinlong Feng
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Correspondence to Xinlong Feng.

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Zhang, M., Xiao, X. & Feng, X. Numerical simulations for the predator-prey model on surfaces with lumped mass method. Engineering with Computers 37, 2047–2058 (2021). https://doi.org/10.1007/s00366-019-00929-4

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