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Analysis on Steady States of a Competition System with Nonlinear Diffusion Terms

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Abstract

Competition is a fundamental force shaping population size and structure as a result of limited availability of resources. In biomathematics, the biological models with competitive interactions exist widely. Furthermore, the nonlinear-diffusion (including self- and cross-diffusions) terms are incorporated to the biological models to better simulate the actual movement of species. Therefore, better compatibility with reality can be achieved by introducing nonlinear-diffusion into biological models with competitive interactions. As a result, a competition system with nonlinear-diffusion and nonlinear functional response is proposed and analyzed in this paper. We first briefly discuss the stability of trivial and semi-trivial solutions by spectrum analysis. Then the boundedness and the non-existence of steady states are studied. Based on the boundedness of the solutions, the existence of the steady states is also investigated by the fixed point index theory in a positive cone. The result shows that the two species can coexist when their diffusion and inter-specific competition pressures are controlled in a certain range.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, China

    Jingjing Wang & Hongchan Zheng

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  1. Jingjing Wang
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  2. Hongchan Zheng
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Correspondence to Jingjing Wang.

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The work is supported in part by the Natural Science Foundations of China (11771262), and by the Natural Science Basic Research Plan in Shaanxi Province of China (2018JM1020).

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Wang, J., Zheng, H. Analysis on Steady States of a Competition System with Nonlinear Diffusion Terms. Acta Appl Math 171, 26 (2021). https://doi.org/10.1007/s10440-021-00393-7

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  • DOI: https://doi.org/10.1007/s10440-021-00393-7

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