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Bifurcation and Pattern Formation in an Activator–Inhibitor Model with Non-local Dispersal

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Abstract

In this paper, by approximating the non-local spatial dispersal equation by an associated reaction–diffusion system, an activator–inhibitor model with non-local dispersal is transformed into a reaction–diffusion system coupled with one ordinary differential equation. We prove that, to some extent, the non-locality-induced instability of the non-local system can be regarded as diffusion-driven instability of the reaction–diffusion system for sufficiently small perturbation. We study the structure of the spectrum of the corresponding linearized operator, and we use linear stability analysis and steady-state bifurcations to show the existence of non-constant steady states which generates non-homogeneous spatial patterns. As an example of our results, we study the bifurcation and pattern formation of a modified Klausmeier–Gray–Scott model of water–plant interaction.

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Acknowledgements

This work was done when the first author visited Department of Mathematics, William & Mary during the academic year 2018-2019, and she would like to thank Department of Mathematics, William & Mary for their support and warm hospitality.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Xiaoli Wang & Guohong Zhang

  2. Department of Mathematics, William & Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

Authors
  1. Xiaoli Wang
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  2. Junping Shi
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  3. Guohong Zhang
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Correspondence to Junping Shi.

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In Memory of Professor Masayasu Mimura.

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Partially supported by Grants from National Science Foundation of China (11701472, 11871403, 11871060), China Scholarship Council and US-NSF Grant DMS-1853598, Fundamental Research Funds for the Central Universities (XDJK2020B050)

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Wang, X., Shi, J. & Zhang, G. Bifurcation and Pattern Formation in an Activator–Inhibitor Model with Non-local Dispersal. Bull Math Biol 84, 140 (2022). https://doi.org/10.1007/s11538-022-01098-0

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