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Diffusive Spatial Movement with Memory

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Abstract

Animal movements and their underlying mechanisms are an extremely important research area in biology and have been extensively studied for centuries. However, spatial memory and cognition, which is the most significant difference between animal movements and chemical movements, has been ignored in the modeling of animal movements. To incorporate cognition and memory of “clever” animals in the simplest and self-contained way, we propose a delayed diffusion model via a modified Fick’s law, whereas in literature the standard diffusion for chemical movements was applied to describe “drunk” animal movements. Our mathematical model is expressed by a reaction–diffusion equation with an additional delayed diffusion term, which makes rigorous analysis intriguing and challenging. We show the wellposedness and analyze the asymptotic stability of steady state in the spatial memory model. It is shown that for the three possible reaction schemes, the stability of a spatially homogeneous steady state fully depends on the relationship between the two diffusion coefficients but is independent of the time delay. Finally, we numerically illustrate possible spatialtemporal patterns when the system is divergent.

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Acknowledgements

The authors thank the anonymous reviewer and the editor for helpful suggestions which improved the initial version of the manuscript. The first author’s research is partially supported by NSF Grants DMS-1313243 and DMS-1715651. The second author’s research is partially supported by Chinese NSF Grants 11671110 and 11201097. The third author’s research is partially supported by an NSERC grant. The fourth author’s research is partially supported by a Chinese NSF grant 61763024 and Foundation of a Hundred Youth Talents Training Program of Lanzhou Jiaotong University 152022.

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Authors and Affiliations

  1. Department of Mathematics, College of William and Mary, Williamsburg, VA, 23187-8795, USA

    Junping Shi

  2. Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, Heilongjiang, China

    Chuncheng Wang

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Hao Wang

  4. Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, 730070, Gansu, China

    Xiangping Yan

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  1. Junping Shi
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  2. Chuncheng Wang
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  3. Hao Wang
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  4. Xiangping Yan
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Correspondence to Hao Wang.

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Shi, J., Wang, C., Wang, H. et al. Diffusive Spatial Movement with Memory. J Dyn Diff Equat 32, 979–1002 (2020). https://doi.org/10.1007/s10884-019-09757-y

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  • DOI: https://doi.org/10.1007/s10884-019-09757-y

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