A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and
- PDF / 3,588,296 Bytes
- 48 Pages / 439.37 x 666.142 pts Page_size
- 7 Downloads / 145 Views
Mathematical Biology
A continuation method for spatially discretized models with nonlocal interactions conserving size and shape of cells and lattices Shin-Ichiro Ei1 · Hiroshi Ishii1 · Makoto Sato2 · Yoshitaro Tanaka3 Miaoxing Wang2 · Tetsuo Yasugi2
·
Received: 10 January 2020 / Revised: 15 July 2020 / Published online: 21 September 2020 © The Author(s) 2020
Abstract In this paper, we introduce a continuation method for the spatially discretized models, while conserving the size and shape of the cells and lattices. This proposed method is realized using the shift operators and nonlocal operators of convolution types. Through this method and using the shift operator, the nonlinear spatially discretized model on the uniform and nonuniform lattices can be systematically converted into a spatially continuous model; this renders both models point-wisely equivalent. Moreover, by the convolution with suitable kernels, we mollify the shift operator and approximate the spatially discretized models using the nonlocal evolution equations, rendering suitable for the application in both experimental and mathematical analyses. We also demonstrate that this approximation is supported by the singular limit analysis, and that the information of the lattice and cells is expressed in the shift and nonlocal operators. The continuous models designed using our method can successfully replicate the patterns corresponding to those of the original spatially discretized models obtained from the numerical simulations. Furthermore, from the observations of the isotropy of the Delta–Notch signaling system in a developing real fly brain, we propose a radially symmetric kernel for averaging the cell shape using our continuation method. We also apply our method for cell division and proliferation to spatially discretized models of the differentiation wave and describe the discrete models on the sphere surface. Finally, we demonstrate an application of our method in the linear stability analysis of the planar cell polarity model. Keywords Continuation method · Nonlocal interactions · Spatially discretized model · Singular limit analysis · Delta–Notch signaling Mathematics Subject Classification 35B36 · 35A35 · 92B05 · 35Q92
Extended author information available on the last page of the article
123
982
S.-I. Ei et al.
1 Introduction The development of multicellular organisms is regulated by intercellular communication and signaling pathways of various types. These include diffusible proteins acting as ligands and cell membrane proteins communicating with the neighboring cells. In the last fifty years, approaches comprising mathematical models and numerical simulations have been extensively used to understand the mechanisms underlying the biological phenomena. It is a common practice to divide a region of interest either into square or hexagonal elements representing cells, as shown in Fig. 1; this allows for the discrete spatial independent variables to be used. We also assume that the unknown dependent variables of the model are uniform on
Data Loading...
