University of Oulu

Jung-Chao Ban, Jyy-I Hong, Cheng-Yu Tsai, Yu-Liang Wu; Topological and random spread models with frozen symbols. Chaos 1 June 2023; 33 (6): 063144.

Topological and random spread models with frozen symbols

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Author: Ban, Jung-Chao1; Hong, Jyy-I1; Tsai, Cheng-Yu1;
Organizations: 1Department of Mathematical Sciences, National Chengchi University, Taipei 11605, Taiwan
2Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland
Format: article
Version: accepted version
Access: open
Online Access: PDF Full Text (PDF, 0.7 MB)
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Language: English
Published: American Institute of Physics, 2023
Publish Date: 2023-08-17


When a symbol or a type has been “frozen” (namely, a type of which an individual only produces one individual of the same type), its spread pattern will be changed and this change will affect the long-term behavior of the whole system. However, in a frozen system, the ξ-matrix and the offspring mean matrix are no longer primitive so that the Perron–Frobenius theorem cannot be applied directly when predicting the spread rates. In this paper, our goal is to characterize these key matrices and analyze the spread rate under more general settings both in the topological and random spread models with frozen symbols. More specifically, we propose an algorithm for explicitly computing the spread rate and relate the rate with the eigenvectors of the ξ-matrix or offspring mean matrix. In addition, we reveal that the growth of the population is exponential and that the composition of the population is asymptotically periodic. Furthermore, numerical experiments are provided as supporting evidence for the theory.

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Series: Chaos
ISSN: 1054-1500
ISSN-E: 1089-7682
ISSN-L: 1054-1500
Volume: 33
Issue: 6
Article number: 063144
DOI: 10.1063/5.0156262
Type of Publication: A1 Journal article – refereed
Field of Science: 111 Mathematics
Funding: This work was supported by the National Science and Technology Council, ROC [grant numbers: NSTC 111-2115-M-004-005-MY3, NSTC 111-2115-M-004-001-MY2].
Copyright information: © Copyright 2023 AIP. This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos 1 June 2023; 33 (6): 063144 and may be found at