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Analysis of spreading speeds for monotone semiflows with an application to CNNs

Part of: Representations of solutions General theory in ordinary differential equations Circuits, networks

Published online by Cambridge University Press:  06 March 2019

ZHI-XIAN YU  and
LEI ZHANG Open the ORCID record for LEI ZHANG [Opens in a new window]
ZHI-XIAN YU
Affiliation:
Mathematics and Science, College, Shanghai Normal University, Shanghai200234, P. R. China e-mail: [email protected] College of Science, University of Shanghai for Science and TechnologyShanghai200093, China e-mail: [email protected]
LEI ZHANG
Affiliation:
Department of Mathematics, Harbin Institute of Technology in Weihai Weihai, Shandong264209, China e-mail: [email protected]
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Abstract

The purpose of this work is to investigate the properties of spreading speeds for the monotone semiflows. According to the fundamental work of Liang and Zhao [(2007) Comm. Pure Appl. Math.60, 1–40], the spreading speeds of the monotone semiflows can be derived via the principal eigenvalue of linear operators relating to the semiflows. In this paper, we establish a general method to analyse the sign and the continuity of the spreading speeds. Then we consider a limiting case that admits no spreading phenomenon. The results can be applied to the model of cellular neural networks (CNNs). In this model, we find the rule which determines the propagating phenomenon by parameters.

Keywords

Spreading speedstravelling wavesmonotone semiflowsCNNs

MSC classification

Primary: 35C07: Traveling wave solutions
Secondary: 34A33: Lattice differential equations 94C99: None of the above, but in this section
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 3 , June 2020 , pp. 369 - 384
Copyright
© Cambridge University Press 2019

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References

Aronson, D. G. & Weinberger, H. F. (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein, J. A. (editor), Lecture Notes in Mathematics, Vol. 446, Springer, pp. 5–49.Google Scholar
Aronson, D. G. & Weinberger, H. F. (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 33–76.CrossRefGoogle Scholar
Chua, L. O. (1998) CNN: A Paradigm for Complexity, Vol. 31, World Scientific Publisher.CrossRefGoogle Scholar
Chua, L. O. & Yang, L. (1988) Cellular neural networks: Applications. IEEE Trans. Circuits Systems I Fund. Theory Appl. 35, 1273–1290.CrossRefGoogle Scholar
Chua, L. O. & Yang, L. (1988) Cellular neural networks: Theory. IEEE Trans. Circuits Systems I Fund. Theory Appl. 35, 1257–1272.CrossRefGoogle Scholar
Ding, W. & Liang, X. (2015) Principal eigenvalues of generalized convolution operators on the circle and spreading speeds of noncompact evolution systems in periodic media. SIAM J. Math. Anal. 47, 855–896.CrossRefGoogle Scholar
Fang, J. & Zhao, X.-Q. (2014) Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. 46, 3678–3704.CrossRefGoogle Scholar
Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7, 355–369.CrossRefGoogle Scholar
Golomb, D. & Amitai, Y. (1997) Propagating neuronal discharges in neocortical slices: Computational and experimental study. J. Neurophysiol. 78, 1199–1211.CrossRefGoogle ScholarPubMed
Golomb, D., Wang, X.-J. & Rinzel, J. (1996) Propagation of spindle waves in a thalamic slice model. J. Neurophysiol. 75, 750–769.CrossRefGoogle Scholar
Kolmogorov, A. N., Petrovsky, I. & Piskunov, N. (1937) Etude de l’équation de la diffusion avec croissance de la quantité de matiere et son applicationa un probleme biologique. Mosc. Univ. Bull. Math. 1, 1–25.Google Scholar
Li, B., Weinberger, H. F. & Lewis, M. A. (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 82–98.CrossRefGoogle ScholarPubMed
Liang, X. & Zhao, X.-Q. (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math. 60, 1–40.CrossRefGoogle Scholar
Liang, X. & Zhao, X.-Q. (2010) Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259, 857–903.CrossRefGoogle Scholar
Liang, X., Yi, Y. & Zhao, X.-Q. (2006) Spreading speeds and traveling waves for periodic evolution systems. J. Diff. Equations 231, 57–77.CrossRefGoogle Scholar
Lui, R. (1989) Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math. Biosci. 93, 269–295.CrossRefGoogle ScholarPubMed
Lutscher, F. (2008) Density-dependent dispersal in integrodifference equations. J. Math. Biol. 56, 499–524.CrossRefGoogle ScholarPubMed
Perez-Munuzuri, V., Pérez-Villar, V. & Chua, L. (1992) Propagation failure in linear arrays of chua circuits. Int. J. Bifurc. Chaos 2, 403–406.CrossRefGoogle Scholar
Weinberger, H. (1982) Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353–396.CrossRefGoogle Scholar
Yu, Z. X. & Zhao, X.-Q. (2018) Propagation phenomena for CNNs with asymmetric templates and distributed delays. Discrete Cont. Dyn. Syst. 38, 905–939.CrossRefGoogle Scholar