Skip to main content Accessibility help

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

  • ZHI-XIAN YU (a1) (a2) and LEI ZHANG (a3)


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.



Hide All
[1]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. 549.
[2]Aronson, D. G. & Weinberger, H. F. (1978) Multidimensional nonlinear diffusion arising in population genetics. Adv. Math. 30, 3376.
[3]Chua, L. O. (1998) CNN: A Paradigm for Complexity, Vol. 31, World Scientific Publisher.
[4]Chua, L. O. & Yang, L. (1988) Cellular neural networks: Applications. IEEE Trans. Circuits Systems I Fund. Theory Appl. 35, 12731290.
[5]Chua, L. O. & Yang, L. (1988) Cellular neural networks: Theory. IEEE Trans. Circuits Systems I Fund. Theory Appl. 35, 12571272.
[6]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, 855896.
[7]Fang, J. & Zhao, X.-Q. (2014) Traveling waves for monotone semiflows with weak compactness. SIAM J. Math. Anal. 46, 36783704.
[8]Fisher, R. A. (1937) The wave of advance of advantageous genes. Ann. Eugenics 7, 355369.
[9]Golomb, D. & Amitai, Y. (1997) Propagating neuronal discharges in neocortical slices: Computational and experimental study. J. Neurophysiol. 78, 11991211.
[10]Golomb, D., Wang, X.-J. & Rinzel, J. (1996) Propagation of spindle waves in a thalamic slice model. J. Neurophysiol. 75, 750769.
[11]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, 125.
[12]Li, B., Weinberger, H. F. & Lewis, M. A. (2005) Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. 196, 8298.
[13]Liang, X. & Zhao, X.-Q. (2007) Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm. Pure Appl. Math. 60, 140.
[14]Liang, X. & Zhao, X.-Q. (2010) Spreading speeds and traveling waves for abstract monostable evolution systems. J. Funct. Anal. 259, 857903.
[15]Liang, X., Yi, Y. & Zhao, X.-Q. (2006) Spreading speeds and traveling waves for periodic evolution systems. J. Diff. Equations 231, 5777.
[16]Lui, R. (1989) Biological growth and spread modeled by systems of recursions. I. Mathematical theory. Math. Biosci. 93, 269295.
[17]Lutscher, F. (2008) Density-dependent dispersal in integrodifference equations. J. Math. Biol. 56, 499524.
[18]Perez-Munuzuri, V., Pérez-Villar, V. & Chua, L. (1992) Propagation failure in linear arrays of chua circuits. Int. J. Bifurc. Chaos 2, 403406.
[19]Weinberger, H. (1982) Long-time behavior of a class of biological models. SIAM J. Math. Anal. 13, 353396.
[20]Yu, Z. X. & Zhao, X.-Q. (2018) Propagation phenomena for CNNs with asymmetric templates and distributed delays. Discrete Cont. Dyn. Syst. 38, 905939.


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed