# American Institute of Mathematical Sciences

February  2018, 38(2): 905-939. doi: 10.3934/dcds.2018039

## Propagation phenomena for CNNs with asymmetric templates and distributed delays

 1 College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, NL A1C 5S7, Canada

* Corresponding author: Zhixian Yu

Received  January 2017 Revised  August 2017 Published  February 2018

The aim of this work is to study propagation phenomena for monotone and nonmonotone cellular neural networks with the asymmetric templates and distributed delays. More precisely, for the monotone case, we establish the existence of the leftward ($c_{-}^*$) and rightward ($c_{+}^*$) spreading speeds for CNNs by appealing to the theory developed in [26,27], and $c_{-}^*+c_{+}^*>0$. Especially, if cells possess the symmetric templates and the same delayed interactions, then $c_{-}^*=c_{+}^*>0$. Moreover, if the effect of the self-feedback interaction $α f'(0)$ is not less than 1, then both $c_{-}^*>0$ and $c_{+}^*>0$. For the non-monotone case, the leftward and rightward spreading speeds are investigated by using the results of the spreading speed for the monotone case and squeezing the given output function between two appropriate nondecreasing functions. It turns out that the leftward and rightward spreading speeds are linearly determinate in these two cases. We further obtain the existence and nonexistence of travelling wave solutions under the weaker conditions than those in [46, 47] and show that the spreading speed coincides with the minimal wave speed.

Citation: Zhixian Yu, Xiao-Qiang Zhao. Propagation phenomena for CNNs with asymmetric templates and distributed delays. Discrete & Continuous Dynamical Systems, 2018, 38 (2) : 905-939. doi: 10.3934/dcds.2018039
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##### References:
The spread of wn, and the left plot shows 3-D graph of wn, and the right one indicates that projection of wn is on the plane (n; t)-plane.
The rightward traveling waves observed for wn(t) in different views.
The leftward traveling waves observed for wn(t) in different views.
The leftward traveling waves observed for $w_n(t)$ in different views.
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