scholarly journals Indentical synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo

2020 ◽  
Vol 8 (2) ◽  
pp. 45-53
Author(s):  
Phan Van Long Em
Keyword(s):  
Theoretical Result ◽  
Coupling Strength ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sufficient Condition ◽  
Linear Coupling ◽  
Natural Systems ◽  
Complete Network ◽  
Complete Networks

Synchronization is a ubiquitous feature in many natural systems and nonlinear science. This paper studies the synchronization in complete network consisting of n nodes. Each node is connected to all other nodes by linear coupling and represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the author seeks a sufficient condition on the coupling strength to achieve synchronization. The result shows that the more easily the nodes synchronize, the bigger the degrees of the networks. Based on this consequence, the author will test the theoretical result numerically to see if there is a compromise.

scholarly journals Synchronization in complete networks of reaction-diffusion equations of FitzHugh-Nagumo type with nonlinear coupling

2021 ◽  
Vol 13 (2) ◽  
pp. 43-51
Author(s):  
Van Long Em Phan
Keyword(s):  
Theoretical Result ◽  
Numerical Simulations ◽  
Coupling Strength ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Sufficient Condition ◽  
Nonlinear Coupling ◽  
Complete Network ◽  
Complete Networks

The synchronization in complete network consisting of  nodes is studied in this paper. Each node is connected to all other ones by nonlinear coupling and is represented by a reaction-diffusion system of FitzHugh-Nagumo type which can be obtained by simplifying the famous Hodgkin-Huxley model. From this complete network, the sufficient condition on the coupling strength to achieve the synchronization is found. The result shows that the networks with bigger in-degrees of nodes synchronize more easily. The paper also presents the numerical simulations for theoretical result and shows a compromise between the theoretical and numerical results.

Numerical simulations of reaction–diffusion equations modeling prey–predator interaction with delay

2018 ◽  
Vol 11 (04) ◽  
pp. 1850054 ◽  
Author(s):  
Ishtiaq Ali ◽  
Ghulam Rasool ◽  
Saleh Alrashed
Keyword(s):  
Dimensional Space ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
One Dimensional ◽  
Natural Systems ◽  
Nonlinear Reaction ◽  
The Rich ◽  
Delay Differential ◽  
Type Ii Functional Response

To model biological systems one often uses ordinary and partial differential equations. These equations can be quite good at approximating observed behavior, but they suffer from the downfall of containing many parameters, often signifying quantities which cannot be determined experimentally. For the better understanding of complicated phenomena, the delay differential equation approach to model such phenomena is becoming more and more essential to capture the rich variety of dynamics observed in natural systems. In this study, we investigated numerically the influence of delay on the dynamics of nonlinear reaction–diffusion equations modeling prey–predator interaction using finite difference scheme subject to appropriate initial and boundary conditions. We first consider the prey–predator with Holling type II functional response where the growth of prey is assumed to be logistic in the sense of predator in one-dimensional space. The effect of delay was investigated with the help of simulations and is compared with the model equation without delay. The proposed method is then extended to two-dimensional space.

scholarly journals Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions

2020 ◽  
Vol 18 (1) ◽  
pp. 1552-1564
Author(s):  
Huimin Tian ◽  
Lingling Zhang
Keyword(s):  
Boundary Conditions ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Neumann Boundary ◽  
Reaction Diffusion ◽  
Neumann Boundary Conditions ◽  
Reaction Diffusion Equations ◽  
Time Dependent ◽  
Diffusion Equations ◽  
Nonlocal Reaction

Abstract In this paper, the blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients are investigated under Neumann boundary conditions. By constructing some suitable auxiliary functions and using differential inequality techniques, we show some sufficient conditions to ensure that the solution u ( x , t ) u(x,t) blows up at a finite time under appropriate measure sense. Furthermore, an upper and a lower bound on blow-up time are derived under some appropriate assumptions. At last, two examples are presented to illustrate the application of our main results.

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