Abstract
The reaction diffusion system is one of the important models to describe the objective world. It is of great guiding importance for people to understand the real world by studying the Turing patterns of the reaction diffusion system changing with the system parameters. Therefore, in this paper, we study Gierer–Meinhardt model of the Depletion type which is a representative model in the reaction diffusion system. Firstly, we investigate the stability of the equilibrium and the Hopf bifurcation of the system. The result shows that equilibrium experiences a Hopf bifurcation in certain conditions and the Hopf bifurcation of this system is supercritical. Then, we analyze the system equation with the diffusion and study the impacts of diffusion coefficients on the stability of equilibrium and the limit cycle of system. Finally, we perform the numerical simulations for the obtained results which show that the Turing patterns are either spot or stripe patterns.
Highlights
As early as 1952s, the famous British mathematician Turing turned his attention to the eld of biology and succeeded with a reaction di usion system
An et al [14] have studied the explicit solution to the initial-boundary value problem of Gierer– Meinhardt model under certain conditions
We note that the system (4) experiences a Hopf bifurcation at = 0 under condition 0 < < 1
Summary
As early as 1952s, the famous British mathematician Turing turned his attention to the eld of biology and succeeded with a reaction di usion system. It is worth nothing that Turing patterns have long been widely found in nature and in many experimental systems, such as real chemical system [3,4,5], spiral galaxies in space [6], spiral wave electrical signals of myocardial tissue [7], biology systems [8, 9], hyper-points in nonlinear optical systems [10], etc To this end, the exploration and analysis of the Turing patterns has attracted the attention of many scholars. E rest of paper is organized as follows: In Section 2, we analyze the existence and stability of the positive equilibrium and the Hopf bifurcation.
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