# American Institute of Mathematical Sciences

## Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme

 1 School of Mathematical Sciences, Anhui University, Hefei 230601, China 2 Department of Mathematics, Hangzhou Normal University, Hangzhou 311121, China

* Corresponding author: Ranchao Wu

Received  September 2020 Revised  March 2021 Published  April 2021

Fund Project: The second author is supported by NSF grants 11971032 and 62073114. The third author is supported by NSF grant 11671114 and NSF of Zhejiang LY20A010002

A depletion-type reaction-diffusion Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme and the homogeneous Neumann boundary conditions is introduced and investigated in this paper. Firstly, the boundedness of positive solution of the parabolic system is given, and the constant steady state solutions of the model are exhibited by the Shengjin formulas. Through rigorous theoretical analysis, the stability of the corresponding positive constant steady state solution is explored. Next, a priori estimates, the properties of the nonconstant steady states, non-existence and existence of the nonconstant steady state solution for the corresponding elliptic system are investigated by some estimates and the Leray-Schauder degree theory, respectively. Then, some existence conditions are established and some properties of the Hopf bifurcation and the steady state bifurcation are presented, respectively. It is showed that the temporal and spatial bifurcation structures will appear in the reaction-diffusion model. Theoretical results are confirmed and complemented by numerical simulations.

Citation: Mengxin Chen, Ranchao Wu, Yancong Xu. Dynamics of a depletion-type Gierer-Meinhardt model with Langmuir-Hinshelwood reaction scheme. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021132
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##### References:
Comparison of $f(h) = h$ and $f(h) = \frac{h}{1+mh}$ in the plane of $h-f(h)$
Upper: there exists a unique $\tilde{\lambda}\in(\pi_u, \pi_v)$ such that $h_1(\tilde{\lambda}) = 0$, we have $h_1(\lambda)>0$ when $\lambda\in(\pi_u, \tilde{\lambda})$, and $h_1(\lambda)<0$ when $\lambda\in(\tilde{\lambda}, \pi_v)$; Lower left: there exists a unique $\widetilde{\lambda}$ satisfying $\lambda_\sharp<\widetilde{\lambda}<\pi_v$ such that $h_3(\widetilde{\lambda}) = 0$, and one has $h_3(\lambda)>0$ when $\lambda\in(\lambda_\sharp, \widetilde{\lambda})$, and $h_3(\lambda)<0$ when $\lambda\in(\widetilde{\lambda}, \pi_v)$; Lower right: there exists a unique $\lambda^\ddagger\in(\lambda^\dagger, \pi_v)$ such that $f_u(\lambda^\ddagger) = 0$, and $f_u(\lambda)>0$ when $\lambda\in(\lambda^\dagger, \lambda^\ddagger)$, and $f_u(\lambda)<0$ when $\lambda\in(\lambda^\ddagger, \pi_v)$. Parameters are chosen as follows. Upper: $r = 0.5, m = 1.3, \alpha = 0.08, \beta = 0.25$; Lower left: $r = 1.15, m = 0.02, \alpha = 0.15, \beta = 3.8$; Lower right: $r = 1.15, m = 0.02, \alpha = 0.15, \beta = 2.8$
Positive constant steady state $E_* = (0.9022568,$ $0.0462763)$ is locally asymptotically stable. Here $d_1 = 1.5, d_2 = 0.5, r = 1.15, m = 0.02, \alpha = 0.1, \beta = 0.05$
Graph of the curve $\phi(\lambda)$. There exists a unique $\underline{\lambda}<\overline{\lambda}<\pi_v$ such that $\phi(\overline{\lambda}) = 0$. Thus, $\phi(\lambda)>0$ when $\lambda\in(\underline{\lambda}, \overline{\lambda})$ and $\phi(\lambda)<0$ when $\lambda\in(\overline{\lambda}, \pi_v)$, and $\phi(\lambda)$ achieves its maximum $\phi(\underline{\lambda}) = \phi_*$ at $\lambda = \underline{\lambda}$ for $\lambda\in(\pi_u, \pi_v)$. Here $r = 0.25, m = 2.7, \alpha = 0.098, \beta = 0.5$
Left: there exist two positive constants $\lambda_*$ and $\lambda^*$ satisfying $\pi_u<\lambda_*<\lambda^*<\pi_v$, such that $\Delta(\lambda_*) = \Delta(\lambda^*) = 0$ and $\Delta(\lambda)>0$ when $\lambda\in(\pi_u, \lambda_*)\cup(\lambda^*, \pi_v)$; Right: the existence of $p_+(\lambda)$ and $p_-(\lambda)$ in $\mathcal{C}_1$, it is found that $p_+(0)<0, p_-(0)<0$ and $\lim_{\lambda\rightarrow \lambda_{min}}p_+(\lambda) = \lim_{\lambda\rightarrow \lambda_{min}}p_-(\lambda)>0$. Here $d_1 = 0.4, d_2 = 1, r = 0.6, m = 0.5, \alpha = 0.2, \beta = 1.25$
There exist spatially homogeneous periodic solutions of system (4)
There exist steady state solutions of system (4). Here $d_1 = 8, d_2 = 0.5, r = 0.058, m = 0.15, \alpha = 4.0, \beta = 1.35$

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