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September  2017, 37(9): 4785-4813. doi: 10.3934/dcds.2017206

## Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics

 1 School of Mathematics and Information Science, Shaanxi Normal University, Xi'an, Shaanxi 710062, China 2 Department of Mathematics, California State University, Northridge, CA 91325, USA

* Corresponding author (J.Wu): jianhuaw@snnu.edu.cn

Received  July 2016 Revised  April 2017 Published  June 2017

Fund Project: The work is supported by the Natural Science Foundations of China(11271236,11671243,61672021), the Shaanxi New-star Plan of Science and Technology of China(2015KJXX-21) and the Fundamental Research Funds for the Central University(GK201701001).

In this paper, a reaction-diffusion system known as an autocatalytic reaction model is considered. The model is characterized by a system of two differential equations which describe a type of complex biochemical reaction. Firstly, some basic characterizations of steady-state solutions of the model are presented. And then, the stability of positive constant steady-state solution and the non-existence, existence of non-constant positive steady-state solutions are discussed. Meanwhile, the bifurcation solution which emanates from positive constant steady-state is investigated, and the global analysis to the system is given in one dimensional case. Finally, a few numerical examples are provided to illustrate some corresponding analytic results.

Citation: Yunfeng Jia, Yi Li, Jianhua Wu. Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4785-4813. doi: 10.3934/dcds.2017206
##### References:

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##### References:
A three dimensional view of spatiotemporal pattern of solution of system (1). The equilibrium $U^*\approx(0.77, 1.3)$. The parameter values are endowed with $d_1=1, d_2=0.05, a=1.3$, and the initial conditions are set as $(u_0, v_0)=(1.2, 0.9)$
A three dimensional view of spatiotemporal pattern of solution of system (1). The equilibrium $U^*\approx(0.91, 1.1)$. The parameter values are endowed with $d_1=1, d_2=0.06, a=1.1$ and the initial conditions are set as $(u_0, v_0)=(1.3, 0.8)$
A three dimensional view of spatiotemporal periodic pattern of solution of system (1). The equilibrium $U^*\approx(1.02, 0.98)$. The parameter values are endowed with $d_1=1.2, d_2=0.08, a=0.98$ and the initial conditions are set as $(u_0, v_0)=(2.1, 1.9)$
A three dimensional view of spatiotemporal periodic pattern of solution of system (1). The equilibrium $U^*\approx(0.97, 1.03)$. The parameter values are endowed with $d_1=1.2, d_2=0.09, a=1.03$ and the initial conditions are set as $(u_0, v_0)=(1.2, 1.1)$
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