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

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)

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  • 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.

    Mathematics Subject Classification: Primary: 35J55, 35K57; Secondary: 35Q80, 92C45.

    Citation:

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  • Figure 1.  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 $(u_0, v_0)=(1.2, 0.9)$

    Figure 2.  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 $(u_0, v_0)=(1.3, 0.8)$

    Figure 3.  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 $(u_0, v_0)=(2.1, 1.9)$

    Figure 4.  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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