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- Open Access
Dynamical properties of a fractional reaction-diffusion trimolecular biochemical model with autocatalysis
- Hongwei Yin^{1, 2}Email author and
- Xiaoqing Wen^{1}
https://doi.org/10.1186/s13662-017-1427-4
© The Author(s) 2017
- Received: 12 July 2017
- Accepted: 16 November 2017
- Published: 25 November 2017
Abstract
In this paper, a reaction-diffusion trimolecular biochemical model with autocatalysis and fractional-order derivative is proposed. We establish the existence and uniqueness of a positive solution to this system in a Besov space. Besides, for this system, we obtain stability, Hopf and Turing bifurcations and spatial patterns. These dynamic behaviors of this system are slightly different from those of its corresponding first-order system. The difference is illustrated by performing some numerical simulations, through which our main results are verified.
Keywords
- Caputo’s derivative
- reaction-diffusion trimolecular model
- Besov spaces
- spatial patterns
- Hopf and Turing bifurcations
1 Introduction
Model (1.2a)-(1.2c) is an integer-order system, that is, the first-order derivative and the second-order derivative with respect to the time variable t and the spatial variable x, respectively. Wherein, the first-order derivative to the variable t implies the transient change rate of these reactions. However, due to the complexity of biochemical reactions, chemical reaction processes are often affected by or depend on the history of chemical reactions. Thus, this phenomenon can be described by fractional-order differential equations.
The outline of this paper follows here. In Section 2, some necessary lemmas and definitions are introduced. In Section 3, the solution to the fractional-order PDEs model (1.3a)-(1.3d) is established in Besov spaces. In Section 4 we study the stability and Hopf bifurcation of system (1.3a)-(1.3d) and perform numerical simulations. In Section 5, we investigate the Turing bifurcation of system (1.3a)-(1.3d), and some numerical simulations are made to show spatial patterns. Finally, we end our study with some conclusions.
2 Preliminaries
Definition 2.1
Lemma 2.2
System (2.7) with origin as a hyperbolic equilibrium point is linearly stable if each eigenvalue λ of A, \(\vert \arg (\lambda)\vert > \frac{\pi \alpha }{2}\); system (2.7) is linearly unstable if \(\vert \arg (\lambda)\vert <\frac{\pi \alpha }{2}\) for some eigenvalue λ of A.
3 Existence and uniqueness of solution to (1.3a)-(1.3d)
Next, we have the following.
Lemma 3.1
Proof
Lemma 3.2
Proof
Theorem 3.3
Let \(1< p<\infty \), \(\frac{2N}{3p}\leq \sigma <2\), \(\sigma \neq1+ \frac{1}{p}\) and \(1\leq q \leq 3p\). Then, given \((u_{0},v_{0})^{T} \in B^{\sigma }_{p,q,\mathcal{N}}\), there exists a constant \(\tau >0\) such that problem (1.3a)-(1.3d) has a unique locally mild positive solution \((u,v)^{T}: [0,\tau ] \rightarrow B^{\sigma }_{p,q, \mathcal{N}} \times B^{\sigma }_{p,q,\mathcal{N}}\).
Proof
According to Theorem 3.3, there exists small enough \(\tau >0\) such that problem (1.3a)-(1.3d) has a unique local mild solution defined in \(\mathcal{B}(\tau,R,u_{0},v_{0})\). This solution is bounded, i.e., \(\Vert u\Vert _{C([0,\tau ]:B^{\sigma }_{p,q,\mathcal{N}})}, \Vert v\Vert _{C([0,\tau ]:B^{\sigma }_{p,q,\mathcal{N}})} \leq \mathcal{R}= \max \{R+\Vert u_{0}\Vert _{B^{\sigma }_{p,q,\mathcal{N}}}, R+ \Vert v_{0}\Vert _{B^{\sigma }_{p,q,\mathcal{N}}}\}\). The mild solution of (1.3a)-(1.3d) at \(t=\tau \) exists, which is denoted by \((u(\tau),v(\tau))^{T}\). We take \((u(\tau),v(\tau))^{T}\) as another initial value of (1.3a)-(1.3d). Repeating the above discussion and by Theorem 3.3, we know that under the conditions in Theorem 3.3 the problem of (1.3a)-(1.3d) with the initial value \((u(0),v(0))^{T}=(u(\tau),v( \tau))^{T}\) has a unique mild solution (denoted by \((u_{1}(t),v_{1}(t))\)) defined on the interval \([\tau,\tau_{1}]\), and this solution is bounded, that is, there exist two positive constants \(\mathcal{R}_{1}\), \(R_{1}\) such that \(\Vert u_{1}\Vert _{C([\tau,\tau_{1}]:B ^{\sigma }_{p,q,\mathcal{N}})}, \Vert v_{1}\Vert _{C([\tau,\tau_{1}]:B^{ \sigma }_{p,q,\mathcal{N}})} \leq \mathcal{R}_{1}= \max \{R_{1}+\Vert u(\tau)\Vert _{B^{\sigma }_{p,q,\mathcal{N}}}, R_{1}+ \Vert v(\tau)\Vert _{B^{ \sigma }_{p,q,\mathcal{N}}}\}\). Repeating this process over and over, a mild solution of (1.3a)-(1.3d) is ultimately established on a maximum interval \((0,T_{\max })\). So, we have the following result.
Theorem 3.4
Let \(1< p<\infty \), \(\frac{2N}{3p}\leq \sigma <2\), \(\sigma \neq1+ \frac{1}{p}\) and \(1\leq q \leq 3p\). Then, given \((u_{0},v_{0})^{T} \in B^{\sigma }_{p,q,\mathcal{N}}\times B^{\sigma }_{p,q,\mathcal{N}}\), problem (1.3a)-(1.3d) has a unique mild positive solution \((u,v)^{T}: [0,T_{\max }) \rightarrow B^{\sigma }_{p,q,\mathcal{N}} \times B^{ \sigma }_{p,q,\mathcal{N}}\).
4 Stability and Hopf bifurcation
Theorem 4.1
- (i)
\((a+b)^{3}> \max \{ b-a,\frac{d_{2}(b-a)}{d_{1}} \} \),
- (ii)
\(\frac{d_{2}(b-a)}{d_{1}}<(a+b)^{3}<b-a\) and \({\frac{ 4 ( a+b) ^{4}}{ ( ( a+b) ^{3}+ a-b) ^{2}}} > \tan^{2} ( \frac{\alpha \pi }{2} ) +1\).
5 Turing pattern
In the above section, we have obtained the stability and Hopf bifurcation of system (1.3a)-(1.3d). However, these results focus on the spatial homogeneity of the dynamical behaviors of system (1.3a)-(1.3d). The spatial heterogeneity (i.e., Turing pattern) is interesting and significant for a reaction-diffusion system, which breaks the homogeneous states because the Turing bifurcation occurs. In this section, we continue to investigate the Turing instability of system (1.3a)-(1.3d). In particular, we will find difference of spatial patterns between system (1.3a)-(1.3d) and its corresponding first-order system. Turing patterns require two conditions. First, a nontrivial homogeneous steady state exists and is stable for spatially homogeneous perturbations. This condition is obtained in Theorem 4.1. Second, the stable steady state is unstable to at least one type of spatially heterogeneous perturbations. The second condition defines the condition for Turing instability, which ensures that local perturbations on the stable homogeneous steady state gradually expand globally.
Next, we will perform numerical simulations of system (1.3a)-(1.3d) in a two-dimensional space with parameters satisfying Turing conditions to obtain Turing patterns. To this end, we should discretize the space and the time of the problem because the dynamical behavior of system (1.3a)-(1.3d) cannot be investigated by using analytical methods or normal forms. We will transform it from an infinite-dimensional (continuous) to a finite-dimensional (discrete) form. System (1.3a)-(1.3d) is solved in a discrete domain with \(M\times N\) lattice sites. The spacing between the lattice points is defined by the lattice constant \(\Delta h=0.25\). In this discrete system, the Laplacian operator describing diffusion is calculated by using a five central difference scheme. The time evolution is also discrete, that is, the time goes by steps of \(\Delta t=0.02\), and is solved by an Adams-type predictor-corrector method for fractional-order equation. In order to avoid numerical artefacts, we checked the sensitivity of the results to the choice of the time and space steps, and their values have been chosen sufficiently small. Both numerical schemes are standard, hence we do not describe them here.
6 Conclusion and discussion
In this paper, for system (1.3a)-(1.3d), which is a fractional-order partial differential equation model, we have discussed the existence and uniqueness of the positive solution in a Besov space. Besides, we have obtained stability, Hopf bifurcation, Turing bifurcation and spatial patterns of this model. From theoretical analysis and numerical simulations, we find some difference between system (1.3a)-(1.3d) and its corresponding first-order system on the stability, Hopf bifurcation and spatial patterns. In detail, the fractional-order derivative enlarges the parameter region of the stability in comparison with its corresponding first-order system. Besides, these two systems with the same parameters can form different spatial patterns (see Figure 16). These results, as far as our knowledge goes, are completely new. So far, there are few literature works to study fractional-order partial differential equations. This kind of equations are still open, such as Hopf bifurcation direction, normal form and their application, which are worth being investigated in the future.
Declarations
Acknowledgements
We thank for helpful comments of referees. This work is supported by grants 61563033, 11563005 and 11563005 from the National Natural Science Foundation of China, and 20161BAB201010 from the Natural Science Foundation of Jiangxi Province.
Authors’ contributions
All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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