Advances in Mathematical Physics
Volume 2020 (2020), Article ID 8949263, 5 pages
https://doi.org/10.1155/2020/8949263

Research Article

Numerical Simulation and Symmetry Reduction of a Two-Component Reaction-Diffusion System

Jina Li and Xuehui Ji

College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

Correspondence should be addressed to Jina Li; lijina@zut.edu.cn

Received 2 June 2020; Accepted 31 July 2020; Published 5 October 2020

Academic Editor: Marin Marin

Copyright © 2020 Jina Li and Xuehui Ji. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, the symmetry classification and symmetry reduction of a two-component reaction-diffusion system are investigated, the reaction-diffusion system can be reduced to system of ordinary differential equations, and the solutions and numerical simulation will be showed by examples.

1. Introduction

The system

where the parameters are the intrinsic growth coefficients, are the coefficients of intraspecific competitions, () are the diffusion rate, and parameters and determine the types of species interactions. When and , this model is competition interaction; when and , this model is mutualism interaction; and when and , this model is prey-predator interaction. Systems (1) and (2) are proposed by Shigesada et al. [1] and include the classical Lotka-Volterra system, diffusive Lotka-Volterra system, and the generalization form [2–4]. The symmetry methods are also known one of the effective methods for construction exact solutions of differential equations; the symmetry method was created by Sophus Lie [5] and was developed by Ovsiannikov [6], Bluman [7], Olver [8], Cherniha [9], and other researchers [10–17]. The authors mainly research the Lie symmetry, exact solution, conditional Lie-Bäcklund symmetry (CLBS) of reaction-diffusion system, or the relevant research work [18–27]. This paper mainly research the symmetry reduction, solutions, and numerical simulation of systems (1) and (2).

2. Symmetry Reduction

In this section, we will illustrate the main feature of the reduction procedure. The systems (1) and (2) admit the conditional Lie-Bäcklund symmetry (CLBS)

when

We mainly consider the following two cases.

Case 1. When , then system can be derived to the following form

and admits the CLBS:

The systems (7) and (8) are a system of ordinary differential equations (ODEs) with respect to variable , so the following forms are the corresponding solutions:

In the following, inserting solutions (9) and (10) into (7) and (8) yields the following ODEs:

We solve the systems (11)–(14); the solutions are shown as below:

Then, the solutions of systems (5) and (6) can be shown by substituting the above functions , , , and into Eqs. (9) and (10).

Case 2. When , the system

admits the CLBS:

The system (19) is a system of ODEs with respect to variable , so the following forms are the corresponding solutions.

In the following, inserting solutions (21) into (16) yields the following ODEs:

3. Numerical Simulation

In the following, we research the numerical simulations of systems (22) and (25).

Systems (22) and (25) have four equilibria and . The Jacobian matrix of systems (22) and (25) at takes the form of

respectively, where

(i)In the case of and , that is and , obviously, the eigenvalues of the matrix are not all negative. So, equilibrium of system is not stable. The eigenvalues of the matrix are , , , and . If , that is , then equilibrium is locally asymptotically stable (please see Figures 1 and 2). The eigenvalues of the matrix are , , and . If , that is ; then, equilibrium is locally asymptotically stable. Due to the complexity of the eigenvalues of the matrix , we do not give a theoretical result for the stability of equilibrium here, and we shall investigate it through numerical simulations (please see Figures 3 and 4).

(ii)In the case of and , that is and , similar to the analysis as in case (i), we can see that equilibrium is unstable, is locally asymptotically stable under the condition , is locally asymptotically stable under the condition , and exists under conditions (iii)In the case of and , that is and , similar to the analysis as in case (ii), we can see that equilibrium is unstable, is locally asymptotically stable under the condition , is locally asymptotically stable under the condition , and exists under conditions .

Figure 1: The equilibrium of system is locally asymptotically stable in the case of and . Here, .

(a)

(b)

Figure 2: In the case of and , the asymptotical behavior of system with equilibrium of the corresponding ODE system is stable. Here, , , , , , , , , and .

Figure 3: The equilibrium of system is locally asymptotically stable in the case of and . Here, , , , , , , , , and .

(a)

(b)

Figure 4: In the case of and , the asymptotical behavior of system with equilibrium of the corresponding ODE system is stable. Here, , , , , , , , , and .

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (under Grant No. 11701594 and 11947093) and Training Plan for Key Young Teachers of Colleges and Universities in Henan Province (No. 2019GGJS143).

References

N. Shigesada, K. Kawasaki, and E. Teramoto, “Spatial segregation of interacting species,” Journal of Theoretical Biology, vol. 79, no. 1, pp. 83–99, 1979.

Y. Lou and W. M. Ni, “Diffusion, self-diffusion and cross-diffusion,” Journal of Differential Equations, vol. 131, no. 1, pp. 79–131, 1996.

K. Kuto, “Bifurcation branch of stationary solutions for a Lotka–Volterra cross-diffusion system in a spatially heterogeneous environment,” Nonlinear Analysis, vol. 10, no. 2, pp. 943–965, 2009.

L. Desvillettes and A. Trescases, “New results for triangular reaction cross diffusion system,” Journal of Mathematical Analysis and Applications, vol. 430, no. 1, pp. 32–59, 2015.

S. Lie, “Uber die integration durch bestimmte integrale von einer klasse linearer partieller differential gleichungen,” Archiv der Mathematik, vol. 6, pp. 328–368, 1881.

L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982.

G. W. Bluman and S. Kumei, “Symmetries and Differential Equations,” in Applied Mathematical Sciences, Springer Verlag, New York, 1989.

P. J. Olver, “Applications of Lie Groups to Differential Equations,” in Graduate Texts in Mathematics, Springer Verlag, New York, 1993.

R. Cherniha and V. Davydovych, Nonlinear Reaction-Diffusion Systems: Conditional Symmetry, Exact Solutions and their Applications in Biology, Springer-Verlag, New York, 2017.

R. Z. Zhdanov, “Higher conditional symmetry and reductions of initial-value problems,” Nonlinear Dynamics, vol. 28, pp. 17–27, 2004.

C. Z. Qu, “Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method,” IMA Journal of Applied Mathematics, vol. 62, pp. 283–302, 1999.

S. F. Tian, “Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation,” Applied Mathematics Letters, vol. 100, article 106056, 2020.

A. G. Johnpillai, A. H. Kara, and A. Biswas, “Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation,” Applied Mathematics Letters, vol. 26, no. 3, pp. 376–381, 2013.

X. E. Zhang and Y. Chen, “Inverse scattering transformation for generalized nonlinear Schrödinger equation,” Applied Mathematics Letters, vol. 98, pp. 306–313, 2019.

S. Y. Lou and X. B. Hu, “Infinitely many Lax pairs and symmetry constraints of the KP equation,” Journal of Mathematical Physics, vol. 38, no. 12, pp. 6401–6427, 1997.

L. N. Ji and C. Z. Qu, “Conditional Lie-Bäcklund symmetries and invariant subspaces to nonlinear diffusion equations,” IMA Journal of Applied Mathematics, vol. 76, pp. 17–55, 2011.

C. R. Zhu and C. Z. Qu, “Maximal dimension of invariant subspaces admitted by nonlinear vector differential operators,” Journal of Mathematical Physics, vol. 52, no. 4, article 043507, 2011.

R. Cherniha, “Lie symmetries of nonlinear two-dimensional reaction-diffusion systems,” Reports on Mathematical Physics, vol. 46, no. 1-2, pp. 63–76, 2000.

R. Cherniha and V. Davydovych, “Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system,” Mathematical and Computer Modelling, vol. 54, no. 5-6, pp. 1238–1251, 2011.

R. Cherniha, V. Davydovych, and L. Muzyka, “Lie symmetries of the Shigesada–Kawasaki–Teramoto system,” Communications in Nonlinear Science and Numerical Simulation, vol. 45, pp. 81–92, 2017.

R. Cherniha and V. Davydovych, “Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 8, pp. 3177–3188, 2012.

G. Groza, S. M. A. Khan, and N. Pop, “Approximate solutions of boundary value problems for ODEs using Newton interpolating series,” Carpathian Journal of Mathematics, vol. 25, pp. 73–81, 2009.

M. Marin, E. M. Craciun, and N. Pop, “Considerations on mixed initial-boundary value problems for micropolar porous bodies,” Dynamic Systems and Applications, vol. 25, pp. 175–195, 2016.

M. M. Bhatti, R. Ellahi, A. Zeeshan, M. Marin, and N. Ijaz, “Numerical study of heat transfer and Hall current impact on peristaltic propulsion of particle-fluid suspension with compliant wall properties,” Modern Physics Letters B, vol. 33, no. 35, article 1950439, 2019.

M. Marin, S. Vlase, R. Ellahi, and M. M. Bhatti, “On the partition of energies for the backward in time problem of thermoelastic materials with a dipolar structure,” Symmetry, vol. 11, no. 7, p. 863, 2019.

A. S. Fokas and Q. M. Liu, “Nonlinear interaction of traveling waves of nonintegrable equations,” Physical Review Letters, vol. 72, no. 21, pp. 3293–3296, 1994.

R. Z. Zhdanov, “Conditional Lie-Bäcklund symmetries and reductions of evolution equations,” Journal of Physics A: Mathematical and General, vol. 128, pp. 3841–3850, 1995.