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

Authors: Jina Li; Xuehui Ji
Journal: Advances in Mathematical Physics
 (2020)

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## 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.


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## Body

## 1. Introduction

The system(1)ut=d1+d2u+d3vuxx+ua11−b11u−c1v,(2)vt=d4+d5u+d6vvxx+va21−b12u−c2v,where the parameters a11,a21>0 are the intrinsic growth coefficients, b11,c2≥0 are the coefficients of intraspecific competitions, di (i=1,2,⋯,6) are the diffusion rate, and parameters c1 and b12 determine the types of species interactions. When c1>0 and b12>0, this model is competition interaction; when c1>0 and b12<0, this model is mutualism interaction; and when c1<0 and b12<0, 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)
(3)η1=uxx−b1ux,η2=vxx−b1vx,when
(4)b11=4d2b12,c1=4d3b12,b12=4d5b12,c2=4d6b12.We mainly consider the following two cases.Case 1.
Whenb1=0, then system can be derived to the following form
(5)ut=d1+d2u+d3vuxx+a11u,(6)vt=d4+d5u+d6vvxx+a21v,and admits the CLBS:
(7)η1=uxx,(8)η2=vxx.The systems (7) and (8) are a system of ordinary differential equations (ODEs) with respect to variable x, so the following forms are the corresponding solutions:
(9)u=ϕ1tx+ϕ2t,(10)v=ψ1tx+ψ2t.In the following, inserting solutions (9) and (10) into (7) and (8) yields the following ODEs:
(11)dϕ1dt=a11ϕ1t,(12)dϕ2dt=2d2ϕ1t2+2d3ϕ1tψ1t+a11ϕ2t,(13)dψ1dt=a21ψ1t,(14)dψ2dt=2d5ϕ1tψ1t+2d6ψ1t2+a21ψ2t.We solve the systems (11)–(14); the solutions are shown as below:
(15)ϕ1t=C2ea11t,ϕ2t=ea11t2C2d3C1a21ea21t+d2C2a11ea11t+C3,ψ1t=C1ea21t,ψ2t=ea21t2C1d6C1a21ea21t+d5C2a11ea11t+C4.Then, the solutions of systems (5) and (6) can be shown by substituting the above functions ϕ1t, ϕ2t, ψ1t, and ψ2t into Eqs. (9) and (10).Case 2.
Whenb1≠0, the system
(16)ut=d1+d2u+d3vuxx+ua11−4d2b12u−4d3b12v,(17)vt=d4+d5u+d6vvxx+va21−4d5b12u−4d6b12v,admits the CLBS:
(18)η1=uxx−b1ux,(19)η2=vxx−b1vx.
The system (19) is a system of ODEs with respect to variable x, so the following forms are the corresponding solutions.
(20)u=ϕ1t+ϕ2teb1x,(21)v=ψ1t+ψ2teb1x.
In the following, inserting solutions (21) into (16) yields the following ODEs:
(22)dϕ1dt=−4ϕ1t2b12d2+−4b12d3ψ1t+a11ϕ1t,(23)dϕ2dt=−6b12d2ϕ2t−3ψ2tb12d3ϕ1t+−3b12d3ψ1t+d1b12+a11ϕ2t,(24)dψ1dt=−4ψ1t2b12d6+−4b12d5ϕ1t+a21ψ1t,(25)dψ2dt=−3b12d5ϕ2t−6ψ2tb12d6ψ1t+−3b12d5ϕ1t+d4b12+a21ψ2t.


## 3. Numerical Simulation

In the following, we research the numerical simulations of systems (22) and (25).Systems (22) and (25) have four equilibria E10,0,0,0,E2a11/4b12d2,0,0,0,E30,0,a21/4b12d6,0 and E4a11d6−a21d3/4b12d2d6−d3d5,0,a21d2−a11d5/4b12d2d6−d3d5,0. The Jacobian matrix of systems (22) and (25) at Eii=1,2,3,4 takes the form of
(26)JE1=a110000d1b12+a110000a210000d4b12+a21,JE2=−a110−d3a11d200−12a11+d1b120−3d3a114d200−d5a11d2+a210000−3d5a114d2+d4b12+a21,JE3=−d3a21d6+a110000−3d3a214d6+d1b12+a1100−d5a21d60−a2100−3d5a214d60−12a21+d4b12,JE4=J110−4b12d3ϕ1∗00J220−3b12d3ϕ1∗−4b12d5ψ1∗0J3300−3b12d5ψ1∗0J44,respectively, where
(27)J11=−4b122d2ϕ1∗+d3ψ1∗+a11,J22=−3b122d2ϕ1∗+d3ψ1∗+d1b12+a11,J33=−4b122d6ψ1∗+d5ϕ1∗+a21,J44=−3b122d6ψ1∗+d5ϕ1∗+d4b12+a21,ϕ1∗=a11d6−a21d34b12d2d6−d3d5,ψ1∗=a21d2−a11d54b12d2d6−d3d5.(i)
In the case ofc1>0 and b12>0, that is d3>0 and d5>0, obviously, the eigenvalues of the matrix JE1 are not all negative. So, equilibrium E1 of system is not stable. The eigenvalues of the matrix JE2 are λ21=−a11<0, λ22=−1/2a11+d1b12, λ23=−d5a11/d2+a21, and λ24=−3d5a11/4d2+d4b12+a21. If a11>max2d1b12,d2/d5a21,4d2/3d5d4b12+a21, that is λ2i<0,i=2,3,4, then equilibrium E2 is locally asymptotically stable (please see Figures 1 and 2). The eigenvalues of the matrix JE3 are λ31=−d3a21/d6+a11, λ32=−3d3a21/4d6+d1b12+a11, λ33=−a21<0, and λ34=−1/2a21+d4b12. If a21>maxd6/d3a11,4d6/3d3d1b12+a11,2d4b12, that is λ3i<0,i=1,2,4; then, equilibrium E3 is locally asymptotically stable. Due to the complexity of the eigenvalues of the matrix JE4, we do not give a theoretical result for the stability of equilibrium E4 here, and we shall investigate it through numerical simulations (please see Figures 3 and 4).(ii)
In the case ofc1<0 and c2<0, that is d3<0 and d5<0, similar to the analysis as in case (i), we can see that equilibrium E1 is unstable, E2 is locally asymptotically stable under the condition a11>max2d1b12,d2/d5a21,4d2/3d5d4b12+a21, E3 is locally asymptotically stable under the condition a21>maxd6/d3a11,4d6/3d3d1b12+a11,2d4b12, and E4 exists under conditions a11d6−a21d3d2d6−d3d5>0,a21d2−a11d5d2d6−d3d5>0(iii)
In the case ofc1>0 and c2<0, that is d3>0 and d5<0, similar to the analysis as in case (ii), we can see that equilibrium E1 is unstable, E2 is locally asymptotically stable under the condition a11>max2d1b12,d2/d5a21,4d2/3d5d4b12+a21, E3 is locally asymptotically stable under the condition a21>maxd6/d3a11,4d6/3d3d1b12+a11,2d4b12, and E4 exists under conditions d2d6−d3d5<0,a21d2−a11d5<0.Figure 1
The equilibriumE2 of system is locally asymptotically stable in the case of d3>0 and d5>0. Here, b1=1,a11=2,a21=1,d1=0.1,d2=0.1,d3=0.1,d4=0.1,d5=0.15,andd6=0.1.Figure 2
In the case ofd3>0 and d5>0, the asymptotical behavior of system with equilibrium E2 of the corresponding ODE system is stable. Here, b1=1, a11=2, a21=1, d1=0.1, d2=0.1, d3=0.1, d4=0.1, d5=0.15, and d6=0.1.
(a)(b)Figure 3
The equilibriumE4 of system is locally asymptotically stable in the case of d3>0 and d5>0. Here, b1=1, a11=10, a21=1, d1=0.1, d2=0.1, d3=0.1, d4=0.1, d5=0.001, and d6=0.1.Figure 4
In the case ofd3>0 and d6>0, the asymptotical behavior of system with equilibrium E4 of the corresponding ODE system is stable. Here, b1=1, a11=10, a21=1, d1=0.1, d2=0.1, d3=0.1, d4=0.1, d5=0.001, and d6=0.1.
(a)(b)



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