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.
Training Plan for Key Young Teachers of Colleges and Universities in Henan Province2019GGJS143National Natural Science Foundation of China11947093117015941. 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.
When b1=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.
When b1≠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.
In the case of c1>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).
In the case of c1<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
In the case of c1>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.
The equilibrium E2 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.
In the case of d3>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.
The equilibrium E4 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.
In the case of d3>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.
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).
ShigesadaN.KawasakiK.TeramotoE.Spatial segregation of interacting species1979791839910.1016/0022-5193(79)90258-32-s2.0-0018783879513804LouY.NiW. M.Diffusion, self-diffusion and cross-diffusion199613117913110.1006/jdeq.1996.01572-s2.0-0030579038KutoK.Bifurcation branch of stationary solutions for a Lotka–Volterra cross-diffusion system in a spatially heterogeneous environment200910294396510.1016/j.nonrwa.2007.11.0152-s2.0-56549113683DesvillettesL.TrescasesA.New results for triangular reaction cross diffusion system20154301325910.1016/j.jmaa.2015.03.0782-s2.0-84929372401LieS.Uber die integration durch bestimmte integrale von einer klasse linearer partieller differential gleichungen18816328368OvsiannikovL. V.1982New YorkAcademic PressBlumanG. W.KumeiS.Symmetries and Differential Equations1989New YorkSpringer Verlag10.1007/978-1-4757-4307-4OlverP. J.Applications of Lie Groups to Differential Equations1993New YorkSpringer Verlag10.1007/978-1-4612-4350-2ChernihaR.DavydovychV.2017New YorkSpringer-Verlag10.1007/978-3-319-65467-6ZhdanovR. Z.Higher conditional symmetry and reductions of initial-value problems200428172710.1023/A:10149626015692-s2.0-0036537869QuC. Z.Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method19996228330210.1093/imamat/62.3.2832-s2.0-6544263763TianS. F.Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation2020100, article 10605610.1016/j.aml.2019.106056JohnpillaiA. G.KaraA. H.BiswasA.Symmetry reduction, exact group-invariant solutions and conservation laws of the Benjamin–Bona–Mahoney equation201326337638110.1016/j.aml.2012.10.0122-s2.0-84869503935ZhangX. E.ChenY.Inverse scattering transformation for generalized nonlinear Schrödinger equation20199830631310.1016/j.aml.2019.06.0142-s2.0-85068234120LouS. Y.HuX. B.Infinitely many Lax pairs and symmetry constraints of the KP equation199738126401642710.1063/1.5322192-s2.0-0031508840JiL. N.QuC. Z.Conditional Lie-Bäcklund symmetries and invariant subspaces to nonlinear diffusion equations2011761755ZhuC. R.QuC. Z.Maximal dimension of invariant subspaces admitted by nonlinear vector differential operators2011524, article 04350710.1063/1.35745342-s2.0-79955458443ChernihaR.Lie symmetries of nonlinear two-dimensional reaction-diffusion systems2000461-2637610.1016/S0034-4877(01)80009-4ChernihaR.DavydovychV.Conditional symmetries and exact solutions of the diffusive Lotka-Volterra system2011545-61238125110.1016/j.mcm.2011.03.0352-s2.0-79957865303ChernihaR.DavydovychV.MuzykaL.Lie symmetries of the Shigesada–Kawasaki–Teramoto system201745819210.1016/j.cnsns.2016.09.0192-s2.0-84991494452ChernihaR.DavydovychV.Conditional symmetries and exact solutions of nonlinear reaction-diffusion systems with non-constant diffusivities20121783177318810.1016/j.cnsns.2011.12.0232-s2.0-84857659590GrozaG.KhanS. M. A.PopN.Approximate solutions of boundary value problems for ODEs using Newton interpolating series2009257381MarinM.CraciunE. M.PopN.Considerations on mixed initial-boundary value problems for micropolar porous bodies201625175195BhattiM. M.EllahiR.ZeeshanA.MarinM.IjazN.Numerical study of heat transfer and Hall current impact on peristaltic propulsion of particle-fluid suspension with compliant wall properties20193335, article 195043910.1142/S0217984919504396MarinM.VlaseS.EllahiR.BhattiM. M.On the partition of energies for the backward in time problem of thermoelastic materials with a dipolar structure201911786310.3390/sym110708632-s2.0-85068585238FokasA. S.LiuQ. M.Nonlinear interaction of traveling waves of nonintegrable equations199472213293329610.1103/PhysRevLett.72.32932-s2.0-12044249804ZhdanovR. Z.Conditional Lie-Bäcklund symmetries and reductions of evolution equations199512838413850