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 .
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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).