Zheng, Sining A reaction-diffusion system of a competitor-competitor-mutualist model. (English) Zbl 0658.35053 J. Math. Anal. Appl. 124, 254-280 (1987). We investigate the homogeneous Dirichlet problem and Neumann problem to a reaction-diffusion system of a competitor-competitor-mutualist model. The existence, uniqueness, and boundedness of the solutions are established by means of the comparison principle and the monotonicity method. For the Dirichlet problem, we study the existence of trivial and nontrivial nonnegative equilibrium solutions and their stabilities. For the Neumann problem, we analyze the constant equilibrium solutions and their stabilities. The main method used in studying of the stabilities is the spectral analysis to the linearized operators. The O.D.E. problem to the same model was proposed and studied by B. Rai, H. I. Freedman, and J. F. Addicott [Math. Biosci. 65, 13-50 (1983; Zbl 0532.92025)]. Cited in 21 Documents MSC: 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35K57 Reaction-diffusion equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B35 Stability in context of PDEs 92D25 Population dynamics (general) Keywords:Dirichlet problem; Neumann problem; reaction-diffusion system; competitor-competitor-mutualist model; existence; uniqueness; boundedness; comparison principle; monotonicity; equilibrium Citations:Zbl 0532.92025 PDFBibTeX XMLCite \textit{S. Zheng}, J. Math. Anal. Appl. 124, 254--280 (1987; Zbl 0658.35053) Full Text: DOI References: [1] Ewer, J. P.G.; Peletier, L. A., On the asymptotic behavior of solutions of semilinear parabolic equations, SIAM J. Appl. Math., 28, 43-53 (1975) · Zbl 0295.35044 [2] Fife, P., Mathematical Aspects of Reacting and Diffusing Systems (1979), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0403.92004 [3] Henry, D., Geometric Theory of Semilinear Parabolic Equations (1980), Springer-Verlag: Springer-Verlag New York [4] L. Hsiao, Y. Su, and Z. P. Xin; L. Hsiao, Y. Su, and Z. P. Xin [5] Leung, A., A study of three species prey-predator reaction-diffusions by monotone schemes, J. Math. Anal. Appl., 100, 583-604 (1984) · Zbl 0568.92016 [6] de Mottoni, P.; Tesei, A., Asymptotic stability results for a system of quasilinear parabolic equations, Appl. Anal., 9, 7-21 (1979) · Zbl 0408.35055 [7] Pao, C. V., Coexistence and stability of a competition-diffusion system in population dynamics, J. Math. Anal. Appl., 83, 54-76 (1981) · Zbl 0479.92013 [8] Pao, C. V., On nonlinear reaction-diffusion system, J. Math. Anal. Appl., 87, 165-198 (1982) · Zbl 0488.35043 [9] Rai, B.; Freedman, H. I.; Addicott, J. F., Analysis of three species models of mutualism in predator-prey and competitive systems, Math. Biosc., 65, 13-50 (1983) · Zbl 0532.92025 [10] Sattinger, D., Monotone methods in nonlinear elliptic and parabolic equations, Indiana U. Math. J., 979-1000 (1972) · Zbl 0223.35038 [11] Serrin, A remark on the preceding paper of Amann, Arch. Rational Mech. Anal., 44, 182-186 (1972) · Zbl 0225.35042 [12] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1983), Springer-Verlag: Springer-Verlag Berlin/Heidelberg/New York · Zbl 0508.35002 [13] Zheng, S. N., A reaction-diffusion system of a predator-prey-mutualist model, Math. Biosc., 78, 217-245 (1986) · Zbl 0634.92016 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.