Li, Mei; Lin, Zhigui; Liu, Jiahong Coexistence in a competitor-competitor-mutualist model. (English) Zbl 1201.35097 Appl. Math. Modelling 34, No. 11, 3400-3407 (2010). Summary: This paper is concerned with a system modeling a competitor-competitor-mutualist three-species Lotka-Volterra model. By Schauder fixed point theory, the existence of positive solutions to a strongly coupled elliptic system is given. Applying the method of upper and lower solutions and its associated monotone iterations, the true solutions are constructed and a numerical simulation is also presented. Our results show that this system possesses at least one coexistence state if cross-diffusions and cross-reactions are weak. Cited in 5 Documents MSC: 35J56 Boundary value problems for first-order elliptic systems 65N99 Numerical methods for partial differential equations, boundary value problems 92D25 Population dynamics (general) Keywords:competition; mutualism; cross-diffusion; coexistence PDFBibTeX XMLCite \textit{M. Li} et al., Appl. Math. Modelling 34, No. 11, 3400--3407 (2010; Zbl 1201.35097) Full Text: DOI References: [1] Rai, B.; Freedman, H. I.; Addicott, J. F., Analysis of three-species models of mutualism in predator-prey and competitive systems, Math. Biosci., 65, 13-50 (1983) · Zbl 0532.92025 [2] Zheng, S., A reaction-diffusion system of a competitor-competitor-mutualist model, J. Math. Anal. Appl., 124, 254-280 (1987) · Zbl 0658.35053 [3] Du, Y., Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 19, 1043-1066 (1996) · Zbl 0858.35057 [4] Fu, S.; Cui, S., Persistence in a competitor-competitor-mutualist diffusion system, J. Math. Anal. Appl., 263, 234-245 (2001) · Zbl 0995.35008 [5] Pao, C. V., The global attractor of a competitor-competitor-mutualist reaction-diffusion system with time delays, Nonlinear Anal., 67, 2623-2631 (2007) · Zbl 1121.35021 [6] Shigesada, N.; Kawasaki, K.; Teramoto, E., Spatial segregation of interacting species, J. Theor. Biol., 79, 83-99 (1979) [7] Ruan, W. H., Positive steady-state solutions of a competing reaction-diffusion system with large cross-diffusion coefficients, J. Math. Anal. Appl., 197, 558-578 (1996) · Zbl 0855.35066 [8] Lou, Y.; Ni, W. M., Diffusion, self-diffusion and cross-diffusion, J. Diff. Equ., 131, 79-131 (1996) · Zbl 0867.35032 [9] Kuto, K.; Yamada, Y., Multiple coexistence states for a prey-predator system with cross-diffusion, J. Diff. Equ., 197, 315-348 (2004) · Zbl 1205.35116 [10] Kim, K. I.; Lin, Z. G., Coexistence of three-species in a strongly coupled elliptic system, Nonlinear Anal., 55, 313-333 (2003) · Zbl 1103.35323 [11] Chen, B.; Peng, R., Coexistence states of a strongly coupled prey-predator model, J. Partial Diff. Eqs., 18, 154-166 (2005) · Zbl 1330.35466 [12] Zhou, H.; Lin, Z. G., Coexistence in a strongly coupled system describing a two species cooperative model, Appl. Math. Lett., 20, 1126-1130 (2007) · Zbl 1260.35043 [13] Pao, C. V., Strongly coupled elliptic systems and applications to Lotka-Volterra models with cross-diffusion, Nonlinear Anal., 60, 1197-1217 (2005) · Zbl 1074.35034 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.