×

The global attractor of a competitor-competitor-mutualist reaction-diffusion system with time delays. (English) Zbl 1121.35021

Large time behavior of solutions of reaction-diffusion systems is of great concern in population dynamics, and since many such equations possess multiple equilibria, the determination of the time-dependent solution in relation to the steady-state solutions is delicate. Thus, the author investigates the asymptotic behavior (stability problem) of a special time-dependent solution of a three-species reaction-diffusion system in a bounded domain under Neumann boundary condition using the method of upper and lower solutions. The system governs the population densities of a competitor, a competitor-mutualist and a mutualist, including possible time delays in the reaction mechanism.
Under a simple condition on the reaction rates, it is shown that the reaction-diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function, the corresponding time-dependent solution converges (globally) to the positive equilibrium solution. Consequently, the trivial solution and all forms of semi-trivial solutions are unstable, and the results apply to systems without delays as well as to their corresponding ordinary differential counterparts with or without delay.

MSC:

35B41 Attractors
35K57 Reaction-diffusion equations
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs
35R10 Partial functional-differential equations
92D25 Population dynamics (general)
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Du, Y., Positive periodic solutions of a competitor-competitor-mutualist model, Differential Integral Equations, 19, 1043-1066 (1996) · Zbl 0858.35057
[2] Feng, W., Permanence effect in a three species food chain model, Appl. Anal., 54, 192-209 (1994) · Zbl 0834.92023
[3] Feng, W.; Lu, X., Some coexistence and extinction results in a three species ecological model, Differential Integral Equations, 8, 617-626 (1995) · Zbl 0832.35070
[4] Fu, S.; Cui, S., Persistence in a periodic competitor-competitor-mutualist diffusion system, J. Math. Anal. Appl., 263, 234-245 (2001) · Zbl 0995.35008
[5] Lakos, N., Existence of steady-state solutions for a one-predator-two-prey system, SIAM J. Math. Anal., 21, 647-659 (1990) · Zbl 0705.92019
[6] Leung, A., A study of 3-species prey-predator reaction-diffusions by monotone schemes, J. Math. Anal. Appl., 100, 583-604 (1984) · Zbl 0568.92016
[7] Lu, C.; Feng, W.; Lu, X., Long-term survival in a 3-species ecological system, Dyn. Contin. Discrete Impuls. Syst., 3, 199-213 (1997) · Zbl 0883.35064
[8] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0780.35044
[9] Pao, C. V., Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198, 751-779 (1996) · Zbl 0860.35138
[10] Pao, C. V., Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48, 349-362 (2002) · Zbl 0992.35105
[11] Pao, C. V., Periodic solutions of parabolic systems with time delays, J. Math. Anal. Appl., 251, 251-263 (2000) · Zbl 0967.35061
[12] Pao, C. V., Global asymptotic stability of Lotka-Volterra 3-species reaction-diffusion systems with time delays, J. Math. Anal. Appl., 281, 186-204 (2003) · Zbl 1031.35071
[13] 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
[14] Tineo, A., Asymptotic behavior of solutions of a periodic reaction-diffusion system of a competitor-competitor-mutualist model, J. Differential Equations, 108, 326-341 (1994) · Zbl 0806.35095
[15] Zhao, X. Q., Global asymptotic behavior in a periodic competitor-competitor-mutualist system, Nonlinear Anal., 29, 551-568 (1997) · Zbl 0876.35058
[16] Zheng, S., A reaction-diffusion system of a competitor-competitor-mutualist model, J. Math. Anal. Appl., 124, 254-280 (1987) · Zbl 0658.35053
[17] Zhou, L.; Fu, Y. P., Periodic quasimonotone global attractor of nonlinear parabolic systems with discrete delays, J. Math. Anal. Appl., 250, 139-161 (2000)
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.