# zbMATH — the first resource for mathematics

Steady state coexistence solutions of reaction-diffusion competition models. (English) Zbl 1164.35351
Summary: Two species of animals are competing in the same environment. Under which conditions do they coexist peacefully? Or under which conditions does either one of the two species become extinct, i.e. is excluded by the other? It is natural to say that they can coexist peacefully if their rates of reproduction and self-limitation are relatively larger than those of competition. In other words, they can survive if they interact strongly among themselves and weakly with others. We investigate this phenomenon from a mathematical point of view. We concentrate on coexistence solutions of the competition model $\begin{cases} \Delta u + u(a - g(u,v)) = 0,\\ \Delta v + v(d - h(u,v)) = 0& \text{in} \;\Omega ,\\ u| _{\partial \Omega } = v| _{\partial \Omega } = 0. \end{cases}$ This system is the general model for the steady state of a competitive interacting system. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, the implicit function theorem and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.

##### MSC:
 35J55 Systems of elliptic equations, boundary value problems (MSC2000) 35J60 Nonlinear elliptic equations
##### Keywords:
elliptic theory; maximum principles
Full Text:
##### References:
 [1] R. S. Cantrell and C. Cosner: On the steady-state problem for the Volterra-Lotka competition model with diffusion. Houston Journal of mathematics 13 (1987), 337–352. · Zbl 0644.92016 [2] R. S. Cantrell and C. Cosner: On the uniqueness and stability of positive solutions in the Volterra-Lotka competition model with diffusion. Houston J. Math. 15 (1989), 341–361. · Zbl 0721.92025 [3] C. Cosner and A. C. Lazer: Stable coexistence states in the Volterra-Lotka competition model with diffusion. Siam J. Appl. Math. 44 (1984), 1112–1132. · Zbl 0562.92012 [4] D. Dunninger: Lecture note for applied analysis at Michigan State University. [5] R. Courant and D. Hilbert: Methods of Mathematical Physics, Vol. 1. Interscience, New York, 1961. · JFM 57.0245.01 [6] C. Gui and Y. Lou: Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model. Comm. Pure and Appl. Math. 12 (1994), 1571–1594. · Zbl 0829.92015 [7] J. L. Gomez and J. P. Pardo: Existence and uniqueness for some competition models with diffusion. [8] P. Hess: On uniqueness of positive solutions of nonlinear elliptic boundary value problems. Math. Z. 165 (1977), 17–18. · Zbl 0352.35046 [9] L. Li and R. Logan: Positive solutions to general elliptic competition models. Differential and Integral Equations 4 (1991), 817–834. · Zbl 0751.35014 [10] A. Leung: Equilibria and stabilities for competing-species, reaction-diffusion equations with Dirichlet boundary data. J. Math. Anal. Appl. 73 (1980), 204–218. · Zbl 0427.35011 [11] M. H. Protter and H. F. Weinberger: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs, N. J., 1967. [12] I. Stakgold and L. E. Payne: Nonlinear Problems in Nuclear Reactor Analysis. In nonlinear Problems in the Physical Sciences and Biology, Lecture notes in Mathematics 322, Springer, Berlin, 1973, pp. 298–307. · Zbl 0259.35025
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.