Steady State Solutions to General Competition and Cooperation Models

Joon Hyuk Kang


Two species of animals are competing or cooperating in the same environment. Under what conditions do they coexist peacefully? Or under what conditions does either one of the two species become extinct, that is, is either one of the two species excluded by the other? We investigate this phenomena in mathematical point of view.

In this paper we concentrate on coexistence solutions of the competition or cooperation model\[\left\{\begin{array}{l} \begin{array}{l}\Delta u + u(g_{1}(u) + h_{1}(v)) = 0\\\Delta v + v(g_{2}(u) + h_{2}(v)) = 0\end{array}\;\;\mbox{in}\;\;\Omega,\\u|_{\partial\Omega} = v|_{\partial\Omega} = 0. \end{array} \right.\]This system is the general model for the steady state of a competitive or cooperative interacting system depending on growth conditions for $g_{1}$, $g_{2}$, $h_{1}$ and $h_{2}$. The techniques used in this paper are elliptic theory, super-sub solutions, maximum principles, and spectrum estimates. The arguments also rely on some detailed properties of the solution of logistic equations.


Competition and cooperation system; Coexistence state

Full Text:



S.W. Ali and C. Cosner, On the uniqueness of the positive steady state for Lotka-Volterra Models with diffusion, Journal of Mathematical Analysis and Application 168 (1992), 329-341.

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.

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.

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.

D. Dunninger, Lecture Note for Applied Analysis in Michigan State University.

C. Gui and Y. Lou, Uniqueness and nonuniqueness of coexistence states in the Lotka-Volterra competition model, Comm. Pure and Appl. Math. XVL2 (12) (1994), 1571-1594.

J.L.-Gomez and R. Pardo, Existence and uniqueness for some competition models with diffusion, C.R. Acad. Sci. Paris, Ser. I Math. 313 (1991), 933-938.

J. Kang and Y. Oh, A sufficient condition for the uniqueness of positive steady state to a reaction diffusion system, Journal of Korean Mathematical Society 39 (39) (2002), 377-385.

J. Kang and Y. Oh, Uniqueness of coexistence state of general competition model for several competing species, Kyungpook Mathematical Journal 42 (2) (2002), 391-398.

J. Kang, Y. Oh and J. Lee, The existence, nonexistence and uniqueness of global positive coexistence of a nonlinear elliptic biological interacting model, Kangweon-Kyungki Math. Jour. 12 (1) (2004), 77-90.

P. Korman and A. Leung, A general monotone scheme for elliptic systems with applications to ecological models, Proceedings of the Royal Society of Edinburgh 102A (1986), 315-325.

P. Korman and A. Leung, On the existence and uniqueness of positive steady states in the Volterra-Lotka ecological models with diffusion, Applicable Analysis 26, 145-160.

L. Li and R. Logan, Positive solutions to general elliptic competition models, Differential and Integral Equations 4 (1991), 817-834.

A. Leung, Equilibria and stabilities for competing-species, reaction-diffusion equations with Dirichlet boundary data, J. Math. Anal. Appl. 73 (1980), 204-218.



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905