Large systems of reaction-diffusion equations describing many interaction species are studied. In each case, two uncoupled related subsystems are constructed and analyzed. It is shown that appropriate properties of the subsystems will insure that the full system has a steady-state solution which is strictly positive in each component. The problem is applicable to the study of the coexistence of all species when two subgroups of species are mixed together. The method of bifurcation and upper-lower solutions are used in the analysis.
Bifurcation theory is used in the construction of lower solutions.
We consider parameter dependent Dirichlet boundary value problems for semilinear eliptic equations whose nonlinear terms are linear at infinity. Results are presented about the asymptotic behaviour of solution continua which bifurcate from infinity. In particular, we obtain the existence of infinitely many solutions in case the parameter equals a simple eigenvalue of the linearized problem.