Stability analysis for free boundary problems in ecology
This paper is concerned with a kind of Stefan-like problems described by reaction-diffusion equations and free boundary conditions in one space dimension. The problem was previously proposed by the authors [Jap. J. Appl. Math. 2, 151-186 (1985; Zbl 0593.92019)], to model the regional partition phenomena arising in ecology. It has been shown there that the free boundary problem has a unique smooth solution and that its \(\omega\)- limit set is contained in the set of stationary solutions. Generally, the uniqueness of stationary solutions is not expected and the number of stationary solutions heavily depends on the nonlinearity appearing in the reaction-diffusion equations. The authors employ the phase plane analysis to construct stationary solutions since the stationary problem is, in a sense, expressed as boundary value problems for ordinary differential equations. The stability theory for stationary solutions is established with the aid of the notion of \(\omega\)-limit set and the comparison principle. Therefore, one can get useful information about the asymptotic behavior of solutions for the free boundary problem.
scientific article; zbMATH DE number 4000603