On the Role of Basic Production Terms in an Activator-Inhibitor System Modeling Biological Pattern Formation

Considered is the asymptotic behavior of solutions to a class of reaction-diffusion systems comprised of an activator and an inhibitor, which includes the system proposed by Gierer and Meinhardt as a model of biological pattern formation. By the basic production terms we mean those independent of the unknown functions. We prove that, when the basic production term for the activator is absent, some solutions with large initial data converge to the trivial state, i.e., the activator vanishes identically. Also, we demonstrate that there exist solutions which start from large initial data and converge to a small stationary solution in the case where the basic production term for the inhibitor is nontrivial and that for the activator is sufficiently small.