Stability of Spiky Solutions in a Reaction-Diffusion System with Four Morphogens on the Real Line

Abstract

We study a reaction-diffusion system with four morphogens which has been suggested in [H. Takagi and K. Kaneko, Europhys. Lett., 56 (2001), pp. 145–151]. This system is a generalization of the Gray–Scott model [P. Gray and S. K. Scott, Chem. Eng. Sci., 38 (1983), pp. 29–43; 39 (1984), pp. 1087–1097] and allows for multiple activators and multiple substrates. We construct single-spike solutions on the real line and establish their stability properties in terms of conditions of connection matrices which describe the interaction of the components. We use a rigorous analysis for the linearized operator around single-spike solutions based on nonlocal eigenvalue problems and generalized hypergeometric functions. The following results are established for two activators and two substrates: Spiky solutions may be stable or unstable, depending on the type and strength of the interaction of the morphogens. In particular, it is shown that these patterns are stabilized in the following two cases. Case 1: interaction of different activators with each other (off-diagonal interaction of activators). Case 2: variation in strength of interaction of activators with different substrates (e.g., each activator has its preferred substrate).