Stationary and oscillatory fronts in a two-component genetic regulatory network model

We investigate a two-component gene network model, originally used to describe the spatiotemporal patterning of the gene products in early Drosophila   development. By considering a particular mode of interaction between the two gene products, denoted proteins AA and BB, we find both stable stationary and time-oscillatory fronts can occur in the reaction–diffusion system. We reduce the system by replacing BB with its spatial average (shadow system) and assume an abrupt “on-and-off” switch for the genes. In doing so, explicit formula are obtained for all steady-state solutions and their linear eigenvalues. Using the diffusion of A,DaA,Da, and the basal production rate, rr, as bifurcation parameters, we explore ranges in which a monotone, stationary front is stable, and show it can lose stability through a Hopf bifurcation, giving rise to oscillatory fronts. We also discuss the existence and stability of steady-state and time-oscillatory solutions with multiple extrema. An intuitive explanation for the occurrence of stable stationary and oscillatory front solutions is provided based on the behavior of AA in the absence of BB and the opposite regulation between AA and BB. Such behavior is also interpreted in terms of the biological parameters in the model, including those governing the connection of the gene network.