Bifurcation analysis for a model of gene expression with delays

Recently applications of mathematical modelings for gene expressions have received much attention. In this paper, we study the following system of gene expressions with delays {(M) over dot(t) = alpha(m)f(P(t- T(m))) - alpha(m)M(t), (P) over dot(t) = alpha(p)M(t- T(p)) - mu(p)P(t), which originated from the pattern mechanism of somites involving oscillating gene expression for zebrafish. The delays on mRNA and protein are due to the time needed for the gene to make the mRNA molecule and for the ribosome to translate mRNA into the protein molecule. The total delay tau = T(m) + T(p) is used as a bifurcation parameter to show that this system can exhibit Hopf bifurcations at certain critical values tau. For T(m) not equal T(p) and T(m) = T(p), the normal form theory for general DDEs developed by Faria and Magalhaes is used to perform center manifold reduction and determine the stability and direction of periodic solutions generated by Hopf bifurcation. The global existence of periodic solutions when T(m) = T(p) and T(p) = 0 is attained by using a result from Wu (1998) [21]. Examples are given to confirm the theoretical results. Published by Elsevier B.V.