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 ˙ ( t ) = α m f ( P ( t - T m ) ) - μ m M ( t ) , P ˙ ( t ) = α p M ( t - T p ) - μ 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 τ = Tm + Tp is used as a bifurcation parameter to show that this system can exhibit Hopf bifurcations at certain critical values of τ. For Tm ≠ Tp and Tm = Tp, the normal form theory for general DDEs developed by Faria and Magalhães 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 Tm = Tp and Tp = 0 is attained by using a result from Wu (1998) [21]. Examples are given to confirm the theoretical results.