Bifurcation analysis for a model of gene expression with delays

摘要

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_mf(P(t-T_m)-μ_mM(t), {P(t)=α_pM(t-T_p)-μ_pP(t), which originated from the pattern mechanism of somites involving oscillating gene expres-sion 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 mol-ecule. The total delay τ = T_m + T_p is used as a bifurcation parameter to show that this system can exhibit Hopf bifurcations at certain critical values of τ. For T_m ≠ T_p and T_m = T_p, the nor-mal 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.