Oscillations in a model of repression with external control

A mathematical model for control by repression by an extracellular substance is developed, including diffusion and time delays. The model examines how active transport of a nutrient can produce either oscillatory or stable responses depending on a variety of parameters, such as diffusivity, cell size, or nutrient concentration. The system of equations for the mathematical model is reduced to a system of delay differential equations and linear Volterra equations. After linearizing these equations and forming the limiting Volterra equations, the resulting linear system no longer has any spatial dependence. Local stability analysis of the radially symmetric model shows that the system of equations can undergo Hopf bifurcations for certain parameter values, while other ranges of the parameters guarantee asymptotic stability. One numerical study shows that the model can exhibit intracellular biochemical oscillations with increasing extracellular concentrations of the nutrient, which suggests a possible trigger mechanism for morphogenesis.