A simple model of gene regulation in response to stochastically changing environmental conditions is developed and analyzed. The model consists of a differential equation driven by a continuous time 2-state Markov process. The density function of the resulting process converges to a beta distribution. We show that the moments converge to their stationary values exponentially in time. Simulations of a two-stage process where protein production depends on mRNA concentrations are also presented demonstrating that protein concentration tracks the environment whenever the rate of protein turnover is larger than the rate of environmental change. Single-celled organisms are therefore expected to have relatively high mRNA and protein turnover rates for genes that respond to environmental fluctuations.

A numerical method for convection dominated diffusion problems, that exploits the use of characteristics, is derived and analyzed. It is shown to preserve positivity of solutions and be locally mass conserving. Stability, consistency and order one convergence are verified. Because of the way in which it determines characteristic pre-images of grid cells, the method can be easily implemented for 1-, 2-, or 3-dimensional problems on rectangular grids. which is to hold for x ∈ , t > 0, for a bounded domain ⊂ R d . Neumann boundary conditions were also specified: ∂u/∂ν = ∂v/∂ν = 0 (x ∈ ∂ ), where ∂/∂ν denotes the outer normal derivative along the boundary ∂ of . The variables are the cell density u(x, t) and a biochemical concentration v(x, t), which influences u(x, t) chemotactically, that is cells are compelled to move up the concentration gradient. This is reflected in the term ∇ · k 2 (u, v)∇v. When this term dominates the first on the right, the equation is said to be convection dominated, and it is this case that models some of the most interesting biological phenomena.

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The bifurcation function for an elliptic boundary value problem is a vector field B(ω) on R d whose zeros are in a one-to-one correspondence with the solutions of the boundary value problem. Finite element approximations of the boundary value problem are shown to give rise to an approximate bifurcation function B h (ω), which is also a vector field on R d . Estimates of the difference B(ω) − B h (ω) are derived, and methods for computing B h (ω) are discussed.

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