A PDE Model

$${dA\over dt} = F(A) +{\delta\over \rho}(E-A),$$ (14)

$${\partial E\over \partial t} = {\partial^{2} E \over \partial x^{2}} + {\delta\over 1-\rho}(A-E) - k_EE.$$ (15)

where

$$E_{x}(L,t) + \alpha E(L,t) = 0, \qquad E_{x}(0,t) = 0.$$ (16)

Steady State Analysis

$$E_{xx} + {\delta \over 1-\rho}(A-E) - k_EE = 0, \qquad {\rm with~} F(A) +{\delta\over \rho}(E-A) = 0$$ (18)

on $0<x<L$ with $E_{x}(0) =0$ and $E_{x}(L) + \alpha E(L) = 0$, and

$$F(A) = {V_AA^2\over K_A+A^2} -k_AA+ A_0.$$ (19)

Figure: Plot of the function $E(A)= A - {\rho\over \delta} F(A)$ for $K_A = 1.0, V_A = 1.0, A_0 = 0.05, \delta = 0.1, \rho = 0.3, k_A= 0.2$.