TABLE OF CONTENTS The underlying structure of chemotaxis is the movement of many particles controlled by the other species. This chapter describes the way to derive dynamical partial differential equations from the statistic model.
Random walk on lattice induces the equation of diffusion. In this section, we describe mostly one-dimensional lattice 
, but n-dimensional lattice 
is treated similarly. Also, we restrict our considerations to the one-step jump process with continuous time.
Thus, we identify 
with
Let p n(t) ?[0, 1] be the conditional probability that the walker stayed on site n = 0 at time t = 0 is on site n = n at time t = t. Then, it holds that
where 
denotes the transition rates that the walker staying on site n jumps to site n 1 in the unit time. Sometimes, equation (4.1) is called the master equation. Because it is regarded as an ordinary differential equation on the infinite dimensional space, the reader may skip over the following exercise first.
Suppose that 
are constants in [0, 1] and study (4.1) in the following way. First, formulate the equation as an abstract linear ordinary differential equation in the Banach space ? 1, the set of absolutely convergent sequences as
where T is a bounded linear operator on ? 1 and P = (p n(t)) ? C 1([0,
