A SHORT COURSE IN THE MODELING OF CHEMOTAXIS

is the ”mean waiting time at the i th site”. Equation (3.1) appears at first like a
Markov process (the current state being independent of the previous state). How-
ever, this is in general not the case since the walker can influence the weights. We
saw this in chapter 2 in the models described there. That is, the organisms may
secrete, eat, or otherwise influence the state of the control species. So, even though
the transitional probabilities are not shown to depend explicitly on the pis, there is
implicit dependence as W can, and usually does, depend on the pis.
In the rest of this chapter, we will consider four different models that determine
the form of the transitional probabilities. These are (1) a local information model,
(2) a barrier model, (3) a nearest neighbor model, and (5) a gradient-based model.
We will also see how these relate, at least formally, to the macroscopic model de-
scribed in the previous chapter. Equations that govern the control species will be
ignored here.
3.1 A Local Information Model
Here, the transitional probability depends only on the weight at the node of
interest—i.e.
T ± n (W ) = λ ˆ T (wn) ∀n 2 .
In this assignment we are also separating the units by letting the parameter λ have
units of 1/time so that ˆ T is dimensionless. Substitution into the master equa-
tion (3.1) yields
∂pi
∂t = λ ˆ T (wi+1)pi+1(t) + λ ˆ T (wi−1)pi−1(t) − 2λ ˆ T (wi)pi(t) (3.2)
2 Note that this means that T + n = T − n .
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