S I I I I S

is a stochastic process with non-linear transition rates in the master equation d B

-p(I,t) = - 1 )(N-(I- 1 ))p(I - 1 ,t) + a(I + l)p(I+l,t)

Since we assume constant population size N, we have S = N — 1. For the dynamics of the mean value (1} := ^ J=0 1p(J) we obtain by inserting the master equation Eq. (7.1)

where now the second moment (12} := ^12 • p(1, t) enters the right hand side of the equation. So we do not obtain a closed system for the mean (1}. However, as will be described in more detail in the following sections, an approximation, called mean field approximation, can help to close the system. Here it consists of

meaning that the variance is neglected, var := (I2} — (I}2 « 0. So now we obtain a closed ordinary differential equation (ODE)

Considering the density x := (1 }/N instead of the absolute numbers (1} we find the simple quadratic ODE

In this form it will appear again later as a result in Section 7.2.3.

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