Clas Blomberg - Physics of life-Elsevier Science (2007)

202 Part V. Stochastic dynamics
The stochastic (master) equation for the probability function P(n, t) is:
dP( n, t)
dt
an ( 1) Pn ( 1, t) 2anPnt ( , ) an ( 1) Pn ( 1, t)
(20.8)
The relations for P(0) (extinction) and P(1) are:
dP(,)
0 t
aP(, 1 t)
dt
dP(, 1 t)
2aP(, 1 t) 2aP( 2, t)
dt
The only time-independent end probability function is the extinct state, n 0. One can see
directly from the stochastic equation that the average is constant: dn/dt 0, that is the average
remains equal to an original value. The system can be extinct, and in fact the probability
that the system becomes extinct goes to one at long times. However, there is a small probability
that the system grows unlimitedly, and this property allows the average to remain constant.
One can get analytic relations for general probability functions. We do not show the derivation
but present results for an initial situation with population 1, that is P(1, 0) 1.
General results are:
t
P(,)
0 t
t 1
1
P(, 1 t)
( t 1)
2
t
P(,)
2 t
( t 1)
⋮
t
n1
Pnt ( , )
( t 1)
3
n1
(20.9)
These expressions can be verified in the equations above.
By direct calculation, one finds:
n 1; n 2 (1 2t) (20.10)
The probabilities go to zero for all n 0 when time goes to infinity, while the average
remains constant and the variance grows proportional to time, as is the case for other martingale
processes we have here, Random walk, diffusion and Brownian motion.