Page 179 - Special Topic Session (STS) - Volume 4

STS579 Daria Balashova
3. Results
We consider the BRW with positive intensity sources > 0 and
negative intensity sources (−) < 0, located at the vertices of the simplex,
| − | = for ≠ .

In this section, we consider an algorithm similar to the one described in
[5]. We call the state of a BRW system the set of pairs {, }, each of which
corresponds to a particle located at the point ∈ ℤ , that appeared at the

point at the time . By evolution we mean a jump to another point, splitting
or death of a particle. In the process of modelling the transition from one state
of the BRW system to another will be carried out by excluding one pair from
the set of states and adding to the seat of states of one or several pairs
corresponding to the result of the simulated particle evolution.
Initialization. First, we set the characteristics of the simulated BRW: choose
the dimension d of the integer lattice, the functions defining the distribution
of the jumps matrix , location of sources, their intensities and the execution
time . At the initial moment of time, the state of the system is determined by
the presence of a single particle at zero coordinate point of a given space.
Step of algorithm. We model the evolution of the particle: the exponential
time dt of staying it in x, then the jump or birth/death event. In the case of a
jump, the transition state is simulated according to the matrix , in the next
state the current pair {, } disappears and appears new pair {′, + }. In the
case of death, the current pair {, } disappears, and in the case of birth
(dividing into several offsprings) the current pair {, } disappears and two new
pairs {, + } are added to the set. After some time we may have several
pairs {x, t} waiting to be processed. We select an arbitrary pair {, } (due to
the independence of the particles) and model the evolution of the
corresponding particle.
Stop condition. The algorithm terminates when all the values of of all
pairs in the system exceed the specified time , or when the set of states
becomes empty (the process has degenerated).
Finally, we count the total of particles at each time point from 0 to . After
a certain number of runs (determined from statistical considerations) of a
simulation program with the same BRW parameters, the collected information
is processed by the Monte Carlo method.
Fig. 1 corresponds to the simulations on ℤ with 3 sources (1,0,0), (0,1,0) and
3
(0,0,1). In cases of = = = 0.4 and = = 0.5, = −0.5 we can
3
2
1
2
1
3
observe subcritical processes. While in cases of = = = 0.5 and =
3
2
2
1
1
= 0.6, = −0.6 the processes are supercritical.
3
The graphs corresponding to the simulations on ℤ with 4 sources (1,0,0,0),
4
(0,1,0,0), (0,0,1,0) and (0,0,0,1) are shown in the fig. 2. The processes are
subcritical for = = 0.7, = = −0.7 and = 0.8, = = = −0.8.
4
3
2
4
3
1
1
2
In cases of = = 0.8, = = −0.8 and = 0.9, = = = −0.9 we
3
1
4
2
4
1
3
2
can observe supercritical processes.
168 | I S I W S C 2 0 1 9