Some models of branching Brownian motion with selection are presented below.
If you know nothing about standard branching Brownian motion, it is better to start here.
Branching Brownian motion with selection
We consider that the position of a particle in the branching Brownian motion describes its fitness or strength
and we want to introduce a natural selection phenomenon, which will kill the lowest particles in the population over time.
Such a model is called branching Brownian motion with selection.
There are several ways of choosing a selection rule so there are various models that can be defined.
We present here firstly the \(N\)-BBM and then we focus on the \(L\)-BBM, which I studied in this
paper.
\(N\)-branching Brownian motion
Fixing some integer \(N\), we add the following selection rule: as soon that there are more than \(N\) particles, the lowest is killed.
\(L\)-branching Brownian motion
Here we fix a real number \(L>0\) and consider the following selection rule:
as soon as a particle is at a distance \(L\) from the highest, it dies.
The following picture represents a \(L\)-BBM with \(L=5\) until time \(t=20\).
Velocity of the \(L\)-branching Brownian motion
In this paper, I studied the velocity of the \(L\)-BBM.
It is proved that the position of the population grows linearly over time, at a deterministic speed \(v_L\) which depends on \(L\).
See the following picture, representing a \(L\)-BBM with \(L=5\) until time \(t=30\).
The natural question which arises is how close \(v_L\) is from the speed \(\sqrt{2}\) of the maximum of the BBM without selection
(see here).
Since there are less individuals in the \(L\)-BBM, it is clear that \(v_L \leq \sqrt{2}\).
The main result of the aforementioned paper is the following asymptotic development as \(L \to \infty\):
\[ v_L = \sqrt{2} - \frac{\pi^2}{2\sqrt{2} L^2} + o \left( \frac{1}{L^2} \right). \]
Note that, in particular, when the effect of selection vanishes, the speed \(v_L\) converges to the speed of the BBM without selection.
More pictures of the \(L\)-branching Brownian motion
The two following pictures represent realizations of a \(L\)-BBM with \(L=7\) until time \(t=30\).
