Rank dependent branching-selection particle systems

We consider a large family of branching-selection particle systems. The branching rate each depends on its rank and is given by function $b$ defined the unit interval. There also killing measure $D$ supported interval as well. At times, chosen among all particles to left one sampling according $D$. allowed have total mass less than one, which corresponds positive probability no killing. Between perform independent Brownian Motions in real line. This setting includes several well known models like Branching Motion (BBM), $N$-BBM, dependent BBM, many others. conjecture scaling limit for this class processes prove such related system. rich enough allow us use behavior solutions limiting equation asymptotic velocity rightmost under minimal conditions turns out be universal only $b(1)$ If number system $N$ conserved velocities $v_N$ converge $\sqrt{2 b(1)}$. When grows up time exponentially fast b(1)}$ independently initial particles.